tag:blogger.com,1999:blog-1621043984309170662024-02-07T07:40:04.865-08:00 الرياضيات بالاعداديرياضيات عربية لغة و رموزا و مصطلحاتا محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-162104398430917066.post-38805308952713446292015-10-13T02:21:00.001-07:002020-06-05T14:47:22.507-07:00المنطق<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">الرموز الرياضياتية المُستعملة في هذا الدرس</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><b>طــ: </b>مجموعة الأعداد الصحيحة الطبيعية</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><b>صـ:</b> مجموعة الأعداد الصحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><b>كـ:</b> مجموعة الأعداد الكسرية</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><b>حـ:</b> مجموعة الأعداد الحقيقية</span><br />
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العدد 1 عدد كسري. نص رياضي يحمل معنى صحيح (ص)٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العدد 2√ عدد كسري. نص رياضي يحمل معنى خاطئا (خ)٠</span><br />
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">يوجد على الأقل عدد كسري ، 0 < خ < 1 . ونرمز لهذه العبارة بِـ ∃ س∈ كـ ، 0 < خ < 1 ٠ :</span><span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif;"><b>∃</b></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: times, times new roman, serif; font-size: medium;"> يوجد عدد صحيح طبيعي خ وحيد أكبر قطعا من 0 و أصغر قطعا من 2 </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">:</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">!∃</span></span><br />
<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ونرمز لهذه العبارة بِـ ∃</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">! </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س ∈ طـ ، 0 < خ < 2 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "trebuchet ms" , sans-serif;">مهما يكن العدد الحقيقي س، فإن مربع العدد س عدد موجب </span></span><span style="font-family: "trebuchet ms" , sans-serif; font-size: medium;"><span style="font-family: times, "times new roman", serif;">:</span><b>∀</b></span><br />
<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "trebuchet ms" , sans-serif;">ونرمز لهذه العبارة بِـ ∀ س ∈ حـ، </span></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: medium;"><sup>2</sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"> </span><span style="font-family: "trebuchet ms" , sans-serif;">≥ </span></span><span style="font-family: "trebuchet ms" , sans-serif; font-size: medium;">0</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "trebuchet ms" , sans-serif;"> </span></span><span style="font-family: "trebuchet ms" , sans-serif; font-size: medium;"> </span><span style="font-family: "trebuchet ms" , sans-serif; font-size: medium;">٠ </span><br />
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">في نص رياضي ترتيب المكممات جد مهم</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العمليات المنطقية: النفي المنطقي؛ الفصل و العطف المنطقي؛ الاستلزام المنطقي؛ التكافئ المنطقي</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium; font-weight: normal;">النفي المنطقي</span></span></h2>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium; font-weight: normal;">مثال</span></span></h2>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"> </span>نعتبر العبارة (عـ) ∃ س ∈ حـ ، س<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: xx-small;"><span lang="AR-SA"><sup>2</sup></span></span></span>+ 2 س + 5 = 0، هذا النص الرياضي يعني أن المعادلة تقبل حلا على الأقل في مجموعة الأعداد الحقيقية حـ ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-size: medium;">لكن المعادلة ليس لها حل في مجموعة الأعداد الحقيقية حـ لأن مميزها سالب قطعا، و هذا يمكن التعبير عنه بِـ : ∀ س ∈ حـ ، </span></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">+ 2 س + 5 ≠ 0</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">هذا النص الجديد يسمى نفي العبارة (عـ) ونرمز له بِـ (¬عـ) ٠</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">قاعدة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفي العبارة ∀س∈ : هـا(س) هي العبارة ∃س∈ : ¬هـا(س) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفي العبارة ∃س∈ : هـا(س) هي العبارة ∀س∈ : ¬هـا(س) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفي العبارة ∀س∈حـ ، س + 1 ≥ 0 هي العبارة ∃س∈ حـ ، س + 1 < ٠0</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الفصل و العطف المنطقي</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">تكون العبارة ((عـ1) أو (عـ2)) صحيحة اذا كانت على الأقل احدى العبارتين (عـ1) أو العبارة (عـ2) صحيحة، ونرمز له بِـ (عـ1) ∧ (عـ2)٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">تكون العبارة ((عـ1) و (عـ2)) صحيحة اذا كانت العبارة (عـ1) و العبارة (عـ2) صحيحتان معا، ونرمز له بِـ (عـ1) ∨ (عـ2)٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red; font-size: medium;">ملاحظة</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفي العبارة (عـ1 أو عـ2) هي العبارة (عـ1 و عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفي العبارة (عـ1 و عـ2) هي العبارة (عـ1 أو عـ2) ٠</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">ملاحظة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">في بعض النصوص الرياضية ، يكون الفصل مخفيا. مثلا</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س عدد حقيقي، الكتابة س ≥ 1 تعني (س = 0 أو س > 1 ) ٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستلزام المنطقي</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">مثال</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نعلم أن كل عدد صحيح طبيعي مضاعف للعدد 4 هو مضاعف للعدد 2 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">يمكننا صياغة ذلك كما يلي: ص عدد صحيح طبيعي مضاعف للعدد 4 ، إذن ص عدد صحيح طبيعي مضاعف للعدد 2 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نقول أن العبارة (عـ1) "ص عدد صحيح طبيعي مضاعف للعدد 4" تستلزم العبارة (عـ2) "ص عدد صحيح طبيعي مضاعف للعد</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">د</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> 2"٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ونكتب (عـ1) ⇐ (عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">بصفة عامة ليكن عـ1 و عـ2 عبارتين</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نقول أن العبارة (عـ1) تستلزم العبارة (عـ2) و نرمز لها بالكتابة (عـ1) ⇐ (عـ2) عندما تتحقق إحدى الحالات</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1)صحيحة و العبارة (عـ2) صحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1) خاطئة و العبارة (عـ2) صحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1) خاطئة و العبارة (عـ2) خاطئة</span></div>
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<span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red; font-size: medium;">مثال</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة ( ن عدد صحيح طبيعي زوجي ⇐ ن + 2 عدد زوجي) عبارة صحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة ( 1 عدد زوجي ⇐ 2 عدد زوجي) عبارة صحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة ( 1 عدد زوجي ⇐ 3 عدد زوجي) عبارة صحيحة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة ( ن عدد صحيح طبيعي زوجي ⇐ ن + 1 عدد زوجي) عبارة خاطئة</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">ملاحظة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1⇐ عـ2) تكون خاطئة فقط إذا كانت عـ1 صحيحة و عـ2 خاطئة</span></div>
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<span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">ملاحظة</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ليكن (عـ1) ⇐ (عـ2) استلزاما صحيحا٠ فإن (عـ1) شرط كافي لِـ (عـ2) ، و (عـ2) شرط لازم لِـ (عـ1) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "georgia" , "times new roman" , serif; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif;"></span><br /></span></span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الاستلزام المضاد للعكس</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">الاستلزام (¬عـ2) ⇐ (¬عـ1) يسمى الاستلزام المضاد للعكس للاستلزام (عـ1) ⇐ (عـ2)٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><br /></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">التكافئ المنطقي</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1) ⇐ (عـ2) و (عـ2) ⇐ (عـ1) تسمى تكافؤ العبارتين (عـ1) و (عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ونكتب (عـ1)⇔(عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">و نقرأ (عـ1) تكافؤ (عـ2) ٠</span></div>
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<span style="color: red;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ملاحظة</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة (عـ1 ⇐ عـ2) و العبارة (¬عـ1 أو عـ2) لهما نفس قيم الحقيقة، من هنا نستنج</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">أن العبارة (عـ1 ⇐ عـ2) تكافؤ العبارة (¬عـ1 أو عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ثم نفي العبارة (عـ1 ⇐ عـ2) يكافؤ العبارة (عـ1 و ¬عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-size: xx-small;"><br /></span></span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستدلال المنطقي: الإستدلال بالخلف؛ الإستدلال بفصل الحالات؛ الإستدلال بالاستلزام المضاد للعكس؛ الإستدلال بالترجح</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;"><br /></span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستدلال بالخُلف</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">إذا اردنا البرهان على صحة عبارة ما، نفترض أنها خاطئة و نحاول أن نجد تناقضا أو عبارة غير صحيحة٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><br /></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="color: red;">مثال:</span> المماس للدائرة يكون عموديا على نصف القطر المار بنقطة التماس٠</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نبدأ البرهان بفرض أن المماس ليس عموديا على نصف القطر وبالسير بالمناقشة</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> الصحيحة نأتي إلى أن المماس يقطع الدائرة في نقطتين وبما أن النتيجة تتعارض مع تعريف المماس، ينتج أن الإفتراض ليس صحيحا ويكون المماس عموديا على نصف القطر المار بنقطة التماس٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium; font-weight: normal;">مثال</span></span></h2>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "arial" , "helvetica" , sans-serif; font-size: xx-small;"> </span>لتكن س عدد صحيح بحيث س<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: xx-small;"><span lang="AR-SA"><sup>2</sup></span></span></span> عدد عدد زوجي٠ بين أن س عدد زوجي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">نفترض أن س عدد فردي ، أي ∃ ش ∈ صـ بحيث س = 2ش + 1، س<span lang="AR-SA"><sup>2</sup></span> = ...، إذن س<span lang="AR-SA"><sup>2</sup></span> عدد فردي٠ و هذا يناقض المعطيات، إذن الإفتراض ليس صحيحا و بالتالي نفيه صحيح أي أن س عدد زوجي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></span></div>
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<span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستدلال بفصل الحالات</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">مثال</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">حل في حـ المعادلة: |س−3 | = 2 ٠</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">الحالة الأولى: س ≥ 3 ..... ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">الحالة الثانية: س < 3 ..... ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "georgia" , "times new roman" , serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"></span><br /></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستدلال بالاستلزام المضاد للعكس</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "georgia" , "times new roman" , serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"></span><br /></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">العبارة ((عـ1) ⇐ (عـ2)) تكافؤ العبارة ((¬عـ2) ⇐ (¬عـ1))٠</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">إذا اردنا البرهان على صحة العبارة (عـ1) ⇐ (عـ2) ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">فإنه يمكننا عوضا عن ذلك أن نبرهن على صحة العبارة (¬عـ2) ⇐ (¬عـ1)٠</span><br />
<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><br /></span>
<span style="color: red;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">مثال</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">سنبين بالاستلزام المضاد للعكس بين أن: ∀ (س ، ص) ∈ طـ </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">،</span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">> 4 ⇐ ( س < −2 أو س > 2 )٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">لنبين عوضا عن ذلك أن (س ≥ −2 و س ≤ 2) ⇐ س<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: xx-small;"><span lang="AR-SA"><sup>2</sup></span></span></span> ≤ 4 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">أي −2 ≤ س ≤ 2 ⇐ </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">≤ 4 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">بالفعل −2 ≤ س ≤ 2 ⇐ |س| ≤ 2 ⇐ |س|<span lang="AR-SA"><sup>2</sup></span> ≤ 4 ⇐ س<span lang="AR-SA"><sup>2</sup></span> ≤ 4 ٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><br /></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "georgia" , "times new roman" , serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"></span><br /></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">الإستدلال بالتراجح</span></span><br />
<span style="font-size: medium;"><span style="font-family: "times" , "times new roman" , serif; vertical-align: sub;">لتكن ي<sub><span style="font-size: xx-small;">ن</span></sub>، عبارية متعلقة بِـ ن، بحيث ن عدد صحيح طبيعي، </span><span style="font-family: "times" , "times new roman" , serif;">ن ≥ ن</span><span style="color: red; font-family: "times" , "times new roman" , serif;"><span style="color: black;"><sub>0</sub></span></span> </span></div>
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<span style="font-size: medium;"><span style="font-family: "times" , "times new roman" , serif; font-size: small;"><b>لنبرهن</b> </span><span style="font-family: "times" , "times new roman" , serif;">على</span><span style="font-family: "times" , "times new roman" , serif;"> أن</span><span style="font-family: "times" , "times new roman" , serif;"> </span><span style="font-family: "times" , "times new roman" , serif;">ي</span><sub style="font-family: times, "times new roman", serif;"><span style="font-size: xx-small;">ن</span></sub><span style="font-family: "times" , "times new roman" , serif;">، صحيحة لكل ن ≥ ن</span><span style="color: red; font-family: "times" , "times new roman" , serif;"><span style="color: black;"><sub>0</sub></span></span></span></div>
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<span style="font-family: "trebuchet ms" , sans-serif; font-size: xx-small;"><span style="color: red;"><span style="color: black; font-family: "times" , "times new roman" , serif; font-size: medium;">أولا نبين أن العبارة صحيحة عند الرتبة ن<span style="color: red;"><span style="color: black;"><sub>0</sub></span></span></span></span></span></div>
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<span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> <span style="color: red;"> </span>ثانيا نفترض أن العبارة صحيحة عند الرتبة ن٬ ثم نبين أن العبارة صحيحة عند الرتبة ن + ٠1</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"></span></span></div>
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<span style="font-size: medium;"><br /></span><span style="font-family: "times" , "times new roman" , serif; font-size: xx-small;"><span style="font-size: medium;"> <span style="color: red;">مثال:</span> ∀ ن ∈ طـ 3<sup style="line-height: 27.6px;"><span style="font-size: xx-small;">ن </span></sup> > ن</span></span><br />
<span style="font-family: "times" , "times new roman" , serif; font-size: xx-small;"><span style="font-size: medium;"> أولا: نبين صحة العبارة عند الرتبة 0</span></span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> بما أن 1 > 0</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> إذن العبارة صحيحة عند الرتبة 0</span><br />
<span style="font-size: medium;"><span style="font-family: "times" , "times new roman" , serif; font-size: small;"></span><br /></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ثانيا: نفترض صحة العبارة عند الرتبة ن٬ ثم نبين صحة العبارة عند الرتبة ن + 1 ٠</span></div>
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<span style="font-size: medium;"><b><span style="font-family: "times" , "times new roman" , serif; font-size: small;">العبارة صحيحة عند الرتبة ن أي</span><span style="font-family: "times" , "times new roman" , serif; font-size: small;"> 3<sup style="line-height: 27.6px;"><span style="font-size: xx-small;">ن </span></sup><span style="color: red;"><span style="color: black; font-size: xx-small;"> </span></span></span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">> </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">ن</span></b></span></div>
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<span style="font-size: medium;"><b><span style="font-family: "times" , "times new roman" , serif; font-size: small;">إذن 3<sup style="line-height: 27.6px;">ن+1 </sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">> 3</span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">ن</span></b></span><br />
<span style="font-size: medium;"><b><span style="font-family: "times" , "times new roman" , serif; font-size: small;">بما أن <span style="color: red;"><span style="color: black;"> </span></span>3ن<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "verdana" , sans-serif;"><span style="color: red;"><span style="color: black;"> </span></span></span></span>><span style="font-size: xx-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "verdana" , sans-serif;"><span style="color: red;"><span style="color: black;"> </span></span></span></span></span>ن+ 1 </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">إذن 3<sup style="line-height: 27.6px;"><span style="font-size: xx-small;">ن+1</span><span style="font-size: xx-small;"> <span style="font-family: "courier new" , "courier" , monospace;"> </span></span></sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">> </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">ن+ 1</span></b></span><br />
<span style="font-size: medium;"><b><span style="font-family: "times" , "times new roman" , serif; font-size: small;">إذن العبارة </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">صحيحة </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">عند الرتبة ن + 1 ٠</span></b></span><br />
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<span style="font-size: medium;"><b><span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-size: x-small;">أي ∀ ن ∈ طـ </span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "verdana" , sans-serif;"><span style="font-family: "arial" , "helvetica" , sans-serif;"> </span></span></span> 3<sup style="line-height: 27.6px;"><span style="font-size: xx-small;">ن </span></sup><span style="color: red;"><span style="color: black; font-size: xx-small;"> </span></span></span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">> </span><span style="font-family: "times" , "times new roman" , serif; font-size: small;">ن</span></b></span><br />
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="color: red;">ملاحظة</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">يمكن البرهان على تكافؤ باستعمال الاستلزام المزدوج٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">كي نثبت على صحة تكافؤ ((عـ1)⇔(عـ2)) يمكننا اثبات صحة الاستلزامين ((عـ2) ⇐ (عـ1)) و ((عـ1) ⇐ (عـ2))٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="color: red;">مثال:</span> س عدد صحيح زوجي تكافؤ </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: medium;"><sup>2</sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">عدد صحيح زوجي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">(⇐) نين أولا صحة الاستلزام</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س عدد صحيح زوجي أي ∃ ش ∈ صـ بحيث س = 2ش ، </span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: medium;">س</span><span lang="AR-SA" style="font-size: medium;"><sup>2</sup></span><span style="font-size: medium;"> </span><span style="font-size: medium;">= 4</span></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: medium;"><sup>2</sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">، إذن </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: medium;"><sup>2</sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> عدد صحيح زوجي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">(⇒) ثم نين صحة الاستلزام</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">أي س<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: xx-small;"><span lang="AR-SA"><sup>2</sup></span></span></span> عدد صحيح زوجي تستلزم س عدد صحيح زوجي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">و نستعمل مثلا الإستدلال بالاستلزام المضاد للعكس: س عدد صحيح فردي ، أي ∃ ش ∈ صـ بحيث س = 2ش + 1، س<span lang="AR-SA"><sup>2</sup></span> = ...، إذن </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: medium;"><sup>2</sup></span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> </span><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">صحيح عدد فردي٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"><span style="font-family: "georgia" , "times new roman" , serif; font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;"></span><br /></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">ملاحظة</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">كي نثبت على صحة التكافؤات (عـ1)⇔(عـ2) ⇔(عـ3) يمكننا اثبات صحة الاستلزامات (عـ1) ⇐ (عـ2) ⇐(عـ3) ⇐ (عـ1)٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">تمارين</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">بين أن كل عدد صحيح طبيعي و مربعه لهما نفس الزوجية٠</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">بين بالإستدلال بالخلف أن 2√ عدد غير كسري٠</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">تمرين2</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">مستعملا الاستلزام المضاد للعكس بين أن: ∀ (س ، ص) ∈ طـ ٬ س + ص < 2 ⇐ ( س≥1 أو ص≥1 )٠</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">تمرين3</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;">بين بالتراجح أن: ∀ ن ∈ طــ<span style="font-size: xx-small;"> </span>٬ 1 − 4<span style="font-family: "courier new" , "courier" , monospace;">ن</span><span style="font-size: xx-small;"> </span>مضاعف للعدد ٠3</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">تمرين4</span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> بين بالتراجح أن: ∀ س ∈ حـ ٬ ( 1 + س + س<sup style="line-height: 27.6px;">2</sup> + س<sup style="line-height: 27.6px;">3</sup> ..... + س<sup style="line-height: 27.6px;">ن</sup> ) ( 1 − س<sup style="line-height: 27.6px;">2</sup> ) = ( 1 − س<sup style="line-height: 27.6px;">ن+1</sup> )٠</span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: medium;">تمرين5</span></div>
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<span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">ليكن ن عدد صحيح طبيعي٬ بين أن ن<sup style="line-height: 27.6px;">2</sup> + ن<span style="font-size: xx-small;"> </span>عدد زوجي٠</span></span></div>
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<span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> ليكن س عدد حقيقي، بين أن إذا كان 2س + 1 مربعا كاملا فإن</span></span><br />
<span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif; font-size: medium;">س + 1 مجموع مربعين كاملين٠</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif; font-size: medium;"> ليكن ن عدد صحيح طبيعي٬ بين أن العدد 1 − 9<sup style="line-height: 27.6px;">ن</sup> يقبل القسمة على ٠8</span></div>
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محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.com0tag:blogger.com,1999:blog-162104398430917066.post-9805631820057747072013-09-13T02:29:00.000-07:002019-10-09T03:00:16.818-07:00عموميات حول الدوال، دوال ابتدائية<div dir="rtl" style="text-align: right;" trbidi="on">
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><b>تمارين</b> </span></span></span></div>
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<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;"><span style="color: red;">تمرين29</span></span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;"><span style="color: #333333;"><span style="line-height: 21.168px;">نعتبر الدالة العددية </span></span>عا(س) <span dir="RTL" lang="AR-SA">= </span> س<sup>2</sup> – 2 س –<span lang="AR-SA"> </span>1</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">أحسب صورة العدد 0</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">أحسب صورة العدد 1</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">حدد سوابق العدد – 2</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">أحسب صورة العدد –1</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">حدد سوابق العدد –1</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;"><br /></span></span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;"></span><span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">بين أن التمثيل المبياني للدلة عـا يمر بالنقط: (0 ؛ ـ1) ، (ـ1 ؛ 2) ٬ (1 ؛ -2)٠</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ادرس إشارة النسبة التالية٬ ثُم ارسم التمثيل المبياني للدلة عـا في المستوى منسوب إلى معــلم ممنظم و متعامد٠</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBwqLuTv6mjVDOZgr2X3sm_kKNJZeupaF-y3MBYCe1uFFVgwSkZN4fb7NIsqIBDGgyB1FVy5V09d8HwKBTmU_-ushCV8dAdFL67rMyHVqHbPFtG769AI0_AUNVoQ66uo7NUHcJGm_XFsA/s1600/TAU.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em;"><img border="0" height="79" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBwqLuTv6mjVDOZgr2X3sm_kKNJZeupaF-y3MBYCe1uFFVgwSkZN4fb7NIsqIBDGgyB1FVy5V09d8HwKBTmU_-ushCV8dAdFL67rMyHVqHbPFtG769AI0_AUNVoQ66uo7NUHcJGm_XFsA/s200/TAU.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="200" /></a></span></span></div>
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">ادرس جبريا و هندسيا تقاطع التمثيل المبياني للدلة عـا مع محور الأفاصيل٠</span></span><br />
<span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;"><br /></span></span><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;"></span><span style="background-color: white; color: #222222; font-family: "times" , "times new roman" , serif; font-size: 13.2px;"><span style="font-size: x-small;">باستعملك للتمثيل المبياني للدلة عـا، اعطي صورة العدد 3، و سوابق العدد –٠1</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;">تمرين30</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA">دائرة الوحدة</span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">بين أن الدوال المثلثية</span></span> جب(<span style="line-height: 18.48px;">س</span>) (جيب) و تجب(<span style="line-height: 18.48px;">س</span> (جيب تمام) قيمها محصورة بين<span style="color: red;"><span style="color: black;"> </span></span></span><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">−1 و 1</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">و أن: جب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (<span style="line-height: 18.48px;">س</span>)<sup style="line-height: 27.6px;"> </sup>+ تجب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (<span style="line-height: 18.48px;">س</span>) =1<span style="color: red;"><span style="color: black;"> ، من أجل كل عدد حقيقي س</span></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">بين أن </span></span>جب و تجب<span style="color: red;"><span style="color: black;"> دوال دورية، و أن دورتهما تساوي 2ط</span></span></span></span></div>
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<span style="background-color: white; color: red; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">تمرين 31</span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">لتكن عـا دالة معرفة على مجموعة الأعداد الحقيقية </span><b style="background-color: white; color: #222222; font-family: arial, tahoma, helvetica, freesans, sans-serif; font-size: 13.2px;">حـ</b><span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;"></span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">بحيث: لكل عددين حقيقيين س و ص، عـا (س + ص) = عـا (س) + عـا (ص)٠</span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">بيّن أن عـا (0) = 0 ، ثم استنتج بأن عـا دالة فردية٠</span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">بيّن بأن لكل ن عدد صحيح نسبي، عـا (ن س) = ن عـا (س)٠</span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">بيّن بأن لكل كـ عدد كسري، عـا (كـ س) = كـ عـا (س)٠</span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;">استنتج بأن عـا (كـ) = عـا(1) كـ٠</span></div>
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<b><span style="color: red;">الدرس</span></b><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نظام الإحداثيات</span></span></b></span><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">،</span></span></b></span><b> <span style="color: red;">معادلة الخط المستقيم</span></b><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نظام الإحداثيات، المستقـيم الـمــدرج</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxBAbEQ-1lAd6qvTp63cShGmVT3KJoVIFfgq6hQOLO9t4FDgQaHId5CBMMMBrYEc7De586J5180JTZwa2d_E8T4X8zFrMNruQ2uyuzTlI9zjg6joYaF7LiLwYIb0VaWeAzmJr9HtdEczI/s1600/dro.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxBAbEQ-1lAd6qvTp63cShGmVT3KJoVIFfgq6hQOLO9t4FDgQaHId5CBMMMBrYEc7De586J5180JTZwa2d_E8T4X8zFrMNruQ2uyuzTlI9zjg6joYaF7LiLwYIb0VaWeAzmJr9HtdEczI/s320/dro.JPG" /></a><img src="https://blogger.googleusercontent.com/img/proxy/AVvXsEglH0nad-VvSxh5GQ5_SCWNYb4dn7lPcXGRIIDgHLqX6Wm-Ar9Ra8LNLErVDWBjWooFnxVSDdfXMZwBmn86Yi5WkV1RqjoY4_c3WatUieWsXWIb_lWc3BE9rqf0XoFa1GMqbSYIJLrRzqdzoGshd1-Jya4jbLxcqfE3aKQV2XCcSnV7V448DCVQsOUFxOgsmXLQ0j1nU-EKwUrgE3qxchQ3vZg=" /></span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">المستقيم (اب) يسمى مستقيم مدرج، وتسمى النقطة ا نقطة الأصل، و طول القطعة [اب] وحدة الطول أو التدرج، نسمي النقطة ا نقطة الأصل</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نسمي العدد 0 أفصــول النقطة ا و العدد 1 أفصــول النقطة ب</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">كل نقطة ن من المستقيم مرتبطة بعدد يسمى إحداثية النقطة ن٬ و نكتب : ن(س) </span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نظام الإحداثيات ثنائي الأبعاد، المعلم في المستوى</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر (اب) و (ات) مستقيمين مدرجين على التوالي بواسطة [اب] و[ات] و متعامدين في النقطة ا٬ يسمى تقاطع المحاور بالنقطة الأصل</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">(طول القطعة [اب] وحدة الطول على المحور (اب)، كما أن طول القطعة [ات] وحدة الطول على المحور (ات))</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نقول أن المستوى منسوب إلى معــلم متعامد</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نرمز لمعلم في المستوى بالرمــز : ( ا ؛ب ؛ ت) </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نسمي المستقيم (اب) : محــور الأفاصيل</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نسمي المستقيم (ات) : محــور الأراتيب</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">إذا كان اب = ات نقول أن المستوى منسوب إلى معــلم ممنظم و متعامد</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">كل
نقطة ن من المستوى مرتبطة بعددين يسميان إحداثيتي النقطة: الإسقاط
العمودي للنقطة ن على محور الفواصل و يسمى أفصول النقطة ن، و الإسقاط
العمودي للنقطة ن على محور الأراتيب و يسمى أرتوب النقطة ن٠ و نكتب: ن( س ؛
ص)٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span><b><span style="color: red;">معادلة الخط المستقيم</span></b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8UdIop4pMXkZu1B2my46RzJG5nzB0dJ7I80uIvlG2RXDwLidOijRmZWkKk98nFqs5X7FxqilmlzOs0U05wF4te8ZxmPhCz8GZvXnsssHQBM8W-bDo4e4Oa9KeUslfum-nKppsQdCdYqk/s1600/droi.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8UdIop4pMXkZu1B2my46RzJG5nzB0dJ7I80uIvlG2RXDwLidOijRmZWkKk98nFqs5X7FxqilmlzOs0U05wF4te8ZxmPhCz8GZvXnsssHQBM8W-bDo4e4Oa9KeUslfum-nKppsQdCdYqk/s1600/droi.JPG" /></a></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">لكل خط مستقيم </span></span></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">( م )</span></span> في المستوى </span></span></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">منسوب إلى معــلم</span></span> توجد علاقة تربط بين الإحداثي السيني والإحداثي الصادي للنقط الواقعة عليه و هذه العلاقة معادلة من الدرجة الأولى ذات مجاهيل إحداثية، حلها يمثل ذلك المستقيم ونكتبها بالصورة العامة : </span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">أ ص + ب س + جـ = 0، حيث أ، ب، جـ أعداد حقيقية معلومة ( أ، ب) ≠ (0، 0) </span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></span><br />
تسمى المعادلة الديكارتية للمستقيم ( م ) الموجه بالمتجهة (<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">–</span></span> ب، أ)<br />
(<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">–</span></span> ب، أ) تسمى المتجه الموجه للمستقيم ( م )<br />
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من أجل كل نقطة هـ و نقطة جـ من المستقيم (م) فإن المتجهة <img border="0" src="https://www.schoolarabia.net/images/modules/physics/level_2/vectors/66.gif" /> متجهة موجه للمستقيم (م)<br />
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<span style="color: red;"><b>ملاحظة</b></span><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>1:</b></span> </span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">في المستوى </span></span></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">منسوب إلى معــلم </span></span></span></span></span></span>الصورة العامة لمعادلة خط مستقيم تُكتب على الشكل: <br />أ ص + ب س + جـ = 0، حيث أ، ب، جـ أعداد حقيقية معلومة ( أ، ب) ≠ (0، 0) </span><span style="font-family: "times" , "times new roman" , serif;"> </span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">- </span>إما الخط المستقيم موازي لمحور الأراتيب، و في هذه الحالة تُكتب معادلته على الشكل: س = أ ، حيث أ عدد معلوم.<br />- إما الخط المستقيم غير موازي لمحور الأراتيب، و في هذه الحالة تُكتب معادلته على الشكل: ص = أ س + ب حيث أ ، ب عددان حقيقيان.</span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">العدد أ يسمى ميل المستقيم أو المعامل الموجه للمستقيم، و العدد ب يسمى الأرتوب عند الأصل</span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">.</span><span lang="AR-SA" style="font-size: 18pt;"> </span></span></span></span><br />
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<b><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">ملاحظة</span></span></span></b><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">2</span></span>:</span></span></b> </span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">- ميل المستقيم ( م ) = التغير في الإحداثيات الصادية ÷ التغير في الإحداثيات السينية</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">- </span>ميل المستقيم ( م ) = ضل الزاوية بين محور الأفاصيل و المستقيم ( م )</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">- مستقيمان متوزيان لهم نفس الميل</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>مثال1:</b></span> </span><br />
<span style="font-family: "times" , "times new roman" , serif;">جد </span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">معادلة</span> المستقيم الذي يمر بالنقطة ن ( 1 ، 3 ) والنقطة د ( 2 ، 5 ).</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;">الحل :</span><br />
<span style="font-family: "times" , "times new roman" , serif;">أولا : نوجد الميل </span><br />
<span style="font-family: "times" , "times new roman" , serif;">ثانيا: نختار أحد النقطتين ونعوض في المعادلة</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>مثال2:</b></span> </span><br />
<span style="font-family: "times" , "times new roman" , serif;">جد </span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">معادلة</span> المستقيم الذي يمر بالنقطة ن ( 2 ، 3 ) والنقطة د ( 2 ، 5 ).</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;">الحل :</span><br />
<span style="font-family: "times" , "times new roman" , serif;">النقطة ن و النقطة د لهما نفس الأفصول ألا و هو العدد 2، إذن معادلة المستقيم المار من النقطة ن و النقطة د هي: س = 2</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;">معادلة مستقيم بمعرفة نقطة تقع عليه ومستقيم موازي له </span><br />
<span style="font-family: "times" , "times new roman" , serif;">مثال:</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;">معادلة مستقيم بمعرفة نقطة تقع عليه ومستقيم عمودي عليه</span><br />
<span style="font-family: "times" , "times new roman" , serif;">مثال:</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "times" , "times new roman" , serif;">معادلة مستقيم بمعرفة كلٍ من الجزء المقطوع من المحورين : السيني وَ الصادي</span><br />
<span style="font-family: "times" , "times new roman" , serif;">مثال:</span><br />
<br />
<span style="color: red;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><b>عموميات حول الدوال</b> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTbZBM2AnxTYOaPXjLttpcbP8mPMEPHTWFtARWMe9LVtWm3IA0GSTwXAosCPyp3mvzgwOksFBJJeeYQUeafMe3t4MWJacEb9xeHQKoUhMAk-W1tfPrO_epLxUrBf9rnc3N8Ro5HYg50_I/s1600/fonct3.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTbZBM2AnxTYOaPXjLttpcbP8mPMEPHTWFtARWMe9LVtWm3IA0GSTwXAosCPyp3mvzgwOksFBJJeeYQUeafMe3t4MWJacEb9xeHQKoUhMAk-W1tfPrO_epLxUrBf9rnc3N8Ro5HYg50_I/s320/fonct3.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">المثال الأول:</span> عندنا علاقة نرمز لها مثلا بِـ عـا تربط كل عنصر من المجوعة فـ = {1 ، 2 ، 3 ، 4 ، 5 ، 6} (تدعى المنطلق) عنصرا واحدا وواحدا فقط من المجموعة قـ = {ا ٬ ب ٬ ج ٬ ح ٬ خ } (تدعى المستقر)٠ تسمى هذه العلاقة تطبيق، ونرمز له بِـ عـا : فـ ← قـ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر ب يسمى صورة العنصر 2 بالتطبيق عـا</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر 2 يسمى سابق العنصر ب بالتطبيق عـا</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">المثال الثاني:</span> عندنا علاقة نرمز لها مثلا بِـ هـا تربط كل عنصر من المجوعة فـ بعنصر على الأكثر من المجموعة قـ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">(العنصر 4 ليس له صورة) )٠ تسمى هذه العلاقة دالة، ونرمز له بِـ هـا : فـ ← قـ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر ج يسمى صورة العنصر 3 الدالة هـا</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر 3 يسمى سابق العنصر ج الدالة هـا </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>تعريف:</b></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التطبيق هو كائن رياضي يمثل علاقة تربط بكل عنصر يسمى المتغير من مجموعة تدعى المنطلق عنصرا واحدا وواحدا فقط من مجموعة تدعى المستقر. </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة أو التابع هو كائن رياضي يمثل علاقة تربط بكل عنصر يسمى المتغير من مجموعة تدعى المنطلق بعنصر على الأكثر من مجموعة تدعى المستقر. </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نرمز للدوال و التطبيقات عادة بالحروف د، ذ، ر، ز، عـا، غـا، هـا.... و للمتغيرات بِالحروف ش، س، ص، ض،... و للدالة و التطببيق عـا من المجوعة <span style="font-family: "times" , "times new roman" , serif;">ك</span> إلى المجموعة <span style="font-family: "times" , "times new roman" , serif;">ل</span> للمتغير س، </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بِـ</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>عـا :</b> <span style="font-family: "times" , "times new roman" , serif;"><b>ك</b></span><b></b> <b>← <span style="font-family: "times" , "times new roman" , serif;">ل</span> </b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <span style="font-family: "times" , "times new roman" , serif;"> </span><b> س ↤ عـا(س)</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">عـا(س) تُقرأ عا لِـ س</span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر عـا(س) يسمى صورة العنصر س بالدالة عـا</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العنصر س يسمى سابق العنصر عـا(س) بالدالة عـا</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نكتب مثلا الدالة العددية هـا للمتغير الحقيقي س ( منطلقها و مستقرها مجموعة الاعداد الحقيقية): التي تربط بكل عدد حقيقي س، العدد الحقيقي </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">2</span></span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> </span>:</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>هـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <span style="font-family: "times" , "times new roman" , serif;"> </span> س ↤ هـا(س) = <span style="font-family: "times" , "times new roman" , serif;">2</span>س</span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أو باختصار الدلة العددية هـا المعرفة بِـ: هـا(س) = 2</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span><br />
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></b></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أمثلة لدوال معرفة بمعادلات جبرية:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">عـا(ش) = 5ش – 1، دالة عددية للمتغير الحقيقي ش </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">عـا(ش)= </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span><sup style="background-color: white; color: #222222;">2</sup></span></span></span></span> + 7ش + 1، دالة عددية للمتغير الحقيقي ش </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أمثلة من الهندسة:</span></span></b></span><br />
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الإنعكاس (الهندسة)</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">- ط نقطة من المستوى خارج المستقيم (هـ و)</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">يتم إسقاط خط عمودي على الخط (هـ و) المستعمل كمحور الإنعكاس، ثم مد الخط بشكل مستقيم في الجهة الأخرى من المحور وبنفس المسافة٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">النقطة التي نصل إليها تسمى صورة النقطة ط بالإنعكاس عن المستقيم (هـ و))٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">- ط نقطة من المستقيم (هـ و)</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">إذا كانت النقطة ا تنتمي إلى محور الإنعكاس (أي المستقيم (هـ و)) فإن صورتها هي نفسها٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia2WpBAh3XIZvzDETb_qQ9WGpxykmvGsZom5eLRfrR6itopN8SfJlKX5WbDLNA0ALy-b124icPszYXEmzhI-shRPi3Us5rqlNBQtVW-YbWmcaED-CdACbiRTwwdiYqk-a2-H92N8EMk3Y/s1600/%25D8%25A7%25D9%2584%25D8%25A5%25D9%2586%25D8%25B9%25D9%2583%25D8%25A7%25D8%25B3.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEia2WpBAh3XIZvzDETb_qQ9WGpxykmvGsZom5eLRfrR6itopN8SfJlKX5WbDLNA0ALy-b124icPszYXEmzhI-shRPi3Us5rqlNBQtVW-YbWmcaED-CdACbiRTwwdiYqk-a2-H92N8EMk3Y/s200/%25D8%25A7%25D9%2584%25D8%25A5%25D9%2586%25D8%25B9%25D9%2583%25D8%25A7%25D8%25B3.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تمرين تطبيقي:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة العددية عـا(</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span>) = </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span><sup style="background-color: white; color: #222222;">2</sup></span></span></span></span></span></span></span></span> + 4ش – 1 <br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أحسب صورة العدد 0 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">حدد سوابق العدد 5 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أحسب صورة 1 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">حدد سوابق العدد –1 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">حدد صورة العدد –1 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مجال تعريف دالة</span></span></b></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"></span></span> رياضية</span></span></b></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">،</span></span></span></span></b></span></span> مدى دالة رياضية</span></span></b></span></span><span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مجال تعريف </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دالة رياضية </span></span>هو مجموعة جزئية من المنطلق، مجموعة العناصر التي لها صورة فى المستقر. </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مدى دالة رياضية هو مجموعة جزئية من المستقر، مجموعة القيمات المحتمل خروجها كناتج للدالة بعد التعويض بالقيم الخاصة بمجال الدالة (مجموعة القيم الفعلية للدالة)٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مث</span></span></b></span><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ا</span></span>ل:</span></span></b></span><br />
<span style="color: red;"><span style="color: black;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة <span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة عـا المعرفة بالرسم البياني التالي</span></span> المعرفة بالرسم البياني التالي</span></span></span><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></b></span><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitFcSE0MPXYr5IGRviDCBLcKerjnznwe4kAe_fW4DI2g1E7QjiVSBC6nQfCXngki8lpwtakoZKQdUm9DyMiC83MnTqhK-V82NOklNBBq_iNXsuS4FcYLPWeBgIxANIoMNGPGEjMfnZqBg/s1600/fonnn.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitFcSE0MPXYr5IGRviDCBLcKerjnznwe4kAe_fW4DI2g1E7QjiVSBC6nQfCXngki8lpwtakoZKQdUm9DyMiC83MnTqhK-V82NOklNBBq_iNXsuS4FcYLPWeBgIxANIoMNGPGEjMfnZqBg/s1600/fonnn.JPG" /></a><br />
مجال تعريف <span style="font-family: "times" , "times new roman" , serif;">الدالة</span> نعتبر الدالة عـا {1 ، 2 ، 3 ، 5 ، 6} ٠ <br />
مدى <span style="font-family: "times" , "times new roman" , serif;">الدالة</span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"></span></span> عـا {ا ٬ ب ٬ ج ٬ خ } ٠ <br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مث</span></span></b></span><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ا</span></span>ل:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مجموعة تعريف الدالة العددية د للمتغير الحقيقي ش المعرفة بما يلي: عـا(ش) = جذر(س)، هي مجموعة الأعداد الحقيقية الموجبة <b>حـ+</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مجموعة تعريف دالة المقلوب، هي مجموعة الأعداد الحقيقية الغير منعدمة <b>حـ*</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تمرين تطبيقي:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">حدد مجموعة تعريف الدالة العددية المعرفة بما يلي: عـا(س) = جذر(س+1)</span></span><br />
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<span style="color: red;"><b>تركيب دالتين</b></span><br />
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<span style="color: red;"><b>تعريف:</b></span><br />
لتكن ك،ل،ي ثلاثة مجموعات٠ لتكن الدالتين د : ك ← ل و ر: ل ← ي٠ نعرف تركيب الدالة د بالدالة ر و نكتب ر<img alt="o" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> د<br />
بِـ: من أجل كل س من ك، ر <img alt="o" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> د(س) = ر(د(س))٠<br />
ننتج إذن دالة جديدة د <img alt="o" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> ر: ك ← ي٠ و نقرأ ر تركيب د أو ر بعد د٠<br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مث</span></span></b></span><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ا</span></span>ل:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة العددية </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">د(س) = </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span><sup style="background-color: white; color: #222222;">2</sup></span></span></span></span></span></span></span></span> <span style="font-family: "times" , "times new roman" , serif;">+</span> <span style="font-family: "times" , "times new roman" , serif;">5</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة العددية </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> ر(س) = 4</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span> – 1</span></span><br />
ر <img src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> د(س) = ر(د(س)) = ر(س2 + 5) = 4(<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span><sup style="background-color: white; color: #222222;">2</sup></span></span></span></span></span></span></span></span> + 5) – 1 = 4س2 + 19 <br />
د <img src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> ر(س) = د(ر(س)) = ر(4س – 1) = (4س – 1)<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><sup style="background-color: white; color: #222222;">2</sup></span></span></span></span></span></span></span></span> + 5<br />
د <img src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> د(س) =.....<br />
ر <img src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> ر(س) = ....<br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">تطبيق </span></span></span></b></span><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">تقابلي<span style="font-family: "courier new" , "courier" , monospace;">،</span></span></span></span> </span></span></span></b></span><span style="color: red;"><b>التقابل العكسي</b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>تعريف:</b></span></span></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">يكون التطبيق تقابلا إذا كان لكل عنصر من المجموعة المستقر،هناك سابق </span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></span></span></span></span><span style="font-family: "times" , "times new roman" , serif;">وحيد</span> من المجموعة المنطلق٠ أي أنها تطبيق تبايني (العناصر في المستقر لا ترتبط بعنصرين مختلفين في المنطلق <span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">ب</span></span></span>التطبيق</span></span></span></span></span></span></span></span></span>) وفي نفس الوقت تطبيق شمولي (لكل عنصر من المجموعة المستقر، هناك سابق<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> من المجموعة المنطلق</span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></span></span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">ب</span></span></span>التطبيق</span></span></span></span></span></span>)٠ <span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></b></span><br />
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أي يكون التطبيق د : ك ← ل<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">،</span></span></span> تطبيق تقابلي إذا تحقق الشرط التالي: من أجل كل ص من ل يوجد سابق وحيد من ك<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">،</span></span></span> <br />
بحيث: د(س) = ص٠<br />
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<span style="color: red;"><b>خاصية:</b></span><br />
ليكن د : ك ← ل، تطبيق تقابلي، التطبيق الذي يربط بكل عنصر من ل بسابقه (الوحيد) بالتطبيق د، تطبيق تقابلي يسمى التطبيق العكسي لِلتطبيق د أو التقابل العكسي لِلتطبيق د ونرمز له <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "courier new" , "courier" , monospace;">د<span style="font-size: x-small;"><span style="font-size: xx-small;"><b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup><b>-</b></sup></span></span></b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>1</sup></span></span></span></span></span> </span>٠ <br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk8er9e7HUj2OA-L6KH5eKMRbE67kZqG6zsBQeWoq0mCSRi6r9tntxoDH3KlTHsV_imTbbzoX2oELbWwWkTTXBht_fWMoH9RwENhSzRI6HX7JiKHM_WG9zCOxOVFlKi1GclJ8zLpLF_24/s1600/bijrec.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="124" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhk8er9e7HUj2OA-L6KH5eKMRbE67kZqG6zsBQeWoq0mCSRi6r9tntxoDH3KlTHsV_imTbbzoX2oELbWwWkTTXBht_fWMoH9RwENhSzRI6HX7JiKHM_WG9zCOxOVFlKi1GclJ8zLpLF_24/s200/bijrec.JPG" width="200" /></a></div>
بالأسود الدالة س ↤ د (س)<span style="font-family: "trebuchet ms" , sans-serif;">،</span> بالأخضر دالتها العكسية د(س) ↤ س.<br />
<span style="font-family: "trebuchet ms" , sans-serif;">من أجل كل س من ك، <span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">د<span style="font-size: x-small;"><span style="font-size: xx-small;"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>-</sup></span></span><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>1</sup></span></span></span></span></span> <span style="color: red;"><span style="color: black;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><img alt="o" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /></span></span></span></span></span> د(س) = س</span><br />من أجل كل س من ل،<span style="font-family: "courier new" , "courier" , monospace;"> د <span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><span style="color: black;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><img alt="o" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/9/5/d95679752134a2d9eb61dbd7b91c4bcc.png" /> </span></span></span></span></span>د<span style="font-size: x-small;"><span style="font-size: xx-small;"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>-</sup></span></span><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>1</sup></span></span></span></span></span></span><span style="font-family: "courier new" , "courier" , monospace;">(س) = س </span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني وتغيرات الدوال</span></span></b></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWcdtbRi-8AkTtdPX0rIZ0icFXNYRmRgWhDOCr1bQ91-NcrNKlO8_c4-wgkbn9jXs2V559ZFi0joqdJzoNaEgN3VBc82nRq37dFrViHy-sfEpSrdSGLAWyi4LIG69rj6Ez-x_qY5m67YA/s1600/courbe.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWcdtbRi-8AkTtdPX0rIZ0icFXNYRmRgWhDOCr1bQ91-NcrNKlO8_c4-wgkbn9jXs2V559ZFi0joqdJzoNaEgN3VBc82nRq37dFrViHy-sfEpSrdSGLAWyi4LIG69rj6Ez-x_qY5m67YA/s320/courbe.JPG" /></a></span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">لتكن ها دالة عددية، و (ا ؛ ب ؛ ت) معلما للمستوى٠</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني للدالة ها، أو منحنى الدالة ها، هو مجموعة نقط المستوى ن(س ؛ ص) بحيث س تنتمي لِمجموعة تعريف الدلة ها، و ص= هـا(س)٠</span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تغيرات الدوال</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تدعى الدالة العددية هـا المعرفة على مجموعة جزئية من الأعداد الحقيقية
دالة تصاعدية (↖) إذا كانت تحافظ على الترتيب٬ أي لكل س ≤ ش يتحقّق أيضًا
هـا(س) ≤ هـا(ش)، (أنظر رسم 1)٠</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تدعى الدالة العددية هـا المعرفة على مجموعة جزئية من الأعداد الحقيقية دالة تنازلية (↙) إذا كانت تعكس آلترتيب٬ أي لكل س ≤ ش يتحقّق أيضًا هـا(س) ≥ هـا(ش)، (أنظر رسم 2)٠ </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWjQZ1-vTgqQBoh1XBGIT3ooQ62ZgHUg8urZ7aW8dXGy0eDTpP57JIxJ55WQbMJj5OlzQtFhsX4oycw36SNAqOUzb961qvOQAqQpCnF0ehvR3Lxrn4PhQOyoohp6UfMPx1YKtDJmKLIM0/s1600/monot.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWjQZ1-vTgqQBoh1XBGIT3ooQ62ZgHUg8urZ7aW8dXGy0eDTpP57JIxJ55WQbMJj5OlzQtFhsX4oycw36SNAqOUzb961qvOQAqQpCnF0ehvR3Lxrn4PhQOyoohp6UfMPx1YKtDJmKLIM0/s1600/monot.JPG" /></a></span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مثال:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر التمثيل المبياني للدالة العددية هـا للمتغير الحقيقيقي س التلي</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg05HPOSQ28lVELnMBphvTMUVRwQcCckO9USADlrPL3QpHZUfbzaO0BK6oazZNIQtBxff1iG_oXCrBgo2phc3Brp4YOY-QU_abMEGoVQkBTnS1_ykKjLrcH50Tu1XG9DIBXVmjXXYt81yY/s1600/gr2.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg05HPOSQ28lVELnMBphvTMUVRwQcCckO9USADlrPL3QpHZUfbzaO0BK6oazZNIQtBxff1iG_oXCrBgo2phc3Brp4YOY-QU_abMEGoVQkBTnS1_ykKjLrcH50Tu1XG9DIBXVmjXXYt81yY/s320/gr2.JPG" /></a></span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العددية هـا دالة تنازلية من أجل كل س ≤ 3 (أي في المجال [3 ٬ +∞[)، و تصاعدية من أجل كل س ≥ 3 (أي في المجال </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">]–∞ ٬ 3])٠ و نلخص هذا كما يلي</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGrLGzYFouDvbfmNfUlcj37OIi5jbvMEu98m0lfle3fn05clekfs9rgC-Oe6ueG_HiGGyMn_t2ykEKfAL00tJJK8LiqpYaO3EPZZq4-MD4HQY3QUPCYIsUI96ASc6CTX50mdoAm5t6K2k/s1600/TAB.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGrLGzYFouDvbfmNfUlcj37OIi5jbvMEu98m0lfle3fn05clekfs9rgC-Oe6ueG_HiGGyMn_t2ykEKfAL00tJJK8LiqpYaO3EPZZq4-MD4HQY3QUPCYIsUI96ASc6CTX50mdoAm5t6K2k/s320/TAB.JPG" /></a></span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">كيف نتعرف على تغيرات الدوال جبريا؟ </span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة العددية عـا للمتغير الحقيقيقي س المعرفة على مجموعة جزئية من الأعداد الحقيقية</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">للتعرف عن تغيراتها في مجال ما من مجموعة تعريفها ندرس إشارة النسبة: (عـا(س) – عـا(ص)) ÷ (س – ص)</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">من أجل كل س، ص، س≠ ص من المجال٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">إذا كانت هذه النسبة موجبة في مجال ما، فالدالة تحافظ على الترتيب أي تصاعدية (في المجال الذي تتحقق فية إشارة النسبة)٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">و إذا كانت هذه النسبة سالبة في مجال ما، فالدالة تعكس الترتيب أي تنازلية (في المجال الذي تتحقق فية إشارة النسبة)٠</span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تمرين تطبيقي:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ادرس تغيرات الدوال التالية </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العددية عا(س) = 4ش – 1 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العددية عا(س) = ج ش – 1 ، حيث ج عدد حقيقي معلوم لا يتعلق بالمتغير </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العددية عا(س) = ش2 ، في مجموعة الأعداد الحقيقية الموجبة، حـ+ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العددية عا(س) = ش2 ، في مجموعة الأعداد الحقيقية </span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دوال ابتدائية</span></span></b></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة الخطيــــة </span></span></b></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">أ</span> عدد معلوم، العلاقة التي تربط العدد س بالعدد <span style="font-family: "times" , "times new roman" , serif;">أ</span> س تسمى الدالة الخطية ذات المعامل <span style="font-family: "times" , "times new roman" , serif;">أ</span>٠ أي الدالة الخطية هي دلة معرفة كالتالي: </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b> هـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ هـا(س) = <span style="font-family: "times" , "times new roman" , serif;">أ</span> س </span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث <span style="font-family: "times" , "times new roman" , serif;">أ</span> عدد معلوم٠ </span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مثال :</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">هـا(س) = 2س </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>التمثيل المبياني لدالة خطية</b></span> </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة الخطية المعرفة كالتالي: </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> عـا: <b>حـ</b> ← <b>حـ </b> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ هـا(س) = </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">أ</span> </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">س</span></span><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث <span style="font-family: "times" , "times new roman" , serif;">أ</span> عدد معلوم٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مِن دراسة إشارة النسبة: (عـا(س) – عـا(ص)) ÷ (س – ص) من أجل كل س، ص، س≠ ص من مجموعة الأعداد الحقيقية، حـ ٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نٌبيِّن الخاصية التالية: </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>خاصية:</b></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني لدالة خطية في المستوى منسوب إلى معلم متعامد، هو مستقيم يمر من أصل المعلم٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة الخطية دالة تصاعدية إذا كان المعامل جـ عدد موجب، و تكون تنازلية إذا كان المعامل جـ عدد سالب٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة التآلفيــــة</span></span></b></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;">أ</span>، <span style="font-family: "times" , "times new roman" , serif;">ب</span> عددان معلومان، العلاقة التي تربط العدد س بالعدد <span style="font-family: "times" , "times new roman" , serif;">أ</span> س + <span style="font-family: "times" , "times new roman" , serif;">ب</span> تسمى الدالة التآلفيــــة ذات المعامل </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <span style="font-family: "times" , "times new roman" , serif;">أ</span></span></span>٠ أي الدالة التآلفيــــة هي دلة معرفة كالتالي: </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>هـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ هـا(س) = <span style="font-family: "times" , "times new roman" , serif;">أ</span> س + <span style="font-family: "times" , "times new roman" , serif;">ب</span> </span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث <span style="font-family: "times" , "times new roman" , serif;">أ</span>، <span style="font-family: "times" , "times new roman" , serif;">ب</span> عددان معلومان٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مثال:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">هـا(س) = 2 س + 1 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني لدالة التآلفيــــة </span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة التآلفيــــة المعرفة كالتالي: </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>عـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ هـا(س) = جـ س + خـ</span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث جـ، خـ عددان معلومان٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مِن دراسة إشارة النسبة: (عـا(س) – عـا(ص)) ÷ (س – ص) من أجل كل س، ص، س≠ ص من مجموعة الأعداد الحقيقية، حـ ٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نٌبيِّن الخاصية التالية: </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">خاصية:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني لدالة تآلفيــــة في المستوى منسوب إلى معلم متعامد، هو مستقيم٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة التآلفيــــة دالة تصاعدية إذا كان المعامل جـ عدد موجب، و تكون تنازلية إذا كان المعامل جـ عدد سالب٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دراسة الدوال الحدودية من الدرجة الثانية</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixZH-ks_T3xklRccNImxTw2EEdt3ElZpkelipNq3kbW9TRgrEX0FsYyqfLkpCxq7qTmPqkh1ZoLnGAJosxON2CUskDE1Zgqp460BUqpFAdLcn2KSZmsXdujZbgDPOzmNEthaTjm24yDws/s1600/fig1.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixZH-ks_T3xklRccNImxTw2EEdt3ElZpkelipNq3kbW9TRgrEX0FsYyqfLkpCxq7qTmPqkh1ZoLnGAJosxON2CUskDE1Zgqp460BUqpFAdLcn2KSZmsXdujZbgDPOzmNEthaTjm24yDws/s200/fig1.JPG" /></a></span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الشلجم أو اللفت</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدالة الحدودية من الدرجة الثانية هي التطبيق الذي يربط بكل عنصر ش من مجموعة الاعداد الحقيقية بالعدد الحقيقي عـا(ش)=ج </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;"></span></span></span> + ح ش + خ ، حيث ج، ح، خ أعداد حقيقية معلومة لا تتعلق بالمتغير ش، ج عدد غير منعدم </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أي الدالة المعرفة كالتالي</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <span style="font-family: "times" , "times new roman" , serif;"> </span> <b>عـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ عـا(ش)=ج </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span><sup style="background-color: white; color: #222222;">2</sup></span></span> + ح ش + خ </span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث ج٬ ح٬ خ أعداد معلومة<b>٠</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بصفة عامة يمكن كتابة د(ش) على الشكل التالي: عـا(ش) = ج(ش + ح÷2ج</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">)</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="background-color: white; color: #222222;"></span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;"></span></span></span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="background-color: white; color: #222222;"></span><span style="background-color: white; color: #222222;"></span></span></span> + </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">خ – </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ح</span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"></span></span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;"></span></span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="background-color: white; color: #222222;"></span></span></span> ÷ 4ج </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني لدالة حدودية من الدرجة الثانية </span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر الدالة الحدودية من الدرجة الثانية المعرفة كالتالي</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>عـا: حـ ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ عـا(ش)=ج </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span></b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><sup style="background-color: white; color: #222222;">2</sup></span></span> + ح ش + خ</b> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بحيث ج٬ ح٬ خ أعداد معلومة٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مِن دراسة إشارة النسبة: (عـا(س) – عـا(ص)) ÷ (س – ص) من أجل كل س، ص، س≠ ص من المجال [–ح÷2ج ٬ +∞[، و المجال ]–∞ ٬ –ح÷2ج]٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نٌبيِّن الخاصية التالية </span></span><br />
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></b></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">خاصية<span style="font-family: "times" , "times new roman" , serif;">:</span></span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">لتمثيل المبياني في المستوى منسوب إلى معلم متعامد للدالة الحدودية من الدرجة الثانية المعرفة بِـ عـا(ش)=ج </span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ش</span></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="background-color: white; color: #222222;"></span><sup style="background-color: white; color: #222222;">2</sup><span style="background-color: white; color: #222222;"></span></span></span></span></span> + ح ش + خ، قطع مكافئ رأسه النقطة ذات الأفصول –ح÷2ج٠ على الشكل الموضح: منحنى متناظر حول المحور ش = –ح÷2ج، رأسه النقطة ذات الأفصول –ح÷2ج٠</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9pE3HaiSLUjO_yYFkYCQBWeCA7HxlTmeUj8qEz6dtbY-t8QAHGE-nTwSFCeDVqAo1f7FsmR0CBD1Aot1S5ZWo1QqaYsdIRnAQAUeHgI6iFj8vqO2Llnt47JGz3gqoWG2g9yi5Ooz204Q/s1600/cour2.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9pE3HaiSLUjO_yYFkYCQBWeCA7HxlTmeUj8qEz6dtbY-t8QAHGE-nTwSFCeDVqAo1f7FsmR0CBD1Aot1S5ZWo1QqaYsdIRnAQAUeHgI6iFj8vqO2Llnt47JGz3gqoWG2g9yi5Ooz204Q/s320/cour2.JPG" /></a></span></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دراسة دالة المقلوب</span></span></b></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span></span></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دالة المقلوب هي التطبيق الذي يربط بكل عنصر ش من مجموعة مجموعة الأعداد الحقيقية الغير منعدمة حـ* بالعدد الحقيقي </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بالعدد الحقيقي الغير منعدم عـا(ش)= 1÷ ش ٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أي الدالة المعرفة كالتالي</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> <b>هـا: حـ* ← حـ </b></span></span><br />
<b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"> س ↤ هـا(س) = 1÷ س </span></span></b><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني لدالة المقلوب</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">مِن دراسة إشارة النسبة: (عـا(س) – عـا(ص)) ÷ (س – ص) من أجل كل س، ص، س≠ ص من المجال ]0 ٬ +∞[، و المجال ]–∞ ٬ 0[٠ </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نٌبيِّن الخاصية التالية</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">خاصية:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دالة المقلوب دالة تنازلية في المجالين ]0 ٬ +∞[، و المجال ]–∞ ٬ 0[ ٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التمثيل المبياني للدالة المقلوب في المستوى منسوب إلى معلم متعامد منحنى متناظر بالنسبة لأصل المعلم، على الشكل الموضح٠</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif9oIqDX5lfK4-fMlDJWcykDAKIRV0JXqdPyhZfNrFDYkF_AYNXytpuoFA_2dRohheWi1wP_coRmJt52RdPg3ei3vL5q5d6R4ICojiVyL0xPRzUUNBb52gYnPdEm4YWIiObHx7oAvzvuA/s1600/INVERSE.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEif9oIqDX5lfK4-fMlDJWcykDAKIRV0JXqdPyhZfNrFDYkF_AYNXytpuoFA_2dRohheWi1wP_coRmJt52RdPg3ei3vL5q5d6R4ICojiVyL0xPRzUUNBb52gYnPdEm4YWIiObHx7oAvzvuA/s320/INVERSE.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التوابع المثلثية</span></span></b></span><br />
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<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الراديان</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الراديان (Radian) هي وحدة قياس للزوايا المستوية وهي الوحدة الرسمية المعتمدة ضمن النظام الدولي للوحدات المستخدمة في الرياضيات والفيزياء</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">"في المستوى الأقليدي" النسبة بين محيط الدائرة وقطرها ثابت رياضي (عدد حقيقي لا يتغير مع تغيير قطرالدائرة) نرمز له بالحرف ط أو π بالحرف اليوناني</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الراديان (واحد) هو قياس زاوية تحدد قوساً (دائريا) طوله مساوي لشعاع الدائرة</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6X8AckicdK-Pm-CL1z3L4wjRUa3fnBCrsZuYyGe9kJdLvQH6Ch8BZ3GIE1-eHPr3tLuzOI08IdfuTW6WY0kzfquZRSeeBjNfdnnwGoVrEXAjHuyHjOmnhsZthQMZZ41Q1UUFCJdmfNnM/s1600/rad.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh6X8AckicdK-Pm-CL1z3L4wjRUa3fnBCrsZuYyGe9kJdLvQH6Ch8BZ3GIE1-eHPr3tLuzOI08IdfuTW6WY0kzfquZRSeeBjNfdnnwGoVrEXAjHuyHjOmnhsZthQMZZ41Q1UUFCJdmfNnM/s200/rad.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">العلاقة المستعملة للتحويل بين الوحدتين الدرجة و الرديان هي</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">( راديان = 180 درجة π) ط رديان = 180 درجة</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">أي راديان واحد يساوي تقريبا 57،29578 درجة </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">ملاحظة:</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بشكل عام نكتب الزاوية بدون أي علامة، و يقصد أن القيمة هي بالراديان</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدائرة المثلثية</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف:</span></span></b></span></span></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">الدائرة المثلثية (دائرة الوحدة) هي دائرة في المستوى الأقليدي منسوب الى معلم متعامد ممنظم (مدرج)، مركزها أصل المعلم و شعاعها يساوي الواحد، حيث الاتجاه الموجب هو اتجاه عكس عقارب الساعة</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbd4rIZXvufz42V5X4aAJW8_1T5kf1nIzIz2EmIOg6U69PIFwXDaZM4YJdLUxKvGhhf58BS3JHVk4-6gjy_kd9Fpu_LyYlYAJQfRDxocAkYx-5P-Katt-OFGqHf2SxJhbIvxI50RH026k/s200/%D8%A7%D9%84%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9+%D8%A7%D9%84%D9%85%D8%AB%D9%84%D8%AB%D9%8A%D8%A9+%D8%AA%D8%B9%D8%B1%D9%8A%D9%81.gif" /></span></span> <br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">التوابع المثلثية</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعرفنا في الدرس السابق (أول عناصر الهندسة، حقائق عن المثلثات و الدوائر) عن الدوال المثلثية للزوايا من 0 إلى 90 درجة، على أنهم نسب بين ضلعين في مثلث قائم الزاوية٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دائرة الوحدة ستمكننا بتوسيع تعريفنا ليشمل كل القيم الحقيقية للزوايا٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">نعتبر المستوى منسوب إلى معلم (ا ؛ ب ؛ ت) ممنظم و متعامد٠</span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_388s1orKKo78B2T8dlbJTa8nVG0Vb0XHszPepYE6GnR-TdLPRQUKZHpftLNJYiYKiNsABI1ZtM1WOFED5VnmZYJWQnP8ar4amzLv_d8sD6dr-OO2AIGRsu4bN0aXkqnZrL1SSgi-r08/s1600/tri.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_388s1orKKo78B2T8dlbJTa8nVG0Vb0XHszPepYE6GnR-TdLPRQUKZHpftLNJYiYKiNsABI1ZtM1WOFED5VnmZYJWQnP8ar4amzLv_d8sD6dr-OO2AIGRsu4bN0aXkqnZrL1SSgi-r08/s200/tri.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">كل نقطة ن من دائرة الوحدة، تشكل زاوية رأسها النقطة (0، 0) و ضلعيها محور الأفاصيل الموجب و الشعاع الذي مبدأ النقطة (0، 0) إلى النقطة ن، حيث الاتجاه الموجب هو اتجاه عكس عقارب الساعة. س قياس هذه الزاوية يساوي طول القوس المقابل لها٬ و محصور بين 0 و 2ط</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">وعكسيا كل عدد حقيقي س محصور بين 0 و 2ط يحدد نقطة وحيدة ن من الدائرة بحيث قياس الزاوية هو العدد س</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">تعريف</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">جيب تمام العدد الحقيقي س هو أفصول النقطة ن، أي الإسقاط العمودي للنقطة ن على محور الفواصل، و نرمز له بِـ تجب(س)٠</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">جيب العدد الحقيقي س هو أرتوب النقطة م، أي الإسقاط العمودي للنقطة م على محور الأراتيب، و نرمز له بِـ جب(س)٠ </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUOU0m9MFcMSfCHGWIhummqvoTo3cR4WQ8T9ALJUXxQk0YmQrmE_NYkQ_oIn5piRuL-WHZtarRKwrJTS7bS6qB6dOrkkTv5FvqCQeOgEFk1jEMJ_0IjGt-_Z99v1DlwdJQYLwOFWkv5Ak/s1600/sincos.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUOU0m9MFcMSfCHGWIhummqvoTo3cR4WQ8T9ALJUXxQk0YmQrmE_NYkQ_oIn5piRuL-WHZtarRKwrJTS7bS6qB6dOrkkTv5FvqCQeOgEFk1jEMJ_0IjGt-_Z99v1DlwdJQYLwOFWkv5Ak/s1600/sincos.JPG" /></a></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">بعض الزوايا الشائعة مع قيم جيب التمام و الجيب و الظل المناظرة لها٠ </span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp1myJnhofjNTyQcQdMdJ0e-tEyW3g_I-PbSQqbJlNIgSbatnohouJIZsBS2WgdVc5fq2FgpbU5a80AYJD4mnY2RVSCOEkwAAKFMvaZgr8GA7ur7cGZKcaT2f2ZY_7kQb7KJsgpj5jcoQ/s1600/ttigooo.JPG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp1myJnhofjNTyQcQdMdJ0e-tEyW3g_I-PbSQqbJlNIgSbatnohouJIZsBS2WgdVc5fq2FgpbU5a80AYJD4mnY2RVSCOEkwAAKFMvaZgr8GA7ur7cGZKcaT2f2ZY_7kQb7KJsgpj5jcoQ/s320/ttigooo.JPG" /></a></span></span><br />
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></b></span>
<span style="color: red;"><b><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">دراسة التوابع المثلثية</span></span></b></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;">........................</span></span><br />
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<!-- Blogger automated replacement: "https://images-blogger-opensocial.googleusercontent.com/gadgets/proxy?url=http%3A%2F%2Fupload.wikimedia.org%2Fwikipedia%2Fcommons%2Fthumb%2Fc%2Fc0%2FNumber_line.svg%2F440px-Number_line.svg.png&container=blogger&gadget=a&rewriteMime=image%2F*" with "https://blogger.googleusercontent.com/img/proxy/AVvXsEglH0nad-VvSxh5GQ5_SCWNYb4dn7lPcXGRIIDgHLqX6Wm-Ar9Ra8LNLErVDWBjWooFnxVSDdfXMZwBmn86Yi5WkV1RqjoY4_c3WatUieWsXWIb_lWc3BE9rqf0XoFa1GMqbSYIJLrRzqdzoGshd1-Jya4jbLxcqfE3aKQV2XCcSnV7V448DCVQsOUFxOgsmXLQ0j1nU-EKwUrgE3qxchQ3vZg=" --> محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.com0tag:blogger.com,1999:blog-162104398430917066.post-60839715379429135192012-12-09T12:09:00.002-08:002020-03-05T04:44:38.089-08:00أول عناصر الهندسة، خصــائص في المثلثات و الدوائر، مبرهنة فيثاغورس<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><br /></span></b></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"></span></span></span></span></span><br />
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><b>تمارين</b> </span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><br /></span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين17</span></span></span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa">لديك مجموعة من المثلثات قوائم الزاوية وعُلِمَ في كل حالة منها طولي ضلعين كما هو مبين في الجدول التالي، احسب طول الضلع الثالث</span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><span lang="ar-sa"><br /></span></span></b></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRHPcpz23H05BbEiAazrsUv7Nzvwjv2nflQ3ubMzQAjFoRYb3y_GEM4nBOp7qDiWxWr565k_lHa1f8uyUWaUD5Ry8Q3ouDpBQcTB8OWlFMF-SPxgBSoxXICAj0UzhLfyWmhtiEwxxVQQQ/s1600/TRI.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRHPcpz23H05BbEiAazrsUv7Nzvwjv2nflQ3ubMzQAjFoRYb3y_GEM4nBOp7qDiWxWr565k_lHa1f8uyUWaUD5Ry8Q3ouDpBQcTB8OWlFMF-SPxgBSoxXICAj0UzhLfyWmhtiEwxxVQQQ/s400/TRI.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="400" /></a></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;">بين</span><span style="color: black; line-height: 18.48px;"> أن </span><span style="line-height: 18.48px;">القطعة [ا ج] قطر الدائرة المحيطة بالمثلث التاني٠</span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: black; line-height: normal; text-align: justify;"><br /></span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين18</span></span></span></span><span lang="ar-sa" style="line-height: 18.48px;"></span></span></span></span></span></span></span></span></div>
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<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: black; line-height: normal; text-align: justify;">أ ب ج مثلث </span> حيث </span></span></span></span></span></span></span></div>
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<div style="line-height: 18.48px; text-align: right;">
<div style="line-height: 18.48px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أ ب] = 6 سنتيمتر</span></span></span></span></span></span></span><br />
<div style="line-height: 18.48px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [ب ج] = 8 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أ <span style="line-height: 18.48px;">ج</span>] = 10 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;">بَيِّن أن المثلث </span><span style="color: black; line-height: normal; text-align: justify;">أ ب ج</span><span style="line-height: 18.48px;"> </span><span style="line-height: 18.48px;">قائم الزاوية</span></span></span></span></span></span></span></span></div>
</div>
</div>
</div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px; text-align: left;"><br /></span></span></span></span></span></span></span></span>
<div style="line-height: 18.48px; text-align: right;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين19 </span></span></span><span lang="ar-sa"><br /></span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: black; line-height: normal; text-align: justify;">أ ب ج مثلث </span> حيث </span></span></span></span></span></span></span></div>
<div style="line-height: 18.48px; text-align: right;">
<div style="line-height: 18.48px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أ ب] = 3 سنتيمتر</span></span></span></span></span></span></span><br />
<div style="line-height: 18.48px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [ب ج] = 5 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أ <span style="line-height: 18.48px;">ج</span>] = 8 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;">هل المثلث</span><span style="line-height: 18.48px;"> </span><span style="color: black; line-height: normal; text-align: justify;">أ ب ج </span><span style="line-height: 18.48px;">قائم الزاوية ؟</span></span></span></span></span></span></span></span></div>
</div>
</div>
</div>
<div dir="RTL" style="font-weight: normal; text-align: justify;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /><span style="color: red;">تمرين20</span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: black;">أ ب ج مثلث قائم الزاوية في </span><span style="color: black;">أ</span></span></span></span></span></span></span></span><br />
<div style="text-align: right;">
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD8yRt0p5sRG5-_NpMPBPnZ-0GRkI4_Lz9bZGRLUBTR6-5NwOF7oJmZhuQ4iTUaI6Tvo5_gxdGoryT_8lSGpERGuA3JnskB078IHEiac8Lzga60RnoNU1EJRYoUQYyLxr58YqfJQ8fYBM/s1600/triarect.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhD8yRt0p5sRG5-_NpMPBPnZ-0GRkI4_Lz9bZGRLUBTR6-5NwOF7oJmZhuQ4iTUaI6Tvo5_gxdGoryT_8lSGpERGuA3JnskB078IHEiac8Lzga60RnoNU1EJRYoUQYyLxr58YqfJQ8fYBM/s320/triarect.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></span></span></span></span></span></span></span></div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بين أن جب(هـ) (جيب) و تجب(هـ) (جيب تمام) قيمها محصورة بين 0 و 1</span></span></span></span></span></span></span></div>
<div style="text-align: right;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بيّن أن: جب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (هـ)<sup style="line-height: 27.6px;"> </sup>+ تجب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (هـ) =1</span></span></span></span></span></span></span></div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;">تمرين21</span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA">دائرة الوحدة</span></span></span></span></span></span></span></span><br />
<div style="text-align: left;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"></span></span></span></span></span></span></span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2DthOeJCIONpzdXTO4MUEF6ASVZFfgmZOJ3_6-APyXJ8HiFCN2u9Bl4IyHN4SQcZgBBdcTrhCPNCR9CA8jG2ZEbngny8gsV61IGZUH3GFsQTa0KxDGnI8Y3jGDr7hZ23U3qwBN8ApKos/s1600/sincos.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="203" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2DthOeJCIONpzdXTO4MUEF6ASVZFfgmZOJ3_6-APyXJ8HiFCN2u9Bl4IyHN4SQcZgBBdcTrhCPNCR9CA8jG2ZEbngny8gsV61IGZUH3GFsQTa0KxDGnI8Y3jGDr7hZ23U3qwBN8ApKos/s320/sincos.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></span></span></span></span></span></span></span></div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">بين أن الدوال المثلثية</span></span><span style="text-align: right;"> جب(</span><span style="line-height: 18.48px; text-align: right;">س</span><span style="text-align: right;">) (جيب) و تجب(</span><span style="line-height: 18.48px; text-align: right;">س</span><span style="text-align: right;"> (جيب تمام) قيمها محصورة بين</span><span style="color: red;"><span style="color: black;"> </span></span></span><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">−1 و 1</span></span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">و أن<span style="text-align: right;">: </span><span style="text-align: right;">جب</span><span style="text-align: right;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (</span><span style="line-height: 18.48px; text-align: right;">س</span><span style="text-align: right;">)<sup style="line-height: 27.6px;"> </sup>+ </span><span style="text-align: right;">تجب</span><span style="text-align: right;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (</span><span style="line-height: 18.48px; text-align: right;">س</span><span style="text-align: right;">) </span><span style="text-align: right;">=1</span><span style="color: red;"><span style="color: black;"> ، من أجل كل عدد حقيقي س</span></span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></span></span></span></span></span>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;">تمرين22</span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><span style="color: black;">أ ب ج مثلث قائم الزاوية في </span></span><span style="color: red;"><span style="color: black;">أ حيث</span> </span></span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أج] = 10 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">قياس الزاوية ب = 60 درجة</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">احسب طول [ب ج]</span></span></span></span></span></span></span></div>
<div dir="RTL" style="font-weight: normal; text-align: justify;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /><span style="color: red;">تمرين23</span></span></span></span></span></span></span></span></div>
<div align="right" style="font-weight: normal;">
<div style="line-height: 18.48px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">في المثلث أ ب ج القائم الزاوية في أ</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [ب ج] = 5 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طول [أ ب] = 3 سنتيمتر</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">احسب قياس الزاوية ب</span></span></span></span></span></span></span></div>
<div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></span></span></span></span></span>
<div dir="RTL" style="color: black; font-family: "Times New Roman"; font-size: medium; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; text-align: justify; widows: 1;">
<div style="margin: 0px;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;">تمرين24</span></span></span></span></span></span></span></span></div>
</div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">أنظر الشكل جانبه ( ا مركز الدائرة التي تمر من النقط ب، ج، ح، خ )، أحسب قيمة الزاوية ب ا ح</span></span></span></span></span></span></span><br />
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> و قيمة الزاوية ح خ ب٠</span></span></span></span></span></span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFXhDxdS9ePhTSW-tbH_BcHYpMhSqhHf3ONvXFo7lmu2hFLUcVPNvd0fgUL17FyHVUQGq1tDct1epzzW1EeH8IkbJDx-tCzC5Vr4iAgQkxdCxuyvOaxJxJlcZUKhz_eahogKeehx2zaPU/s1600/ANG.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="245" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFXhDxdS9ePhTSW-tbH_BcHYpMhSqhHf3ONvXFo7lmu2hFLUcVPNvd0fgUL17FyHVUQGq1tDct1epzzW1EeH8IkbJDx-tCzC5Vr4iAgQkxdCxuyvOaxJxJlcZUKhz_eahogKeehx2zaPU/s320/ANG.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></span></span></span></span></span></span></span></div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ارسم المماس (ح م) للدائرة في نقطة ح، و احسب قيمة الزاوية ب ح م</span></span></span></span></span></span></span></div>
<div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><br /></span></span></span></span></span></div>
<div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><br /></span></span></span></span></span></div>
</div>
</div>
<span style="font-size: large;"><span style="color: blue;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">
</span></span></span></span></span></div>
<div>
<span style="font-size: medium;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"></span></span></span></span><br />
<h3 style="text-align: right;">
<span style="font-size: medium;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">
<span style="font-size: small;">الدرس</span></span></span></span></span></h3>
<span style="font-size: medium;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">
<h3 style="text-align: right;">
<span style="font-size: small;">أول عناصر الهندسة</span></h3>
</span></span></span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>النقطة</b></span></span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">في الهندسة الرياضية،
النقطة عبارة عن كائن رياضي عديم الأبعاد يمثل مفهوما أساسيا في الهندسة
الرياضية. تتميز النقطة بأنها تملك موقعا في الفراغ لكن بدون حجم ومساحة
ولا أبعاد فهي تمثل معلومات عن الموقع فقط دون أي خواص رياضية أخرى٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">المستقيم</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المستقيم خط مستقيم يمتد إلى ما لا نهاية من الجهتين٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"> </span> المستقيم ليس له نقطة بداية ولا نقطة نهاية</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المستقيم لا يمكن قياس طوله</span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">الترقيم</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نرمز للمستقيم الدي يمر بالنقطة ا والنقطة ب بالرمز: (ا ب)٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="28" src="https://www.schoolarabia.net/images/asasia_im/duroos_math/geometry/line/3.gif" width="197" /></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">نصف مستقيم</span></b></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نصف مستقيم هو جزء من خط مستقيم يكون محدودا من جهة واحدة فقط </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نصف مستقيم لا يمكن قياس طوله</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> <span style="color: red;">الترقيم</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نرمز للمستقيم المحدود بالنقطة ا، و يمربالنقطة ب بالرمز: [ا ب)٠</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJZOcxhkEziL-HI1uPfPHW_lRbjHelLhKcV-e5HZ-iIDhh1Db9KcCbo1V6Xus3S3RqGt2PQc80my-IFd69uD3RYqlVY51ZFaOhW859mwmgLBhXEQjy36oxAUO_IJsOmtFS5yo-RZcYQ8s/s1600/CCCC.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJZOcxhkEziL-HI1uPfPHW_lRbjHelLhKcV-e5HZ-iIDhh1Db9KcCbo1V6Xus3S3RqGt2PQc80my-IFd69uD3RYqlVY51ZFaOhW859mwmgLBhXEQjy36oxAUO_IJsOmtFS5yo-RZcYQ8s/s1600/CCCC.JPG" /></a></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>القطعة المستقيمة</b></span></span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">القطعة
المستقيمة هي جزء من خط مستقيم محددة بنقطتين تسميان نقطتي النهاية وتضم
جميع النقاط الواقعة على المستقيم بين هاتين النقطتين٠</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> القطعة هي جزء من خط مستقيم محدود من طرفيه و يمكن قياس طولها٠</span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">الترقيم</span></span></span> </div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نرمز للقطعة المستقيمة التي تمر بالنقطة ا والنقطة ب بالرمز: [ أ ب ] أو [ ب أ ]، والنقطتان أ وَ ب هما
طرفاها٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">و نرمز لطول القطعة المستقيمة [ا ب] بالرمز: ا ب</span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
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<table border="0" dir="ltr" id="table4" style="width: 80%px;"><tbody>
<tr><td style="text-align: center;" width="360"><div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" height="38" src="https://www.schoolarabia.net/images/asasia_im/duroos_math/geometry/dots/3.gif" width="200" /></span></span></div>
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<td style="text-align: center;" width="360"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td width="360"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td width="360"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td width="360"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td>
<td width="54"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td>
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<td style="text-align: right;" width="360"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td></tr>
</tbody></table>
<h3 style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">
<span style="color: purple;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red;"> </span></span><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red;"><span style="color: #005500;"><span style="color: red;">الزاوية</span></span></span></span><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red;"><u><span style="color: #005500;"><br /></span></u></span></span></b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red;"><span style="color: black;">الزاوية هي الشّكل الهندسي الناتج عن التقاء شعاعين بنقطة بدايتهما تسمى رأس الزاوية.</span></span></span><b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red;"> </span></span></b></span></span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="http://www.amblesideprimary.com/ambleweb/mentalmaths/protractor.html"><img alt="" class="aligncenter size-full wp-image-206" height="200" src="https://nonstoplearning.edublogs.org/files/2011/10/Angle-20imwiy.png" title="Angles" width="200" /></a></span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><br /></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>التعامد</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">يعتبر مستقيمان متعامدين على بعضهما إذا شكلا زوايا متجاورة متطابقة. تسمى زوايا قائمة (قياس الزاوية القائمة يساوي 90° درجة)٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">جميع الزوايا المكونة من تعامد خطين مستقيمين هي زوايا قائمة. وبالعكس فإن أي</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> خطين مستقيمين يشكلان زوايا قائمة فهما متعامدان٠</span></span></div>
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<h3 style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 21.528px;">يرمز لعملية التعامد بين خطين بالعلامة </span></span><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 21.528px;"><img alt="\perp" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/d/2/1/d21e3d53f78769beee8ff7b6c4a9a92c.png" /> </span></span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihJrP5h9iOVlxX8sNWjg1KB5zNFknmVCF2v6007SplDv_t3kC2kPD6ODp-a56S6w6CBCgn1wIM0VP6fysdCdqSWcRGzz6gJ3pgq8xaFjCGOxEJeUtiiKUnR00rTn5l3TGkSf8HnxRjZoY/s1600/PERP2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihJrP5h9iOVlxX8sNWjg1KB5zNFknmVCF2v6007SplDv_t3kC2kPD6ODp-a56S6w6CBCgn1wIM0VP6fysdCdqSWcRGzz6gJ3pgq8xaFjCGOxEJeUtiiKUnR00rTn5l3TGkSf8HnxRjZoY/s1600/PERP2.JPG" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="color: red;">التوازي</b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">يعتبر مستقيمان (في المستوى) متوازيين إذ استحالة التقاءهما٠ </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">يرمز لعملية التوازي بين خطين بالعلامة<span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> </span></span><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><img alt="\parallel" class="mwe-math-fallback-image-inline tex" src="https://upload.wikimedia.org/math/6/d/1/6d1776b6521615edcdd32490a027fd75.png" /></span></span></span></span></div>
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<h3 style="text-align: right;">
<span style="font-size: small; font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuep0vSWC56jH1qpwpVP7tIKHyLAVzWEA93GaDbnYzgRcHRdsTkHRiUdjevXc1kWxLfGhF4VRWaMQytH36O5RE0tgi8aGvhz2lnwhu629Rq22NIgEtIXNuu7_UEdWByF-EyOEEsBGpk24/s1600/PARR.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuep0vSWC56jH1qpwpVP7tIKHyLAVzWEA93GaDbnYzgRcHRdsTkHRiUdjevXc1kWxLfGhF4VRWaMQytH36O5RE0tgi8aGvhz2lnwhu629Rq22NIgEtIXNuu7_UEdWByF-EyOEEsBGpk24/s1600/PARR.JPG" /></a></div>
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<span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><br /></span></span></span></div>
</span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 21.528px;">إذا كان مستقيمان متعامدان، فكل مستقيم موازي لأحدهما يكون عموديا على الآخر.<br />إذا كان مستقيمان متوازيان فكل مستقيم عمودي على أحدهما يكون عموديا على الآخر.</span></span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 21.528px;"><br /></span></span></span></span></div>
<h3 style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">المسافة بين نقطة وخط مستقيم </span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvK9uRgrqB8sSPaX9spJ4-MsQ9afzlnEpWzw79GQLcM_DtIJHKUksHg8gyqgwm7QXSLOAEPZ2HOwbC44_AJ26AXAMkLDT4zq68rOYsGhuXVh_00Rr5iH7sn7u9TFweNS72_esiGpOUUzQ/s1600/proj.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvK9uRgrqB8sSPaX9spJ4-MsQ9afzlnEpWzw79GQLcM_DtIJHKUksHg8gyqgwm7QXSLOAEPZ2HOwbC44_AJ26AXAMkLDT4zq68rOYsGhuXVh_00Rr5iH7sn7u9TFweNS72_esiGpOUUzQ/s1600/proj.JPG" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">ط نقطة من المستوى٬ أقرب مسافة تفصل النقطة ط عن الخط المستقيم (م) هي طول القطعة [ط ا]٠ تسمى المسافة أو (المسافة الدنيا) بين النقطة ط و الخط المستقيم (م)، و النقطة ا تسمى الإسقاط العمودي للنقطة ط على المستقيم (م)٠</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red; line-height: 27.6px;">تعريف</span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="line-height: 27.6px;">المسافة أو (المسافة الدنيا) بين نقطة ما من المستوى و خط مستقيم ما من المستوى</span></span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="line-height: 27.6px;">هي أقرب مسافة تفصل النقطة عن الخط المستقيم٠</span></span></span></div>
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<h3>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: purple;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="line-height: 21.528px;"><span style="color: red;">الدائرة</span></span></b></span></span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">الدائرة
هي المحل الهندسي للنقاط المتصلة ببعضها البعض والواقعة في المستوى من على
بعد ثابت من نقطة ثابتة ما، والتي تسمى مركز الدائرة. المسافة الفاصلة بين
مركز الدائرة وأي نقطة منها تسمى شعاعا أو نصف قطر<span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">٠</span></span><span style="color: purple;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> </span></span></span></span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: purple;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: purple;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><a class="image" href="http://ar.wikipedia.org/w/index.php?title=%D9%85%D9%84%D9%81:Circle_-_Arabic.png&filetimestamp=20071220110311"><img alt="" class="thumbimage" height="188" src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Circle_-_Arabic.png/250px-Circle_-_Arabic.png" width="250" /></a></span></b></span></span> </span></b></span></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span lang="ar-bh"><span style="color: red;">المماس للدائـرة<b> </b></span></span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span lang="ar-bh">المماس هو ذلك المستقيم الذي يلاقي الدائرة في نقطة واحدة تعرف بنقطة تماسه معها (نقطة التماس)</span><b><span lang="ar-bh"><span style="color: red;"><span style="color: black;"><b>٠</b></span> </span></span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><img alt="http://upload.wikimedia.org/wikipedia/ar/thumb/c/c9/CIRCLE_LINES_Arabic.svg/440px-CIRCLE_LINES_Arabic.svg.png" class="transparent" height="200" src="https://upload.wikimedia.org/wikipedia/ar/thumb/c/c9/CIRCLE_LINES_Arabic.svg/440px-CIRCLE_LINES_Arabic.svg.png" width="198" />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">المماس للدائـرة يكون عمودياً على نصف القطر المار بنقطة التماس٠<br />المستقيم العمودي على نصف القطر في دائرة عند نهايته يكون مماساً للدائرة٠ </span><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><span style="color: red;"><b> </b></span></span></b></span></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><span style="color: red;"><b>المثلث</b></span> </span></b></span></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تعريف </span></b></span><span style="font-weight: normal;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><span style="color: red;"><b>المثلث</b></span> </span></b>المثلث هو أحد الأشكال الأساسية في الهندسة، وهو شكل ثنائي الأبعاد مكون من ثلاثة رؤوس تصل بينها ثلاثة أضلاع، وتلك الأضلاع هي قطع مستقيمة٠</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;"><span class="mw-headline" id=".D8.A3.D9.86.D9.88.D8.A7.D8.B9_.D8.A7.D9.84.D9.85.D8.AB.D9.84.D8.AB.D8.A7.D8.AA">أنواع المثلثات</span></span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">مثلث متساوي الأضلاع: هو مثلث جميع أضلاعه متساوية٠<br />مثلث متساوي الضلعين: ويسمى أيضا متساوي الساقين، هو مثلث فيه ضلعان متساويان٠<br />مثلث قائم: له زاوية قياسها 90 درجة (زاوية قائمة)، يدعى الضلع المقابل للزاوية القائمة بالوتر، وهو أطول أضلاع هذا المثلث<b>٠</b><span style="color: red;"><span class="mw-headline" id=".D8.A3.D9.86.D9.88.D8.A7.D8.B9_.D8.A7.D9.84.D9.85.D8.AB.D9.84.D8.AB.D8.A7.D8.AA"> </span></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>حساب زوايا المثلث الداخلية: </b><span style="color: black;">مجموع قياسات زوايا المثلث الداخلية تساوي 180 درجة٠</span></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-weight: normal;"><b><span style="color: red;"><span class="mw-headline" id=".D8.AD.D8.B3.D8.A7.D8.A8_.D9.85.D8.B3.D8.A7.D8.AD.D8.A9_.D8.A7.D9.84.D9.85.D8.AB.D9.84.D8.AB">حساب مساحة المثلث</span>:</span></b> المساحة = ½ القاعدة × الارتفاع </span></span></span></h2>
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<span style="color: red;"><span class="mw-headline" id=".D9.86.D9.82.D8.B7_.D9.88.D9.85.D8.B3.D8.AA.D9.82.D9.8A.D9.85.D8.A7.D8.AA_.D9.88.D8.AF.D9.88.D8.A7.D8.A6.D8.B1_.D9.85.D8.B1.D8.AA.D8.A8.D8.B7.D8.A9_.D8.A8.D8.A7.D9.84.D9.85.D8.AB.D9.84.D8.AB"> </span></span></span></span></h2>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span class="mw-headline" id=".D9.86.D9.82.D8.B7_.D9.88.D9.85.D8.B3.D8.AA.D9.82.D9.8A.D9.85.D8.A7.D8.AA_.D9.88.D8.AF.D9.88.D8.A7.D8.A6.D8.B1_.D9.85.D8.B1.D8.AA.D8.A8.D8.B7.D8.A9_.D8.A8.D8.A7.D9.84.D9.85.D8.AB.D9.84.D8.AB">نقط ومستقيمات ودوائر مرتبطة بالمثلث</span></span></span></span></h2>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b> </b></span><b><span style="color: red;">المتوسطات العمودية</span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><b><span style="color: #6c5f1f;"><b><b> </b></b></span></b></b><b><b><span style="color: #6c5f1f;"><b><b><span style="color: black;"> </span></b></b></span></b></b><b><b><span style="color: #6c5f1f;"><b><b> </b></b></span></b></b><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تعريف</span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b> </b></span>المتوسط العمودي لمثلث هو مستقيم يمر من أحد أضلاع المثلث في منتصفه ويكون عمودياً عليه</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">خاصية </span></b></span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">تتقاطع المتوسطات العمودية الثلاثة في مركز الدائرة المحيطة بالمثلث وهي الدائرة التي تمر من رؤوس المثلث ويكون تقاطع متوسطين عموديين فقط كافياً لمعرفة مركز هذه الدائرة</span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a class="image" href="http://ar.wikipedia.org/w/index.php?title=%D9%85%D9%84%D9%81:Triangle.Circumcenter.svg&filetimestamp=20060506175042"><img alt="" class="thumbimage" height="198" src="https://upload.wikimedia.org/wikipedia/commons/thumb/7/74/Triangle.Circumcenter.svg/198px-Triangle.Circumcenter.svg.png" width="198" /></a></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">تقاطع المتوسطات العمودية في مركز الدائرة المحيطة بالمثلث</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">ا</span><b><span style="color: red;">لمثلث القائم و الدائرة المحيطة به</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><br /></span></b></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">خاصية</span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"> </span></b>إذا كان مركز الدائرة المحيطة بالمثلث على ضلع من أضلاع المثلث فإن الزاوية المقابلة لهذا الضلع تكون قائمة<b><span style="color: red;"> </span></b>أي المثلث قائم<b> ٬ </b>كما أن إذا كان المثلث قائم فإن مركز الدائرة المحيطة به هو منتصف وتره٠ </span></span></div>
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src="data:image/png;base64,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" /></b><span style="color: red;"><b><br /></b></span></span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>الارتفاعات</b></span></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><br /></span></b></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تعريف</span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"> </span></b>الارتفاع
هو مستقيم يمر براّس من رؤوس المثلث ويكون عمودياً غلى الضلع المقابل
للرأس</span></span>
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<span style="color: red;"><b><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">خاصية</span></b></b></span></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><span style="color: black;"><b> </b></span><span style="color: black;"> </span></span><span style="color: black;">يمثل الارتفاع البعد بين الرأس والضلغ المقابل له كما تتقاطع
الارتفاعات في نقطة تسمى مركز قائم</span><span style="color: black;"> </span></span> </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a class="image" href="http://ar.wikipedia.org/w/index.php?title=%D9%85%D9%84%D9%81:Triangle.Orthocenter.svg&filetimestamp=20060318164615"><img alt="" class="thumbimage" height="146" src="https://upload.wikimedia.org/wikipedia/commons/thumb/9/93/Triangle.Orthocenter.svg/182px-Triangle.Orthocenter.svg.png" width="182" /></a></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">نقطة تقاطع الارتفاعات في مثلث تسمى المركز القائم</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">منصفات الزوايا</span></b></span></span> <br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">تعريف</span></b></span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"> </span></b>منصف الزاوية هو مستقيم يمر من أحد رؤس المثلث ويقسم الزاوية إلى نصفين</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">خاصية </span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">تتقاطع المنصفات الثلاثة في مركز الدائرة المحيطة بالمثلث وهي الدائرة التي تمس أضلاع المثلث الثلاثة ويكون تقاطع منصفين فقط كافياً لمعرفة مركز هذه الدائرة </span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a class="image" href="http://ar.wikipedia.org/w/index.php?title=%D9%85%D9%84%D9%81:Triangle.Incircle.svg&filetimestamp=20060506184546"><img alt="" class="thumbimage" height="157" src="https://upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Triangle.Incircle.svg/182px-Triangle.Incircle.svg.png" width="182" /></a></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">تقاطع منصفات الزوايا في مركز الدائرة المحاطة بالمثلث</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;"><br /></span></b></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">المتوسطات</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">تعريف</span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المتوسط هو قطعة مستقيم تنطلق من أحد رؤس المثلث وتمر من منتصف الضلع المقابل لهذا الرأس </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"><br /></span></b></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">خاصية</span></b><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"> </span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">تتقاطع المتوسطات الثلاثة في نقطة تسمى مركز ثقل المثلث ويكون تقاطع متوسطين فقط كافياً لمعرفة مركز الثقل </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;"> </span></b></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">كما يكون البعد بين رأس المثلث ومركز الثقل مساوياً لـ<img alt="\frac{2}{3}" class="tex" src="https://upload.wikimedia.org/math/c/d/1/cd1820f286fec69f1c54a381d8fa7d93.png" /> من طول المتوسط الصادر من ذلك الرأس</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المتوسطات ومركز الثقل</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">مبرهنة فيثاغورس</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 18.4px;">خاصية </span></b></span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">مبرهنة فيثاغورس: في المثلث ا ب ت القائم الزاية في النقطة ا، مربع طول الوتر [ب ت] يساوي مجموع مربعي طولي الضلعين القائمين [ا ب] ، [ا ت]٠</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">أي<b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><b> </b></span></span></b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86">:</span></span><b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><b> </b></span></span></b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"> </span></span><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86">ب ت<sup>2</sup></span></span><b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"> = </span></span></b>ا ب<b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>2</sup> + </span></span></b>ا ت<b><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup>2</sup></span><span class="mw-headline" id=".D9.85.D8.A8.D8.B1.D9.87.D9.86.D8.A9_.D9.81.D9.8A.D8.AB.D8.A7.D8.BA.D9.88.D8.B1.D8.B3"> </span></span></b></span></span></div>
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<span style="font-size: 18.72px;"><span style="color: red;">مبرهنة فيثاغورس العكسية</span></span></h3>
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<span style="font-size: 18.72px;"><span style="color: red;">خاصية</span> </span></h3>
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<span style="font-size: 18.72px;">في مثلث، إذا كان مربع طول أطول ضلع يساوي مجموع مربعي طولي الضلعين الآخرين، فإن هذا المثلث قائم الزاوية. الزاوية القائمة هي الزاوية المقابلة لأطول ضلع، والضلع الأطول هو الوتر٠</span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">الحساب المثلثي، تعميم نظرية فيثاغورث</span></b></span></span></div>
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<sup><b><b><span style="color: red;"> </span></b></b></sup><b><span style="color: red;">الحساب المثلثي</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">في الرياضيات، الدوال المثلثية هي دوال ترتبط بالزاوية ، ويمكن تعريف الدوال المثلثية على أنهم نسب بين ضلعين في مثلث قائم الزاوية<br /><br />في المثلث أ ب ج القائم الزاوية في </span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">أ</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">جيب التمام الزاوية هـ = الضلع المجاور للزاوية مقسوما على الوتر<br />جيب الزاوية هـ = الضلع المقابل للزاوية مقسوما على الوتر<br />ظل الزاوية هـ = الضلع المقابل للزاوية مقسوما على الضلع المجاور لها</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">كما أن: ظل الزاوية = جيب الزاوية مقسوما جيب تمام الزاوية</span></span><br />
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<tr><td style="background-color: #ffff33; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">الرمز</span></span></td><td style="background-color: #ffff33; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المصطلح باللغة الفرنسية</span></span></td><td style="background-color: #ffff33; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">المصطلح باللغة العربية</span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="color: #222222; line-height: 18.48px; text-align: right;">جب</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">sinus</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">جيب</span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="color: #222222; line-height: 18.48px; text-align: right;">تجب</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">cosinus</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">جيب تمام </span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="text-align: right;">ظل</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">tangente</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">ظل</span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">Le coté opposé</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">الضلع المقابل</span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">Le cote adjacent</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">الضلع المجاور</span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">L'hypoténuse</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">الوتر</span></span></span></td></tr>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>من خلال التعريفات يمكن أن نُبيِّن أن</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b>جيب و جيب تمام زاوية ما عددان بين 0 و 1 -</b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b>ظل الزاوية = جيب الزاوية مقسوما جيب تمام الزاوية -</b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b>تجب(هـ) = جب(90 <span style="background-color: white; color: #222222; line-height: 25.2px;">–</span> هـ)٠ -</b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 27.6px;">خاصية</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b>جب(هـ) (جيب) و تجب(هـ) (جيب تمام) قيمها محصورة بين 0 و 1</b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b>جب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (هـ)<sup style="line-height: 27.6px;"> </sup>+ تجب<span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup></span></span></span></span></span> (هـ) </b><b>=</b>1</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">في المثلث ا ب ت القائم الزاوية في ا</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">طول [ا ت] = 15 سنتيمتر</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">طول [ا ب] = 5 سنتيمتر</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">فيكون ظل الزاوية ب: ا ت ÷ ا ب = 15÷ 5 = 3 </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">عند معرفة قيم ضلعين أو قيم ضلع وزاوية، يمكن إيجاد قيم باقي عناصر المثلث (زوايا واضلاع) باستخدام قوانين الجيب وقوانين جيب تمام وقوانين ظل الزاوية و الدوال العكسية</span></span></span></div>
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<tr><td style="background-color: #ffff33; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">رمز الدالة العكسية </span></span></span></span></span></span> </span></span></td><td style="background-color: #ffff33; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">الدالة العكسية</span></span></span></span></span></span></span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"></span></span></td><td style="background-color: #ffff33; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">الدالة </span></span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="color: #222222; line-height: 18.48px; text-align: right;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">ق</span></span>جب</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">قوس جيب</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">جيب</span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="color: #222222; line-height: 18.48px; text-align: right;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">ق</span></span>تجب</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">قوس <span style="color: #222222; line-height: 18.48px; text-align: right;">جيب تمام</span></span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">جيب تمام </span></span></span></td></tr>
<tr><td style="background-color: #ccffff; text-align: center; width: 194px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b style="text-align: right;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">ق</span></span>ظل</b></span></span></td><td style="background-color: #e7d6ff; text-align: center; width: 173px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">قوس ظل</span></span></td><td style="background-color: #f9e2a4; color: red; font-weight: bold; text-align: center; width: 265px;"><span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-weight: normal; text-align: right;">ظل</span></span></span></td></tr>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>مثال</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">في المثلث ا ب ت القائم الزاوية في ا</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">طول [ا ت] = 15 سنتيمتر</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">طول [ا ب] = 5 سنتيمتر</span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWMjWp0IHt-T4OYbkEFwPIU7wdAMCicJbhyhAMhLNLL5ThIExR-DWgEms50G5RQWdPvckXn4VjYHuj5YzsH_Xieea39uyYqXPvGuMtVVa-9dBrHbVQiI23thfWy8peynxnIvcQvZsDuW4/s1600/TRIRR.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWMjWp0IHt-T4OYbkEFwPIU7wdAMCicJbhyhAMhLNLL5ThIExR-DWgEms50G5RQWdPvckXn4VjYHuj5YzsH_Xieea39uyYqXPvGuMtVVa-9dBrHbVQiI23thfWy8peynxnIvcQvZsDuW4/s200/TRIRR.JPG" width="200" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">فيكون ظل الزاوية ب: ا ت ÷ ا ب = 15÷ 5 = 3 </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> (و يكون قياس الزاوية ب = قوس ظل الزاوية (3</span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> <b><span style="color: red;">مبرهنة الكاشي (تعميم نظرية فيثاغورث):</span></b> مربع طول الضلع يساوي مجموع مربعي الضلعين الآخرين مطروح منه ضعف حاصل ضروب طولي الضلعين الآخرين في جيب تمام "الزاوية المحصورة بينهما"٠<br /><br /><span style="color: red;"><b> </b></span></span></span></div>
<h3 style="text-align: center;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>الزوايا المركزية والزوايا المحيطية</b></span></span></span></h3>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>1) الزاوية المركزية</b></span><span style="color: red;">،</span><span style="color: red;"><b><span style="color: red;"><b> الزاوية المحيطية</b></span>:</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>تعريف:</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">الزاوية المركزية هي كل زاوية رأسها مركز دائرة و ضلعاها يقطعان الدائرة.</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">الزاوية المحيطية هي كل زاوية رأسها ينتمي إلى دارة و ضلعاها يقطعان الدائرة.</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">نعتبر الشكــل جانبه:</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br />لدينا الزاوية س م ص زاوية مركزية تحصر القوص س ص <br />لدينا الزاوية س ع ص زاوية محيطية تحصر القوص س ص <br />لدينا الزاوية س ب ص زاوية محيطية تحصر القوص س ص</span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5ofU5Q6ROppGPIdETsVy_Wmd3JlABAAGIoknHwXIhsew9lxpFxZ62sqrB03tTTBLiGSwgrSxfutl2J6HOhJI5lMpMoaW2JFBvCf7dbep_4BSdARcvc2B9byeJJgsEjK1ajTu5RtanwA8/s1600/ang3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5ofU5Q6ROppGPIdETsVy_Wmd3JlABAAGIoknHwXIhsew9lxpFxZ62sqrB03tTTBLiGSwgrSxfutl2J6HOhJI5lMpMoaW2JFBvCf7dbep_4BSdARcvc2B9byeJJgsEjK1ajTu5RtanwA8/s200/ang3.JPG" width="200" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: blue;"><b><span style="color: red;">حالة خاصة</span></b></span><span style="color: blue;"><b><span style="color: red;">، الزاوية المماسية:</span></b></span></span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: blue;"><b><span style="color: red;"><br /></span></b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">لاحظ الشكــل جانبه بحيث المستقيم (ب ت) مماس للدئرة في النقطة ب</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">الزاوية ج ب ت زاوية محيطية تحصر القوس ج ب</span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjITlimeIgJccQlXYMnJz5yV8cj1UzwB3ZoQT6UoESZhrcFPG7DDE4GIcqgL1TrFvOXC_3JUgh7O-NsfWdzUA_vDerdpZGgeUXo0jaIXVLSJ6Khy8jI4DqFWHWlDZ6vHirUXukplGv7mQw/s1600/ang2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjITlimeIgJccQlXYMnJz5yV8cj1UzwB3ZoQT6UoESZhrcFPG7DDE4GIcqgL1TrFvOXC_3JUgh7O-NsfWdzUA_vDerdpZGgeUXo0jaIXVLSJ6Khy8jI4DqFWHWlDZ6vHirUXukplGv7mQw/s1600/ang2.JPG" /></a></span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">3</span></b><span style="color: red;"><b>) خصــائص:</b></span></span></span><br />
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">تعريف :</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">تكون زاوية مركزية مرتبطة بزاوية محيطية إذا كانتا تحصران نفس القــوس.</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: red;">خــاصية :</span></b></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">زاويتان محيطيتان تحصران نفس القوس تكونان مقايستين</span></span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;">قياس زاوية محيطية يساوي نصف قياس الزاوية المركزية المرتبطة بها</span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><b>مثال :</b></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzdh3eIKrkFcRJUtzF4uvNufBd2vap2gpvErtX4OeQOm_JCgiC05Y2MNo7hp6Iwf_wU1ybucpe08nW-XsU83wpYF1BEdsBhowXyEJSxTS6czJ9h1xJuTN9nIeXij9YXqdCOkFMX4Zypxc/s1600/ang3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzdh3eIKrkFcRJUtzF4uvNufBd2vap2gpvErtX4OeQOm_JCgiC05Y2MNo7hp6Iwf_wU1ybucpe08nW-XsU83wpYF1BEdsBhowXyEJSxTS6czJ9h1xJuTN9nIeXij9YXqdCOkFMX4Zypxc/s200/ang3.JPG" width="200" /></a></span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"> </span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">لدينا الزاوية س م ص زاوية مركزية تحصر القوص س ص </span></span></div>
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<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">لدينا الزاوية س ع ص زاوية محيطية تحصر القوص س ص <br />لدينا الزاوية س ب ص زاوية محيطية تحصر القوص س ص</span></span><br />
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</div>
<div dir="RTL" style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><span style="color: black;"><span style="color: black;">إذن:</span></span><span style="color: black;"><span style="color: black;"> </span></span><span style="color: black;"><span style="color: black;">م = 2 </span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span style="color: black;">×</span></span> </span></span></span></span></span><span style="color: red;"><span style="color: black;"><span style="color: black;">ع </span></span></span><span style="color: red;"><span style="color: black;"><span style="color: black;"><span style="color: red;"><span style="color: black;"><span style="color: black;">= 2 </span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span style="color: black;">×</span></span> </span></span></span></span></span></span><span style="color: black;"><span style="color: black;">ب</span></span> </div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;">حالة خاصة</span></span></span></div>
<div style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">لاحظ الشكــل جانبه بحيث المستقيم (ب ت) مماس للدئرة في النقطة ب</span></span><br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;">لدينا: الزاوية ت بج زاوية محيطية تحصر القوس بج</span></span></div>
<div dir="RTL" style="text-align: right;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;"><span style="color: black;"><span style="color: black;"><span style="color: black;">إذن:</span></span><span style="color: black;"><span style="color: black;"> </span></span></span></span></span></span><span style="color: black;">الزاوية المركزية </span> ب م<span style="color: black;"> </span><span style="color: black;">ج مرتبطة بالزاوية المحيطية </span>ت ب<span style="color: black;"><span dir="LTR">ج </span></span><span style="color: black;"> </span><span style="color: black;"><span style="color: black;">لأنهما </span> </span><span style="color: black;"><span style="color: black;"> تحصران نفس</span> </span>القوس بج <br />
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikdhC6_tBGAKO-T_XYGnxtTnrjbhj84_VbfQ6d7ROucNFUGADDnLHEnO_ImkNH8sbSVDR5EkYjstqP4QGLqhC8_3g8hohlSsx14061OQNX15OA7CyPsWYEQ_bKhB0Ubmb3G2c3oT7QmoA/s1600/ang.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikdhC6_tBGAKO-T_XYGnxtTnrjbhj84_VbfQ6d7ROucNFUGADDnLHEnO_ImkNH8sbSVDR5EkYjstqP4QGLqhC8_3g8hohlSsx14061OQNX15OA7CyPsWYEQ_bKhB0Ubmb3G2c3oT7QmoA/s1600/ang.JPG" /></a></span></span></div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></span>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black;"><span style="color: black;">إذن:</span></span><span style="color: black;"><span style="color: black;"> </span></span><span style="color: black;"><span style="color: black;">م = 2 </span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span style="color: black;">×</span></span> </span></span></span></span><span style="color: black;"><span style="color: black;">ب</span></span> </div>
<h3 style="text-align: center;">
</h3>
<div>
<span style="font-size: small;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: red;"><br /></span>
</span></span><span style="font-family: "georgia" , "times new roman" , serif;"><span style="font-size: large;"><span style="color: red;"><br /></span></span></span>
</div>
<h3 style="text-align: left;">
</h3>
<div>
<span style="font-family: "georgia" , "times new roman" , serif; text-align: justify;">alfarjimohammed@gmail.com</span></div>
<div style="text-align: right;">
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محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.com0tag:blogger.com,1999:blog-162104398430917066.post-59298545736265847132012-12-09T11:37:00.003-08:002020-03-05T04:49:01.975-08:00المتجهات، تحويلات في المستوى، مبرهنة طاليس<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="color: red;">تمارين</span><br />
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red; line-height: 18.48px;"><br /></span><span style="line-height: 18.48px;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين25</span></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;"> أنظر الشكل جانبه</span><br style="line-height: 18.48px;" />المستقيمان (ب خ)، (ج ح) متقاطعان في النقطة ا<br style="line-height: 18.48px;" /> ا ب = 4م، ا خ = 6م، ح خ = 7م</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">احسب طول القطعة [ب ج]٠</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">احسب النسبة: مساحة المثلث ا ح خ / مساحة المثلث ا ج ب٠</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwWLTYJC_d9Jw3qmyFm06lWg0H45jGAqMK6l3flvCuDg6GFZ3PSvIlpuo2x9XNpNWywp7viwsSGEBWxITx18uVoB-Xu_iWjHX9rHy45OIFnId3XGTMgQyjAc6YYuBa0mrFV5nUnkqKqsE/s1600/TH2.JPG" style="color: #888888; margin-left: 1em; margin-right: 1em; text-decoration-line: none;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwWLTYJC_d9Jw3qmyFm06lWg0H45jGAqMK6l3flvCuDg6GFZ3PSvIlpuo2x9XNpNWywp7viwsSGEBWxITx18uVoB-Xu_iWjHX9rHy45OIFnId3XGTMgQyjAc6YYuBa0mrFV5nUnkqKqsE/s320/TH2.JPG" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px; position: relative;" width="320" /></a></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><br /></span><span style="color: red;"><br /></span><span style="color: red;"><br /></span><span style="line-height: 18.48px;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين26</span></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> أنظر الشكل جانبه</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">المستقيمان (ب ح)، (ج خ) متقاطعان في النقطة ا</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> ا ب = 2م، ب ح = 10م، ب ج = 0.91م</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بين أن: ح خ = 5.45م</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><br /></span><span style="color: #222222; line-height: 18.48px;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">27تمرين </span></span></span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> أنظر الشكل جانبه</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">المستقيمان (ب ح)، (ج خ) متقاطعان في النقطة ا</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ح خ = 5.45م ،ا ب = 2م، ب ح = 10م، ب ج = 0.91م</span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بيِّن إن : المستقيمان (م ج) و (ح خ) متوازيان </span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">استنتج إن : المستقيمان (م ج) و (م ح) متعامدان</span></span><br />
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<span style="background-color: white; font-size: 13.2px;"><span style="color: red; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif;"><b>الدرس</b></span></span><br />
<span style="background-color: white; color: #222222; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13.2px;"><br /></span>
<span style="color: blue; font-family: "times" , "times new roman" , serif; font-weight: 700;"><span style="color: red;">1. <span style="background-color: white; color: #222222; font-size: 13.2px; font-weight: 400;"><span style="font-size: x-small;"><b><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span style="color: red;">المتجهات </span></span></span></span></span></span></span></span></span></b></span></span></span></span><span dir="RTL" lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-weight: 700;"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span style="color: red;"> </span></span></span></span></span></span></span></span></span><br />
<b><span style="color: red;"><br /></span></b>
تقسم الكميات الفيزيائية الى نوعين:<br />
1. الكميات العددية ( القياسية ) وهي الكميات التي يلزم لتعريفها عدد حقيقي، (رقم ) ووحدة فيزيائية٠ و من الامثلة على الكميات العددية: الطول٬ درجة الحرارة، الحجم، الكتلة، الزمن، الشغل٬ الطاقة ٠٠٠<br />
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2. الكميات المتجهة و هي الكميات التي يلزم لتعريفها مقدار عددي (عدد حقيقي موجب) و وحدة فيزيائية و اتجاه و منحى٠ ولا يتم تعريفها إلا اذا اكتملت هذه العناصر٠ و من الامثلة على الكميات المتجهة: السرعة٬ التسارع٬ القوة، الإزاحة٠٠٠<br />
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كل كمية فيزيائية متجهة يمكن تمثيلها رياضياتيا بمتجه٠</div>
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فمثلاً ، إذا قلنا تحركت سيارة بسرعة 60 كم/ ساعة فقط ، فهذا لايتم المعنى ، لأن النتيجة تكون مختلفة بإختلاف المنحى أو خط الإتجاه٠</div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJm6mE6Ev5xLVLv1H0LFs0t96UnyvQPwfOLrGfRSnKEvHqZm_Vzv_YUKD42rPeKxI8IWrzVpxcxx456QbJ2ujuFfc_iXnIlJT7IPvOR4ScXol-OvDlWyWYOadrq90uTB-9Cg8asZ49tZY/s1600/FLECH2.JPG" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGIkZs6z-MJcrfZgqPbZNH5bjfRzgeO2VzjVpIixV0_g-7BtPlM-Lds3kf54LZRZxQKNzTHO2LkQl6Wk1U4azktCVR018exjmPOvwNvbcO1rITMtiV02oGMMnaNVPedZUiKnTq7E3-EB4/s1600/vec.JPG" imageanchor="1"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhGIkZs6z-MJcrfZgqPbZNH5bjfRzgeO2VzjVpIixV0_g-7BtPlM-Lds3kf54LZRZxQKNzTHO2LkQl6Wk1U4azktCVR018exjmPOvwNvbcO1rITMtiV02oGMMnaNVPedZUiKnTq7E3-EB4/s400/vec.JPG" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIYrrfFUQ-gM9AmVBEP86jiRxRpq4Xp2jxJaDzgzEDZnH-31Ib8FHVe_1bDEgQYCXzpDhWdtTRxoJs-LgnK71If4LXtOdIRC_uio4eSgt7pardIz7YMdGOfPMk9vBREW5FnnT2k1qR0oo/s1600/FLECH.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIYrrfFUQ-gM9AmVBEP86jiRxRpq4Xp2jxJaDzgzEDZnH-31Ib8FHVe_1bDEgQYCXzpDhWdtTRxoJs-LgnK71If4LXtOdIRC_uio4eSgt7pardIz7YMdGOfPMk9vBREW5FnnT2k1qR0oo/s1600/FLECH.JPG" /></a></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><br /></span></span></span></span></b></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><br /></span></span></span></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><b><br /></b></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><b>يُمثل
المتجه هندسياً بقطعة مستقيمة ، كما في الشكل المجاور ، حيث النقطة </b><b>ا </b><b>هي
نقطة البداية (التأثير) للمتجه والنقطة </b><b>ب </b><b>هي نقطة النهاية ، ويُرمز
لها</b></span></span></span><b style="background-color: white; text-align: start;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" imageanchor="1" style="font-weight: 400; margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" /></a>.</b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b style="background-color: white; text-align: start;">كما يُرمز لطول المتجه</b></span></span><b style="background-color: white; font-family: times, "times new roman", serif;"> بالرمز <b>ا ب</b>.</b><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b style="background-color: white; text-align: start;"><br /></b></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b style="background-color: white; text-align: start;"><span style="color: red;">مميزات المتجهة</span></b></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfDXQaFOoAccW77PU6O694L0x6wL0heCH4O1PwB9XsUs6m1sPwEu83ngspnV22v2kXlZVzAmdttJYn7krTsWfEGfmcM4ouVtjnPOYxIU9otBL6Nww82IfbmET2Y2FHabgvCjpO6jnGNCU/s1600/flech3.JPG" style="margin-left: 1em; margin-right: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIptd_6BWYv2ZEYl7RR-ikLFrrdDW4f7UhlX7Y6gMGnSY1jmONXIsHBkklcbzyNSSId3NRy-ilNijuJEPXo1ywBZfcu3dCfU3vPsZiHGj5AqHkM4CKIofQwIDrRjBFEEkxphpGTHMeuzw/s1600/vec.JPG" imageanchor="1"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIptd_6BWYv2ZEYl7RR-ikLFrrdDW4f7UhlX7Y6gMGnSY1jmONXIsHBkklcbzyNSSId3NRy-ilNijuJEPXo1ywBZfcu3dCfU3vPsZiHGj5AqHkM4CKIofQwIDrRjBFEEkxphpGTHMeuzw/s400/vec.JPG" /></a><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfDXQaFOoAccW77PU6O694L0x6wL0heCH4O1PwB9XsUs6m1sPwEu83ngspnV22v2kXlZVzAmdttJYn7krTsWfEGfmcM4ouVtjnPOYxIU9otBL6Nww82IfbmET2Y2FHabgvCjpO6jnGNCU/s1600/flech3.JPG" /></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>تتميز المتجهة </b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>إتجاه المتجهة </b><b> </b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" /></a><b> </b><b>هو كل مستقيم يوازي (</b><b>ا ب</b><b>)</b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>منحى المتجهة </b><b> </b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" /></a><b> هو من النقطة ا إلى النقطة</b></span></span><span style="font-family: "times" , "times new roman" , serif;"><b> ب</b></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="background-color: white;"> </span></b><b>طول</b><b> أو منظم المتجهة </b><b> </b><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" /></a><b> </b><b>هو </b></span></span><span style="font-family: "times" , "times new roman" , serif;"><b>طول المستقيم [ا ب]</b></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><br /></b></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: red;">ملاحظة</span> </b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>المتجهة المنعدمة ليس لها إتجاه</b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><br /></b></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b>تساوي متجهتين</b></span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b><br /></b></span></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>نقول عن متجهتين أنهما متساويتان إذا كان لهما</b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>نفس الإتجاه </b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>نفس المنحى </b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b>نفس المنظم</b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b><br /></b></span></span></span>
<span style="color: red;"><span style="font-size: small;"><span style="color: red;"><b>عمليات على المتجهات: مجموع متجهتين، ضرب متجهة في عدد حقيقي، الجداء السُلمي:</b></span></span><b><br /> </b></span><br />
<span style="color: red;"><b><br /> </b></span><br />
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<span style="color: red;"><b>خاصية</b></span><br />
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من أجل كل نقطة أ و نقطة ب و نقطة جـ من المستوى فإن <img border="0" height="27" src="https://www.schoolarabia.net/images/modules/physics/level_2/assaseat/vectors_new/66.gif" width="128" /><br />
<span style="color: red;"><b><br /> </b></span><br />
<span style="color: red;"><b>2. </b></span><b style="color: red; font-family: times, "times new roman", serif;"><span dir="RTL" lang="AR-SA" style="line-height: 18.4px;">التحاكي</span></b><br />
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">التحاكي ذو المركز النقطة ا والنسبة العدد الحقيقي ح هو التحويل النقطي للمستوي (التآلفي) الذي ينقل كل نقطة ب من المستوي إلى نقطة أخرى من المستوي، نرمز لها بـِ تحا(ب)٠</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">بحيث: <br />النقاط ا، ب، تحا(ب) واقعة على مستقيم واحد<br />النسبة الجبرية </span></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdOOdbQ3E8kqXIGMx9c447rFUUbU_ZqlClmljkQ4-UPeCtsxEc4BQaCqEaaZOrPaaasNZ2F-xcyGB1rOQSMHLcjVSwKocKIZw7y2HWs4gat7HhvXvSeIVs9elHvl2EXSM3TcujSaSyyJA/s1600/c3.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdOOdbQ3E8kqXIGMx9c447rFUUbU_ZqlClmljkQ4-UPeCtsxEc4BQaCqEaaZOrPaaasNZ2F-xcyGB1rOQSMHLcjVSwKocKIZw7y2HWs4gat7HhvXvSeIVs9elHvl2EXSM3TcujSaSyyJA/s1600/c3.JPG" /></a></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">= </span></span></span><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ح</span></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><br /></span></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">التحاكي ذو المركز النقطة ا والنسبة العدد الحقيقي ح هو التحويل النقطي للمستوي (التآلفي) الذي ينقل كل نقطة ب من المستوي إلى نقطة أخرى تحا(ب) من المستوي المعرفة بالعلاقة بالشعاعية </span></span></h3>
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<div class="separator" style="clear: both; font-size: medium; font-weight: 400;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPEaSIbi8wjE-Ge0tN8RozdjkSA2YSwfna5wLwR_ZnohBkckEf928UPqr6tvuQ8DVlj6OOgR8hmV0BG88RK7gkzIO_u7BDIt7F9lGom2kaveZryd0ebUVkSyZ7Q4-mPr9KYFgVAcqpp-w/s1600/c2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPEaSIbi8wjE-Ge0tN8RozdjkSA2YSwfna5wLwR_ZnohBkckEf928UPqr6tvuQ8DVlj6OOgR8hmV0BG88RK7gkzIO_u7BDIt7F9lGom2kaveZryd0ebUVkSyZ7Q4-mPr9KYFgVAcqpp-w/s1600/c2.JPG" /></a>= <span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 18.4px;">ح</span></span></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6o4eU-ItP7jhbnUNb0onQt_c1tpG3_yjn-zlqFx2pnTYkA1tSn9tsOlWR1HZSfvf7tljTkK6Rk1HgpgMDhCOYr4tHuuLgyl5L97vRbcu6A_Qw-rAxQCZFw1J6R1rRATM-9BF2hFQR3HE/s1600/c1.JPG" /></a></span></span></div>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">ملاحظة</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">التحاكي تحويل (تآلفي) نقطي للمستوي (التآلفي) ينقل كل نقطة من المستوي إلى نقطة أخرى من المستوي، بحيث </span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">- يحافظ على التوازي٠</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">- العلاقة الخطية بين النقاط، أي أن أي ثلاث نقاط واقعة على مستقيم واحد تبقى واقعة على مستقيم بعد التحويل٠</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="font-weight: 400;"><span dir="RTL" lang="AR-SA" style="font-family: "times" , "times new roman" , serif; font-size: small; line-height: 115%;"></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">- النسب على طول المستقيم، أي من أجل ثلاث نقاط ا، ب، ج، تقع على مستقيم، فإن النسبة |ا ء ب| \ |ب ء ج| تكون محققة قبل وبعد التحويل٠</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"></span></span></span></span></span></h3>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">ملاحظة</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">التحاكي هو عملية نقل تآلفي و يمكن اعتبارها كنقل خطي في البنية الإتجاهية ذات المركز النقطة ا٠ .</span></span></span></span></span></h3>
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<span style="color: red; font-family: "times" , "times new roman" , serif; font-size: small;"><span style="font-weight: 400;">ملاحظة</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">التحاكي هو عملية نقل تشابه حيث تضاعف كل المسافات بـ |ح| </span></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">وكل المساحات بـ </span></span></span></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: xx-small;"><span style="font-family: "times" , "times new roman" , serif;">ح</span><sup style="line-height: 27.6px;">2</sup></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">يسمى العدد |ح| نسبة التكبير٠ </span></span></span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b>مبرهنة طاليس المباشرة :</b></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><br /></b></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><span dir="LTR"> (م<sub>1</sub>)</span>و <span dir="LTR">(م<sub>2</sub>)</span> مستقيمان متقاطعان في نقطة ا</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><span dir="LTR">ب</span> </span><span style="color: black;"><span style="color: black;"><span style="color: black;"> و<span dir="LTR">ت </span> نقطتان من المستقيم </span></span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span dir="LTR"><span style="color: black;"><span dir="LTR">(م<sub>1</sub>)</span></span></span></span></span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ج و <span dir="LTR">ح</span> نقطتان من المستقيم <span dir="LTR">(م<sub>2</sub>)</span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">بحيث : <span dir="LTR">(ب ج) // (ت ح)</span> .</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><br /></span></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">الحالات الثلاث لإنشاء الشكل الهندسي</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnNzDxIzzOqEgHiW-8xENvIWqpLXKBNuf2PCcDvUmaiQBFRDOWnDkamaYIif5uETcT2Bl1j_19IomZaD48pfXEjRPOFGIB2sprbpmSTAbc1MAKOe62gDi8rYq3FD1pXm39EtxkZYiuyCU/s1600/THALES.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="144" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnNzDxIzzOqEgHiW-8xENvIWqpLXKBNuf2PCcDvUmaiQBFRDOWnDkamaYIif5uETcT2Bl1j_19IomZaD48pfXEjRPOFGIB2sprbpmSTAbc1MAKOe62gDi8rYq3FD1pXm39EtxkZYiuyCU/s320/THALES.JPG" width="320" /></a></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><span style="font-weight: normal;"><span style="font-weight: normal;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> </span></span></span></span></span></span></h3>
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<span style="font-size: small;"><span style="color: red;">التحاكي</span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><b> </b></span></span></span><span style="color: red;">و مبرهنة طاليس المباشرة والعكسية</span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ليكن </span><span dir="LTR"> (م<sub>1</sub>)</span>و <span dir="LTR">(م<sub>2</sub>)</span> مستقيمان متقاطعان في نقطة ا</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><span dir="LTR">ب</span> </span><span style="color: black;"><span style="color: black;"><span style="color: black;"> و<span dir="LTR">ت </span> نقطتان من المستقيم </span></span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span dir="LTR"><span style="color: black;"><span dir="LTR">(م<sub>1</sub>)</span></span></span></span></span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ج و <span dir="LTR">ح</span> نقطتان من المستقيم <span dir="LTR">(م<sub>2</sub>)</span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;">إذا كان : <span dir="LTR">(ب ج) // (ت ح)</span> <span style="color: black;">فإن :</span><span style="text-align: center;"> </span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWB8oWyCzxXZgEuCp5y-SLwOG598tuj3CVzjbAjuaJO-2e5k6zylfFv5ulASISdwKxQJur70FuU3S3TInR8uzkNCFKmw0E6DOk74QH1_SQdj49KY1lTLvo7lMzZBx4DO2OVgq0y-p0BRQ/s1600/c4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWB8oWyCzxXZgEuCp5y-SLwOG598tuj3CVzjbAjuaJO-2e5k6zylfFv5ulASISdwKxQJur70FuU3S3TInR8uzkNCFKmw0E6DOk74QH1_SQdj49KY1lTLvo7lMzZBx4DO2OVgq0y-p0BRQ/s1600/c4.JPG" /></a></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b><span style="color: black;">و</span></b><span style="color: black;"> إذا كان</span></span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBTi7_cx6JRrQdj-o0ZNirQdtmdoUWjxtflzha03HJitk3BKj2wixxIonwPgsFvpZthITV8QcHYPIP2FRNxuYZlGwI7RTlvXSf1RbHroNZqjiOqIoixYKHZnpDxI6ouWUWYbwlC5b3u8/s1600/c4.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhmBTi7_cx6JRrQdj-o0ZNirQdtmdoUWjxtflzha03HJitk3BKj2wixxIonwPgsFvpZthITV8QcHYPIP2FRNxuYZlGwI7RTlvXSf1RbHroNZqjiOqIoixYKHZnpDxI6ouWUWYbwlC5b3u8/s1600/c4.JPG" /></a></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"> <span style="color: black;">فإن : </span><span dir="LTR">(ب ج) // (ت ح)</span> </span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b> </b></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b>خاصية </b>(<b>مبرهنة طاليس المباشرة</b>) <b>:</b></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ليكن </span><span dir="LTR"> (م<sub>1</sub>)</span>و <span dir="LTR">(م<sub>2</sub>)</span> مستقيمان متقاطعان في نقطة ا</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><span dir="LTR">ب</span> </span><span style="color: black;"><span style="color: black;"><span style="color: black;"> و<span dir="LTR">ت </span> نقطتان من المستقيم </span></span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span dir="LTR"><span style="color: black;"><span dir="LTR">(م<sub>1</sub>)</span></span></span></span></span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ج و <span dir="LTR">ح</span> نقطتان من المستقيم <span dir="LTR">(م<sub>2</sub>)</span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;">إذا كان : <span dir="LTR">(ب ج) // (ت ح)</span> </span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">فإن :</span><span style="text-align: center;"> </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq7T202X7FTEwawbmW0zzyeU2L9oX2wE2EOyOCfLv_9agJKxl0thJviC_av8eNn7lxacD9LGXpGeYnLcKvEYNNxCKSKzzW7F21v1W4uVsjP68QSAUFSbVBgcuVTggyVyB7fiX35LKfyCY/s1600/THALES2.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgq7T202X7FTEwawbmW0zzyeU2L9oX2wE2EOyOCfLv_9agJKxl0thJviC_av8eNn7lxacD9LGXpGeYnLcKvEYNNxCKSKzzW7F21v1W4uVsjP68QSAUFSbVBgcuVTggyVyB7fiX35LKfyCY/s1600/THALES2.JPG" /></a></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"> </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b>مبرهنة طاليس العكسية:</b></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><b>خاصية</b> (<b>مبرهنة طاليس العكسية)</b> <b>:</b></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;">ليكن <span dir="LTR"> (م<sub>1</sub>)</span>و <span dir="LTR">(م<sub>2</sub>)</span> مستقيمان متقاطعان في نقطة ا</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;"><span dir="LTR">ب</span> </span><span style="color: black;"><span style="color: black;"><span style="color: black;"> و<span dir="LTR">ت </span> نقطتان من المستقيم </span></span></span><span style="color: black;"><span style="color: black;"><span style="color: black;"><span dir="LTR"><span style="color: black;"><span dir="LTR">(م<sub>1</sub>)</span></span></span></span></span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">ج و <span dir="LTR">ح</span> نقطتان من المستقيم <span dir="LTR">(م<sub>2</sub>)</span> تختلفان عن <span dir="LTR">ا</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">إذا كانت النقط <span dir="LTR">ا</span> و <span dir="LTR">ب</span> و <span dir="LTR">ت</span> و النقط <span dir="LTR">ا</span> و <span dir="LTR">ج</span> و <span dir="LTR">ح</span> في نفس الترتيب</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: black;">تحقق : </span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSCgBn0_yfz-B7JjgBsSSV6v7MR1PIqkkEcSwGi3bA45HcvJYNj86mtFJOV1hvn2CbvEDIs54ETgtniUee_IvITT5PV24v0gsBPf2qzt_8osXL_Jt53tLfkzosgGshnzOPw5GDAdsfG20/s1600/THQLES3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSCgBn0_yfz-B7JjgBsSSV6v7MR1PIqkkEcSwGi3bA45HcvJYNj86mtFJOV1hvn2CbvEDIs54ETgtniUee_IvITT5PV24v0gsBPf2qzt_8osXL_Jt53tLfkzosgGshnzOPw5GDAdsfG20/s1600/THQLES3.JPG" /></a><span style="color: black;"> فإن : <span dir="LTR">(ب ج) // (ت ح)</span>.</span></span></span><br />
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<span style="color: red;"><b>3. مساوي القياس</b></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;"><span lang="AR-SA"> </span></span><b><span style="color: red;"><span lang="AR-SA">تعريف</span></span></b><span style="color: red;"><span lang="AR-SA">:</span></span><span dir="RTL" lang="AR-SA"> مساوي القياس، في الهندسة، هو بشكل عام تماثل بين الاشكال. أي حركة جامدة، في المستوى أو في الفراغ ، لا تشوه بأي شكل من الأشكال</span></span></span><span dir="RTL" lang="AR-SA">٠</span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">الإنعكاس</span></span></span></span></b><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">:</span></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">الإنعكاس بالمستقيم (م) هو تحويل أو نسخ، يُحوّل كل نقطة في المستوي إلى " صورتها</span><span lang="AR-SA"> في المرآة " بالنسبة للمستقيم (م)</span></span></span><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">٠</span></span></span> </span></span></span></span></span><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">المستقيم (م) يُسمى خط الإنعكاس٠</span></span></span><br /><br /><br />لإيجاد الانعكاس بالمستقيم (م) لنقطة ما، يتم إسقاط خط عمودي على الخط (م) المستعمل كمحور الانعكاس ، ثم مد الخط بشكل مستقيم في الجهة الأخرى من المحور وبنفس المسافة٠<br />ولتحديد الانعكاس بالمستقيم (م) لرسم ما، يتم تحديد انعكاسات كل النقاط المؤلفة له على الناحية الأخرى من محور الانعكاس المستقيم (م)٠</span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span style="color: red;"><span style="color: black;"><a href="https://sites.google.com/site/taghredmath/Reflection/an3ekas2.jpg?attredirects=0"><img border="0" height="150" src="https://sites.google.com/site/taghredmath/_/rsrc/1333310672642/Reflection/an3ekas2.jpg?height=240&width=320" width="200" /></a></span></span></span></span></span></span></span></b></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /><b><span style="color: red;">التماثل</span><span style="color: red;"><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">:</span></span></span></span></span> </b>محور التماثل وهو المستقيم الذي يقسم الشكل إلى جزئين متماثلين، مثال على ذلك</span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: black;"><a href="https://sites.google.com/site/taghredmath/Reflection/face.jpg?attredirects=0"><img border="0" height="80" src="https://sites.google.com/site/taghredmath/_/rsrc/1333310986512/Reflection/face.jpg?height=129&width=320" width="200" /></a></span></b></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: black;"><span style="line-height: 27.6px;"> </span></span></b></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><span style="color: black;"><span style="line-height: 27.6px;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">وقد يكون للشكل أكثر من محور تماثل واحد، مثال على ذلك: للدائرة عدد لايحصى من محاور التماثل ( وهي أقطارها) لأن كل واحد منها يقسم الدائرة إلى جزئين متماثلين</span></span></span><span style="color: black;"><span style="line-height: 27.6px;"><span style="-moz-font-feature-settings: normal; -moz-font-language-override: normal; font-size-adjust: none; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;">٠</span></span></span></b></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><b><br /></b><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA" style="line-height: 18.4px;"><b>الإنزلاق</b></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><b><span style="color: red;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA">:</span></span></span></span></b><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span lang="AR-SA"><span style="color: red;"><span style="color: black;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;"><span lang="AR-SA" style="line-height: 27.6px;">الإزاحة أو الإنزلاق هي تحريك كل نقاط المستوي في اتجاه معين، وبمقدار طولي معين، اتجاه الإزاحة وبُعد الإزاحة يحددان بحسب سهم يسهى سهم الإزاحة. يمكن وصف سهم الإزاحة على انه " أمر" ننفذ الإزاحة حسبه٠</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /></span></span>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;">الإزاحة تنسخ كل نقاط المستوي على النحو الآتي:<br />1- كل نقاط المستوي تُنسخ بنفس الاتجاه٠<br />2- كل النقاط تتحرك بنفس المقدار٠</span></span></span></span></span></span></span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;"><br /></span></span></span></span></span></span></span></span></span>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;"><span lang="AR-SA" style="line-height: 27.6px;">أمثلة في إزاحة الشكل:</span></span></span></span></span></span></span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;"><br /></span></span></span></span></span></span></span></span></span>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA"><span style="color: #741b47;"><span style="color: black;"><a href="https://sites.google.com/site/taghredmath/Remove/ezaha.jpg?attredirects=0"><img border="0" height="86" src="https://sites.google.com/site/taghredmath/_/rsrc/1333310383304/Remove/ezaha.jpg?height=108&width=400" width="320" /></a></span></span></span></span></span></span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 27.6px;">(كل الأبعاد بين النقاط الأصلية والنقاط الناتجة بعد الإزاحة هي أبعاد متساوية )</span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><br /><span lang="AR-SA" style="line-height: 27.6px;"><span style="color: black;"><span style="color: red;"><b>الدوران</b></span></span></span><span lang="AR-SA" style="line-height: 27.6px;"><span style="color: black;"><b><span style="color: red;"><span lang="AR-SA" style="line-height: 27.6px;">:</span></span></b></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 27.6px;">الدوران: يعني دوران شكل باتجاه معين ( مع أو ضد عقارب الساعة)، حول نقطة معينة (هي مركز الدوران)، بزاوية معينة ( هي مقدار هذا الدوران).<br /> </span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 27.6px;">يحافظ الدوران على شكل الجسم الذي نقوم بتدويره وعلى حجمه. والشكل الناتج من الدوران مطابق تماما للشكل قبل الدوران.<br /> </span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 27.6px;">تحويل الدوران يُدير كل المستوي حول نقطة معينة وبزاوية معينة، كل نقاط المستوي تدور حول نفس النقطة وبنفس الزاوية، لذا عند وصف الدوران لا بد من ذكر زاويته ومركزه.<br /> </span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 27.6px;">يمكن تمييز التحويل الدوراني بأمرين:<br />1. نقطة دوران.<br />2. زاوية دوران.</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span style="color: red;">تمرين</span><span style="color: red; line-height: 18.48px; text-align: justify;"><b> </b></span><b style="color: #222222; line-height: 18.48px; text-align: justify;"><b><span style="color: red;"><span style="color: mediumblue;"><span style="color: red;">(</span></span>تعميم نظرية </span></b></b><b style="color: #222222; line-height: 18.48px; text-align: justify;"><b><span style="color: red;"><b>طاليس</b>):</span></b></b><span style="color: red; line-height: 18.48px; text-align: justify;"> </span></span></span></h3>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;">في الشكل التالي المستقيمان بالأسود يقطعان الثلاثة مستقيمات بالأحمر المتوازية<span style="color: black; line-height: normal;">٠</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;">قارن نسب الأطوال<span style="color: black; line-height: normal;">٠</span><span style="line-height: 18.48px;"> </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 115%;"><a class="image" href="http://ar.wikipedia.org/wiki/%D8%B5%D9%88%D8%B1%D8%A9:Thales_theorem_3.png" style="background-color: white; color: #888888; line-height: 18.48px; text-decoration: none;" title="Thales theorem 3.png"><img alt="" border="0" height="197" src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Thales_theorem_3.png/300px-Thales_theorem_3.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.0980392) 1px 1px 5px; background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.0980392) 1px 1px 5px; padding: 5px; position: relative;" width="200" /></a><span style="color: black;"> </span></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 115%;"><span style="color: black;"><br /></span></span></span></span>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: small;"><span lang="AR-SA" style="line-height: 115%;"><span style="line-height: normal; text-align: justify;">alfarjimohammed@gmail.com</span></span></span></span></div>
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محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.com0tag:blogger.com,1999:blog-162104398430917066.post-75439195370909688122012-12-09T11:23:00.001-08:002019-10-09T03:03:39.685-07:00 الحساب الحرفي، المعادلات، المتراجحات ، معادلات مترابطة<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><b>تمارين</b></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;"> </span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">6</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA"><span style="line-height: 15.18px;">بسط االعبارات التالية :</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"><span style="color: black;">ش = 15س – 5 ص+ 7 – س + 10</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">س – 10 – (7س – 13) = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">س – 10 + (7س – 13) = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"></span></span><br />
<div style="line-height: normal;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"><span style="text-align: justify;"><span lang="ar-sa" style="line-height: 20.7px;">انشر العبارات التالية</span></span></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"></span></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">7(5س – 2) = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">(س – 13)(2 س – 3) = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"><span style="line-height: 15.18px;"><br /></span></span></span></span></div>
</div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">عمل العبارات التالية </span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 15.18px;">نتعرف على العامل المشترك ثم نعمل</span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 6ش </span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">15س – 2 = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;">7(15س – 21) = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"><span style="line-height: 15.18px;">4(3 س – 1) – (7س + 13)(3 س – 1)</span> = </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"></span></span></span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0cm;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"><span style="line-height: 15.18px;">4(6 س – 2) – (7س + 13)(3 س – 1)</span> =</span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 15.18px;"></span></span></span></div>
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<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA">حل المعادلة </span><span lang="AR-SA"> 4(3</span><span style="line-height: 15.18px;"> س – 1) – (7س + 13)(3 س – 1)</span><span style="line-height: 15.18px;"> =</span><span lang="AR-SA"><span lang="AR-SA"> 0</span> و تحقق من صحة الحل </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="background-color: transparent; line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><br /></span></span><span style="background-color: transparent; line-height: 20.7px; text-align: justify;"><span lang="ar-sa"> </span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">7</span></span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA">حل المعادلة </span><span lang="AR-SA"> </span><span lang="AR-SA"><span lang="AR-SA">5</span></span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA">س – 2 = 8</span> </span> و تحقق من صحة الحل </span></span></span></div>
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"></span></span><br />
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<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA">حل المعادلة 3</span><span lang="AR-SA">س </span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"> </span></span></span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA">– </span></span><span lang="AR-SA"><span lang="AR-SA">2</span></span></span></span></span></span><span lang="AR-SA"> </span><span lang="AR-SA"> </span><span lang="AR-SA"> = 5س + 4 </span><span dir="LTR" lang="AR-SA"> </span><span lang="AR-SA">و تحقق من صحة الحل </span></div>
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<div class="MsoNormal" dir="RTL" style="direction: rtl; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0cm 0cm 0cm 6.75pt; unicode-bidi: embed; widows: 1;">
<div style="margin: 0px;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA">حل المعادلة </span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA">(3س</span><span lang="AR-SA"><span lang="AR-SA"> </span></span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"> </span></span></span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA">– </span></span><span lang="AR-SA"><span lang="AR-SA">2)</span></span></span></span></span></span></span></span><span lang="AR-SA"> </span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA"><span style="font-size: xx-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;">÷</span></span></span> 3</span><span lang="AR-SA"> </span><span lang="AR-SA"> = (5س + 4) </span></span></span><span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA"><span style="font-size: xx-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;">÷</span></span></span> 2 </span><span dir="LTR" lang="AR-SA"> </span><span lang="AR-SA">و تحقق من صحة الحل </span></span></span></div>
</div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><br /></span></span><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;"> </span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">8</span></span></span></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA"> حل المعادلات التالية</span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA" style="line-height: 10.35px;">:</span></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> 0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif;"> ش</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86" style="font-family: "times" , "times new roman" , serif;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span><span style="font-family: "times" , "times new roman" , serif;">+ 2ش</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> </span><span style="font-family: "times" , "times new roman" , serif;">+</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> </span><span style="font-family: "times" , "times new roman" , serif;">1 </span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"> ش<sup style="line-height: 27.6px;">2</sup><span lang="AR-SA"> </span></span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"><span lang="AR-SA">−</span><span lang="AR-SA"> </span>1 </span></div>
</div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: purple;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"></span></span></span><br />
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"> ش<sup style="line-height: 27.6px;">2</sup><span lang="AR-SA"> </span></span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"><span lang="AR-SA">−</span><span lang="AR-SA"> 5</span> </span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 6ش<span lang="AR-SA"> =</span> 0 </span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> </span></span><br />
<span style="font-family: times, "times new roman", serif; font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">9</span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بكاتبة العبارة د = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 6ش<span lang="AR-SA"> </span></span><span style="font-size: x-small;">− 7<span lang="AR-SA"> </span> على الشكل التالي</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">د = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 3<span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA">×</span></span></span></span></span>2ش<span lang="AR-SA"> </span>+ 9 </span><span style="font-size: x-small;">− 9<span lang="AR-SA"> </span></span><span style="font-size: x-small;"><span lang="AR-SA">−</span> 7</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 27.6px;">استنتج أن</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">د<span dir="RTL" lang="AR-SA"> =</span> (ش + <span dir="RTL" lang="AR-SA">3)</span><sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>- 16</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA">تم عمل وحل المعادلة</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 6ش<span lang="AR-SA"> </span></span><span style="font-size: x-small;"><span lang="AR-SA">−</span> 7</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">10</span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">بكاتبة العبارة د = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 10ش<span lang="AR-SA"> </span></span><span style="font-size: x-small;">− 7<span lang="AR-SA"> </span> على الشكل التالي</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">د = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 5<span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA">×</span></span></span></span></span>2ش<span lang="AR-SA"> </span>+ 25 </span><span style="font-size: x-small;">− 25<span lang="AR-SA"> </span></span><span style="font-size: x-small;">− 7</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 27.6px;"> استنتج أن</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">د<span dir="RTL" lang="AR-SA"> =</span> (ش + <span dir="RTL" lang="AR-SA">5)</span><sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span></span><span style="font-size: x-small;">− 32</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA">تم عمل وحل المعادلة</span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 = ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 10ش<span lang="AR-SA"> </span></span><span style="font-size: x-small;"><span lang="AR-SA">−</span> 7</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> 0 = استنتج حل المعادلة 5ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 50ش<span lang="AR-SA"> </span></span><span style="font-size: x-small;"><span lang="AR-SA">−</span><span lang="AR-SA"> </span>35</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="color: red;"><br /></span></span></span>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;"> </span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">11</span></span></span></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="AR-SA"> حل المعادلات التالية</span><span lang="AR-SA"><span lang="AR-SA"><span lang="AR-SA"><span dir="RTL" lang="AR-SA" style="line-height: 10.35px;">:</span></span></span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> 0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif;"> ش</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86" style="font-family: "times" , "times new roman" , serif;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span><span style="font-family: "times" , "times new roman" , serif;">+ 2ش</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> +</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> 10</span><span style="font-family: "times" , "times new roman" , serif;"> </span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"> ش<sup style="line-height: 27.6px;">2</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+ 2ش<span lang="AR-SA"> </span></span><span style="font-family: "times" , "times new roman" , serif; font-size: x-small;"><span lang="AR-SA">−</span><span lang="AR-SA"> </span>1 </span></div>
</div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">0 </span></span><span style="font-family: "times" , "times new roman" , serif;">=</span><span style="font-family: "times" , "times new roman" , serif;"> ش</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">4</sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86" style="font-family: "times" , "times new roman" , serif;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span><span style="font-family: "times" , "times new roman" , serif;">+ 2ش</span><sup style="font-family: times, "times new roman", serif; line-height: 27.6px;">2</sup><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> +</span><span lang="AR-SA" style="font-family: "times" , "times new roman" , serif;"> 10</span><span style="font-family: "times" , "times new roman" , serif;"> </span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></div>
<div style="color: black; font-family: "Times New Roman"; font-size: medium; font-variant-east-asian: normal; font-variant-numeric: normal; line-height: normal; margin: 0px; widows: 1;">
<span style="color: #222222; font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="color: #222222; font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">12</span></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> حل المتراجحات التالية</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> ش < 5</span>−</span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> ش < 2ش + 2</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> ش </span><span style="font-size: x-small;">−<span lang="AR-SA"> </span>1 > 2ش + 2</span></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></div>
<span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small; text-align: center;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">13</span></span></span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> حل المتراجحات التالية</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> س<sup>2</sup> <span style="line-height: 15.18px;"><span style="line-height: 15.18px;">(3 س – 1 )</span></span> < 0</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> <span style="line-height: 15.18px;"><span style="line-height: 15.18px;">س + 1)(3 س – 1 )</span></span> < 0)</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;"><br /></span></span></span><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">14</span></span></span></span></span></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">حل النظمة التالية بالطرق الثلاثة</span></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ـ2 س + 3 ش = – 1</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">ـ3 س – 5 ش = 2</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><br /></span></b></span></div>
<div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طريقة التعويض. مـن إحـدى الـمعادلتين؛ نجد قيمة أحـد المجهولين بدلالة الآخـر؛ ثم نعوضه</span></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">في المعادلة الأخرى٠</span></span></div>
</div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><br /></span></b></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"> طريقة التآلفية الخطية. لـكي نـحـتفظ بـأحـد الـمجـهوليـن ( لـكي نتمكن من حساب قيمته) نـضرب المعادلة من معادلتي النظمة في معامل مناسب لنحصل على معاملين متقابلين بالنسبة للمجهول الآخر؛ ثم نجمع المعادلتين الـمحصل عليهما طرفا بطرف٠</span></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><br /></span></span></div>
<div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">طريقة الحل المبياني (هندسية). ننشئ المستقيمين في معلم متعامد ممنظم؛</span></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;">المعادلة من الدرجة الأولى بمجهولين لها عدد لا نهائي، يمثلها خط مستقيم في المستوى٠</span></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><br /></span><span style="font-family: "times" , "times new roman" , serif;"><br /></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين15</span></span></span></span></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;"></span></span></span></span><span style="font-size: x-small;">لتكن س٬ </span><span style="font-size: x-small;">ص٬ عددين حقيقيين بحيث ص < س<br />قارن العددين س </span><span style="font-size: x-small;">−</span><span style="font-size: x-small;"> 10 و ص </span><span style="font-size: x-small;">−</span><span style="font-size: x-small;"> 10<br />قارن العددين </span><span style="font-size: x-small;"><span lang="AR-JO">− </span>س</span><span style="font-size: x-small;"> و </span><span style="font-size: x-small;">−</span><span style="font-size: x-small;"> ص</span><b><span style="font-size: x-small;"></span></b></span><br />
<div>
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"> <span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;"> </span></span></span></span></b></span><br />
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">تمرين</span></span></span></span><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;">16</span></span></span></span></span></div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: x-small;"><span lang="ar-sa">لتكن س٬ ص٬ عددين حقيقيين بحيث </span></span><span style="font-size: x-small;"><span lang="ar-sa"><span lang="ar-sa">س < 1</span></span></span><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: red;"><span style="color: black;"> ،</span></span> </span></span><span style="font-size: x-small;"><span lang="ar-sa"><span lang="ar-sa"> ص</span> < 2</span><span lang="ar-sa"></span></span><b><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;"><br /></span></span></span></span></b></span></div>
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;"><span lang="ar-sa"><span style="color: #bb8081;"><span style="color: red;"> </span></span></span></span></span></span></span><span style="font-size: x-small;"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86">بيٌِن أن </span></span></span><span style="font-size: x-small;"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86">2 س + 3 </span></span></span><span style="font-size: x-small;"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span lang="ar-sa"><span lang="ar-sa">ص</span></span> </span></span></span></b></span><b><span style="font-size: x-small;"><span style="font-weight: normal;"><span lang="ar-sa"><span lang="ar-sa"><</span></span> 8</span></span></b><br />
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><br /></span></b></span></div>
</div>
<div>
<span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-size: x-small;"><br /></span></b></span></div>
</div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><span style="color: purple;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>الدرس</b></span></span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><span style="color: purple;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b><br /></b></span></span></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: small;"><b style="color: purple;"><span style="color: red; font-family: "courier new" , "courier" , monospace;">1) الحساب الحرفي</span></b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: small;"><span style="color: red;"><b><br /></b></span></span><span style="font-size: small;">
<span style="font-family: "courier new" , "courier" , monospace;"><b><span style="line-height: 115%;"><span style="color: red;"><span lang="AR-SA"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">عبارة جبرية</span>، تعريف
وأمثلة</span></span></span></b></span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span><span style="font-size: small;">
<span style="font-family: "courier new" , "courier" , monospace;"><b><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="color: red;">أمثلة:</span></span></b></span></span></span><br />
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ک = 3س – 4 ٠</span><span lang="AR-SA" style="line-height: 115%;"> </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ش</span> = س – 5</span><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> </span><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ص</span> – 7س + 13</span></span></div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span style="color: red;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;"> </span></span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تبسيط عبارة
جبرية</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تعريف:</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="color: red; font-family: "courier new" , "courier" , monospace; line-height: 115%;"><span style="color: black;">تبسيط عبارة جبرية
يعني كتابتها بأقل ما يمكن من الحدود٠</span> </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">مثل:</span></b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="color: red; font-family: "courier new" , "courier" , monospace; line-height: 115%;"><span style="color: black;">بسط العبارة: ش = 10س – 5 ص+ 7 – س + 13</span><span style="color: black;"><br />بعد التبسيط <br />ش = 9س – 5 ص + 20 </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">خاصية حذف
الاقواس التي تسبقهما اشارة ناقص أو زائد:</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">إشارة ناقص أو زائد التي تسبق
القوسين ليست هي إشارة الحد الاول الموجود داخل القوس وانما هي إشارة المجموع او
الفرق المكتوب داخل القوس ككل ٠</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">خاصية</span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">في عبارة جبرية يمكن حذف القوسين المسبوقين
بالإشارة زائد و ذلك دون ان تغيروا إشارة الحدود الموجودة بين القوسين٠</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">في عبارة جبرية يمكن حذف القوسين
المسبوقين بالإشارة ناقص مع تغير إشارة كل حد موجود بين القوسين٠</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">عند تبسيط عبارة جبرية يمكن
استعمال خاصية توزيع الضرب على الجمع و على الطرح كما يمكن استعمال قاعدتي حذف
الاقواس٠</span><span lang="EN-US" style="line-height: 115%;"><br />
</span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;"> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">أ</span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">مثلة</span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></b></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">س – 1 – (7س – 13) = س – 1 – 7س
+ 13
=
–6س + 12</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">س – 1 + (7س – 13) = س – 1 + 7س
+ 13 = 6س – 14</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">نشر عبارة
جبرية</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">تعريف</span><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></b><span style="line-height: 115%;"> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">نشر جداء هو كتابته على شكل مجموع أو فرق</span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">خاصية توزيع
الضرب على الجمع والطرح:</span></b><span style="line-height: 115%;"><br />
</span><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ا٬ ب٬ س٬ ش٬ أعداد
حقيقية</span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ا(س + ش)
= ا س + ا ش</span><span lang="AR-SA" style="line-height: 115%;"> </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ا(س – ش)
= ا س – ا ش</span><span lang="AR-SA" style="line-height: 115%;"> </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">(ا + ب)(س
+ ش) = ا س + ا ش + ب س + ب ش</span><span lang="EN-US" style="line-height: 115%;"> </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span lang="EN-US" style="line-height: 115%;">(</span><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ا + ب)(س – ش) = ا س – ا ش + ب س – ب ش</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;"><span lang="EN-US" style="line-height: 115%;">(</span>ا – ب)(س
+ ش) = ا س + ا ش – ب س – ب ش</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;"><span lang="EN-US" style="line-height: 115%;">(</span>ا – ب)(س
– ش) = ا س – ا ش – ب س + ب ش</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">أ</span><span dir="RTL" lang="AR-SA" style="line-height: 115%;">مثلة</span></span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></b><span lang="AR-SA" style="line-height: 115%;"> </span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">7(3س – 5 ) = 21س – 35</span></span></span></div>
</div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="text-align: center;">(</span>ش – 13)(3 س – 5 ) = 3 </span></span><span style="text-align: center;">ش </span><span dir="RTL" lang="AR-SA" style="line-height: 115%; text-align: center;">س – 5<span style="text-align: -webkit-right;">ش</span> –
39س + 65 </span><span style="line-height: 115%; text-align: center;"> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><b><span dir="RTL" lang="AR-SA" style="color: red;">التعميل</span></b><br />
<span dir="RTL" lang="AR-SA" style="color: red;"> </span></span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="color: red;">تعريف</span></span></b><span style="line-height: 115%;"><b><span dir="RTL" lang="AR-SA" style="color: red;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></b><span dir="RTL" lang="AR-SA"> </span><span dir="RTL" lang="AR-SA"> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">تعميل مجموع أو فرق هو كتابته على شكل جداء</span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><br /></b></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">التعميل بِواسطة
العامل المشترك</span></b></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">عند</span><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> </span><span style="line-height: 115%;">تعميل
عبارة جبرية نتعرف على العامل المشترك ثم نعمل<br />
<span dir="RTL" lang="AR-SA" style="color: red;"> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="color: red;">أمثلة</span></span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></b></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA">ک = </span></span><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">12س – 20</span><span style="line-height: 115%;"> </span> </span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">نتعرف على العامل المشترك ثم نعمل</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">العدد 4 هو العامل المشترك</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;">ک = 4 × 3س – 4 × 5 = 4(3 س – 5 )٠</span><span lang="AR-SA" style="line-height: 115%;"> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><br /></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;">ک = 30(3 س – 5 ) – (7س + 13)(3 س – 5 )</span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; line-height: 115%;"> = (3 س – 5 ) [30 – (7س + 13)] </span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;"> = (3 س – 5 ) [30 – 7س – 13] </span></span></span></div>
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"> = (3 س – 5 ) (7س – 17)٠</span><span style="line-height: 115%;"> <br style="mso-special-character: line-break;" />
<br style="mso-special-character: line-break;" />
</span></span></div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><b><span style="color: red;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">المتطابقات الهامة</span></span></b></span><span style="line-height: 115%;"><span dir="RTL" lang="AR-SA" style="line-height: 115%;"><span style="font-family: "courier new" , "courier" , monospace;"><b> </b>(يتبع)</span> </span></span></span></div>
<div style="line-height: normal;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b><br />2) المعادلات من الدرجة الاولى:</b></span></span></div>
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">معادلة، تعريف
وأمثلة:</span></span></b></span></span></div>
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>تعريف:</b> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المعادلة في الرياضيات، هي عبارة رياضياتية مؤلفة من رموز رياضياتية، تنص على مساواة تعبيرين رياضيتين بِمتغيرات جبرية، ويعبر عن هذه المساواة عن طريق إشارة التساوي٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المجاهيل هي اعداد غير معلومة يرمز لها بـ (ش، س، ص ،ع ،ل...الخ)٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ونقول أن المعادلة من الدرجة الأولى إذا كانت أعلى قوة لِ المجاهيل تظهر في المعادلة هي واحد٠
</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">أمثلة</span></span><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></span></b><span style="mso-list: Ignore;"> </span></span></span></span></span></div>
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace;">3س = 15 </span></div>
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace;">س <span style="font-family: "times" , "times new roman" , serif;">−</span> 1 = 11 </span><br />
<span style="font-family: "courier new" , "courier" , monospace;">س <span style="font-family: "times" , "times new roman" , serif;"><span style="font-size: xx-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;">÷</span> </span></span></span> 3 = 11</span><br />
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">معادلات
فى مجهول واحد أو معادلات فى متغير واحد س٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س تعنى
ان العدد 3 مضروب فى الرمز س٠</span></div>
<div class="MsoNormal" dir="RTL" style="direction: rtl; line-height: normal; margin-bottom: 0cm; unicode-bidi: embed;">
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">حل
معادلة</span></span><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></span></b></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">تعريف</span></span><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></span></b><span lang="AR-SA"><span style="mso-spacerun: yes;"> </span></span><span dir="RTL" lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"> </span></span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span dir="RTL" lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">حل المعادلة يعنى إيجاد المجاهيل التى تحقق المعادلة، ندعو هذه القيم
جذور المعادلة٠</span><span style="mso-spacerun: yes;"> </span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span style="font-family: "courier new" , "courier" , monospace; mso-tab-count: 1;"> </span></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">أمثلة</span></span><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></span><span style="mso-list: Ignore;"><span style="font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"> </span></span></b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">نعتبر المعادلة
التالية: 3س = 15</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;">حل
المعادلة 3س = 15 تعني ما هو العدد المجهول الذي اذا ضربناه فى العدد 3 كان الناتج هو العدد 15</span><span style="text-align: center;">؟</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المعادلة 3س = 15 معادلة غير صحيحة من أجل معظم القيم التي
يمكن أن تعطى لِ س، لكنها
تكون صحيحة فقط في س= 3، إذن
جذر المعادلة أو حل المعادلة
هو العدد٠3 </span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">نعتبر المعادلة
التالية: س + 4 = 7</span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;">حل
المعادلة س + 4 =
7، تعني ما هو العدد المجهول الذي اذا اضيف الى العدد 4 كان الناتج 7</span><span style="text-align: center;">؟</span><span style="text-align: center;"> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">نعتبر المعادلة
التالية: 2س + 1 = 11 </span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ففي
المعادلة 2س + 1 = 11 نقول أن العدد 5 هو حلاً لهذه المعادلة لأن (2 × 5) + 1 =
11 </span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">نعتبر المعادلة
التالية<span style="font-family: "courier new" , "courier" , monospace;">:</span> </span></span><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><span dir="LTR"><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">5</span></span> </span>س – 1 = 7</span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;">وفي
المعادلة 5س – 1 = 7 نقول أن العدد 1 هو ليس حلاً للمعادلة لأن <span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">(5 × 1) </span><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">– 1 = 4 </span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="color: red;"><b>خاصية:</b></span> </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;">تتحقق الخصائص التالية على أي معادلة محققة، وذلك من أجل الحصول على
معادلة جديدة</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">· من
الممكن إضافة أي كمية إلى طرفي المعادلة</span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">· من
الممكن طرح أي كمية من طرفي المعادلة</span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">· من
الممكن ضرب طرفي المعادلة بأي كمية</span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;">· من
الممكن قسمة طرفي المعادلة على أي كمية لاتساوي الصفر</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">· بشكل عام
من الممكن تطبيق أي دالة على طرفي المعادلة</span><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="mso-list: Ignore;"> </span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">الطرق
الجبرية لحل المعادلات فى مجهول واحد من الدرجة الاولى:</span></span></b></span></span></span></div>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red; font-family: "courier new" , "courier" , monospace;">
</span></b></span></span><br />
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
</span><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">أمثلة</span></span><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></span></span></b></span><span style="mso-list: Ignore;"><span style="font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal;"><b><span style="font-family: "courier new" , "courier" , monospace;"> </span> </b>
</span></span><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">نعتبر
المعادلات التالية: 3س – 2 = 13</span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;">حل المعادلة 3س – 2 = 13 والتحقق من صحة
الحل :</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">الحل : </span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ما هي العملية الحسابية التي نقوم بها لجعل
هذه المعادلة على صورة أس = ب </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذا أضفنا العدد 2 إلى طرفي المعادلة </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س – 2 + 2 = 13 + 2</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">فإننا نحصل على :</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س =
15 </span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المعادلة الآن على صورة أس = ب </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ما هي العملية الحسابية التي نقوم بها لجعل
هذه المعادلة على صورة س = … </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">نقسم طرفي المعادلة على 3 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س 3 : 15 =
3 :</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذن
س = 5، إذن جذر
المعادلة 3س – 2 = 13 هو العدد٠5 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">التحقق من صحة الحل: </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">(3 × 5) – 2 = 13.</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;"><br /></span><span style="font-size: x-small;"><span style="font-family: "courier new" , "courier" , monospace;">نعتبر
المعادلات التالية: 3س <span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">– 2</span></span> = س + 4 </span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;">حل المعادلة 3س <span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">– 2</span></span> = س + 4 والتحقق
من صحة الحل :</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">الحل : </span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س – 2 = س + 4 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: x-small;">3س – 2 + 2 = س <span lang="AR-SA" style="font-family: "arial" , "sans-serif"; font-size: xx-small; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">+ 4 + 2 </span></span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س
= س + 6</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س- س = س + - 6 س</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">2س 6 = </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">2س 2 : 6 =2 : </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذن
س = 3، إذن
جذر المعادلة 3س – 2 = س + 4 هو العدد٠3 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">التحقق من صحة الحل: </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3 × 7 = 2 – 3 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"> 7 = 3 + 4</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذن
س = 3 تحقق المعادلة
3س – 2 = س + 4<span dir="LTR" lang="AR-SA">
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "arial" , "sans-serif"; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">تمرين</span></span><span dir="RTL" lang="AR-SA" style="color: red; line-height: 115%;">:</span></b></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">حل المعادلات التالية</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">س – 4 = 0</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س = 6</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">3س – 4 = 0</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">5س – 2 = –س + 4 </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span style="font-family: "courier new" , "courier" , monospace; line-height: 20.7px;"><br /></span></span></b></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span style="font-family: "courier new" , "courier" , monospace; line-height: 20.7px;">3) نـظمـة مـعادلـتـين من الـدرجـة الأولى بـمجهـوليـن</span></span></b></span></div>
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<span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;"><br /></span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace;">الصورة العامة لنـظمـة مـعادلـتـين من الـدرجـة الأولى بـمجهـوليـن س و ش: </span></span></span></div>
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<span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">أس + ب ش = د</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">جـ س + حـ ش = ط</span></span><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;"></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-size: small;"><span style="font-family: "courier new" , "courier" , monospace;">بحيث المعاملات أ ، ب ، جـ ، حـ ، د ، ط ، تمثل أعداد معلومة.</span></span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>ا</b></span></span>لطرق الجبرية لحل نـظمـة مـعادلـتـين من الـدرجـة الأولى بـمجهـوليـن:</b></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>طريقة التعويض:</b></span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">مـن إحـدى الـمعادلتين؛ نجد قيمة أحـد المجهولين بدلالة الآخـر؛ ثم نعوضه</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">في المعادلة الأخرى٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>طريقة التآلفية الخطية:</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">لـكي نـحـتفظ بـأحـد الـمجـهوليـن ( لـكي نتمكن من حساب قيمته) نـضرب المعادلة من معادلتي النظمة في معامل مناسب لنحصل على معاملين متقابلين بالنسبة للمجهول الآخر؛ ثم نجمع المعادلتين الـمحصل عليهما طرفا بطرف٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>الطرق الهندسية لحل نـظمـة مـعادلـتـين من الـدرجـة الأولى بـمجهـوليـن:</b></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>طريقة الحل المبياني (الهندسية). ننشئ المستقيمين في معلم متعامد ممنظم:</b></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المعادلة من الدرجة الأولى بمجهولين لها عدد لا نهائي، يمثلها خط مستقيم في المستوى٠ انظر درس الدول (الدرس الخامس)٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b>مثال:</b></span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">حل النظمة </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">س + 2 ش = 1</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">2 س + 4 ش = 3</span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "courier new" , "courier" , monospace;">4</span>) المتراجحات من الدرجة الاولى:</b></span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span><span style="font-size: small;">
<span style="font-family: "courier new" , "courier" , monospace;"> لتكن ا٬ ب٬ عددين حقيقيين</span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ا < ب تكافئ ب – ا > 0 </span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">لتكن ا٬ ب٬ ك٬ ثلاثة أعداد حقيقية</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ا < ب تكافئ ا + ك < ب + ك</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">لتكن ا٬ ب٬ ك٬ ثلاثة أعداد حقيقية بحيث ك عدد موجب</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ا < ب تكافئ ك ا < ك ب</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">لتكن ا٬ ب٬ ك٬ ثلاثة أعداد حقيقية بحيث ك عدد سالب</span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ا < ب تكافئ ك ا > ك ب</span></td></tr>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>تعريف:</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المعادلة في الرياضيات، هي عبارة رياضياتية مؤلفة من رموز رياضياتية، تنص على مساواة تعبيرين رياضيتين بِمتغيرات جبرية، ويعبر عن هذه المساواة عن طريق إشارة التساوي٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">المجاهيل هي اعداد غير معلومة يرمز لها بـ (ش، س، ص ،ع ،ل...الخ)٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">ونقول أن المعادلة من الدرجة الثانية إذا كانت أعلى قوة لِ المجاهيل تظهر في المعادلة هي إثنان٠</span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">الصورة العامة لمعادلة الدرجة الثانية في مجهول واحد هي : </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">أ س<sup>2</sup> + ب س + جـ </span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 21.6px;">- </span><span lang="AR-SA" style="line-height: 20.4px;"> المعاملات أ ، ب ، جـ تمثل أعداد .</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 21.6px;">- </span><span lang="AR-SA" style="line-height: 20.4px;"> المجهول فيها واحد هو س .</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: 21.6px;">- </span><span lang="AR-SA" style="line-height: 20.4px;"> أكبر أس مرفوع له المجهول س هو 2 أي أن هذه المعادلة ذات مجهول واحد من الدرجة <span style="line-height: 20.4px;">الثانية .</span></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">الطرق الجبرية لحل المعادلات فى مجهول واحد من الدرجة </span></span></b><b>الثانية</b><b><span style="color: red;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-bidi-font-family: Arial; mso-bidi-theme-font: minor-bidi; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">:</span></span></b></span></span></span></div>
<div style="line-height: normal;">
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<tr style="height: 28.5pt;"><td width="100%"><span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;">خاصية</span></b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">من أجل كل عدد حقيقي أ، ب</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">أ ب = 0 فإما أ = 0 أو ب = 0 ٠</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="background-color: white; line-height: 30px;"><span style="color: red;"><br /></span></span></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="background-color: white; line-height: 30px;"><span style="color: red;">طريقة تعميل حدودية من الدرجة الثانية (</span></span></b></span></span><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>اكمال المربع).</b></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;">بصفة عامة يمكن كتابة أ س<sup>2</sup> + ب س + جـ على الشكل التالي</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"> أ س<sup>2</sup> + ب س + جـ <span dir="RTL" lang="AR-SA"> =</span> أ(س + <span dir="RTL" lang="AR-SA">ب:2</span> أ<span dir="RTL" lang="AR-SA">)</span> <sup>2 </sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+<span style="color: red;"> </span>جـ –<span lang="AR-SA" style="color: red;"> </span>ب <sup style="line-height: 27.6px;">2</sup><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span style="font-weight: normal;"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"> </span></span></span></span>: 4 أ</span><br />
<div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إما المعادلة لا تنعدم من أجل أي عدد حقيقي</span></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">أو تنعدم في مجموعة الأعداد الحقيقية و في هذه الحالة نعمل بإستعمال المتطابقة الهامة الثالثة و نطبق الخاصية أ ب = 0 فإما أ = 0 أو ب = 0 ٠ </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>طريقة حل معادلة من الدرجة الثانية بمجهول واحد</b></span></span></div>
</td></tr>
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<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody>
<tr style="height: 28.5pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"><td style="height: 28.5pt; padding: .75pt .75pt .75pt .75pt; width: 75.6pt;" width="101"><div class="MsoBodyText" dir="RTL" style="text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red;"><b><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">مثال:</span></b></span></span></div>
<div class="MsoBodyText" dir="RTL" style="text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">حل المعادلة س<sup>2</sup> + 2 س + 5 = 0 </span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">س<sup>2</sup> + 2 س + 5 <span dir="RTL" lang="AR-SA"> =</span> (س + 1 <span dir="RTL" lang="AR-SA">)</span> <sup>2 </sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+<span style="color: red;"> </span>5 –<span lang="AR-SA"><span style="color: red;"> </span>1</span><span lang="AR-SA" style="color: red;"> </span><span dir="RTL" lang="AR-SA"> </span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span dir="RTL" lang="AR-SA"> =</span> (س + 1 <span dir="RTL" lang="AR-SA">)</span> <sup>2 </sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>+<span style="color: red;"> </span>4</span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;"><span style="text-align: start;">إذن </span><span style="text-align: start;"> </span><span style="text-align: start;">المعادلة</span><span style="text-align: start;"> </span><span style="text-align: start;"> </span><span style="text-align: start;">س<sup>2</sup> </span>+<span style="text-align: start;"> 2 س </span>+<span style="text-align: start;"> 5 = 0 </span>لا تنعدم من أجل أي عدد حقيقي </span></span></div>
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<tr style="height: 28.5pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"><td style="height: 28.5pt; padding: .75pt .75pt .75pt .75pt; width: 31.6pt;" width="42"><table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody>
<tr style="height: 28.5pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"></tr>
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<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<div style="text-align: right;">
<div class="MsoBodyText" dir="RTL" style="margin: 0px; text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="color: red;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">مثال:</span></span></b></span></div>
<div class="MsoBodyText" dir="RTL" style="margin: 0px; text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">حل المعادلة س<sup>2</sup> <span style="text-align: right;">–</span> 4 س <span style="text-align: right;">–</span> 13 = 0 </span></span></div>
</div>
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<table border="0" cellpadding="0" class="MsoNormalTable" style="margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody>
<tr style="height: 28.5pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"><td style="height: 28.5pt; padding: .75pt .75pt .75pt .75pt; width: 75.6pt;" width="101"><div class="MsoBodyText" dir="RTL" style="margin: 0px; text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span lang="AR-SA" style="font-family: "courier new" , "courier" , monospace;">س<sup>2</sup> – 4 س – 13 <span dir="RTL" lang="AR-SA">= </span>(س – 2 <span dir="RTL" lang="AR-SA">)</span> <sup>2 </sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>–<span style="color: red;"> </span>13 <span lang="AR-SA"><span style="color: black;">+</span><span style="color: red;"> </span>4</span></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace; font-size: xx-small;"><span dir="RTL" lang="AR-SA"> =</span> (س + 1 <span dir="RTL" lang="AR-SA">)</span> <sup>2 </sup><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><span class="mw-headline" id=".D8.AA.D8.B4.D8.A7.D8.A8.D9.87_.D9.85.D8.AB.D9.84.D8.AB.D9.8A.D9.86"><sup> </sup></span></span>–<span style="color: red;"> </span>9</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span lang="AR-SA"> </span><span dir="RTL" lang="AR-SA"> =</span> (س + 1 – 3)(س + 1 + 3 <span dir="RTL" lang="AR-SA">)</span></span> </span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="line-height: normal;"><span dir="RTL" lang="AR-SA"> <span style="font-family: "courier new" , "courier" , monospace;"> <span style="font-family: "courier new" , "courier" , monospace;"> </span> </span> =</span> </span><span style="line-height: normal;">(</span><span style="line-height: normal;">س</span><span style="line-height: normal;"> </span><span style="line-height: normal;"> </span><span style="line-height: normal;">– 2)</span><span style="line-height: normal;">(</span><span style="line-height: normal;">س</span><span style="line-height: normal;"> </span><span style="line-height: normal;">+</span><span style="line-height: normal;"> 4</span><span style="line-height: normal;"> </span><span style="line-height: normal;"><span dir="RTL" lang="AR-SA">)</span></span></span><br />
<div style="text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذن<span style="text-indent: 48px;"> </span><span style="text-align: right;">س</span><span style="text-align: right;"> </span><span style="text-align: right;">– 2 = 0 </span><span style="text-align: right;">أو </span><span style="text-align: right;">س</span><span style="text-align: right;"> </span><span style="text-align: right;">+</span><span style="text-align: right;"> 4 = 0</span></span></div>
<div style="text-align: start;">
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;">إذن: <span style="text-align: right;">س</span><span style="text-align: right;"> </span><span style="text-align: right;">= 2 </span><span style="text-align: right;">أو</span><span style="text-align: right;"> </span><span style="text-align: right;">س</span><span style="text-align: right;"> </span><span style="text-align: right;">=</span><span style="text-align: right;"> </span><span style="text-align: right;">–</span><span style="text-align: right;">4</span></span></div>
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<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="color: red; font-family: "courier new" , "courier" , monospace;"><b>6) المتراجحات من الدرجة الثانية:</b></span></span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="background-color: white; line-height: 30px; text-align: right;"><span style="color: red;">جدول اشارات حدوية من الدرجة الثانية</span></span></b></span></span></div>
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<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: normal; margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: normal; margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: normal; margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"></table>
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<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="font-size: small;">
</span></span>
</span></div>
<span style="font-family: "courier new" , "courier" , monospace;"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><br /></span>
<span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><b><span style="font-family: "times" , "times new roman" , serif;">
</span></b></span>
</span><br />
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: normal; margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody></tbody></table>
<table border="0" cellpadding="0" class="MsoNormalTable" style="line-height: normal; margin-left: auto; margin-right: 0px; text-align: right; width: 599px;"><tbody>
<tr style="height: 28.5pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;"><td style="height: 28.5pt; padding: .75pt .75pt .75pt .75pt; width: 27.15pt;" width="36"><span style="font-family: "courier new" , "courier" , monospace; font-size: small;"><span style="font-family: "courier new" , "courier" , monospace; text-align: justify;">alfarjimohammed@gmail.com</span></span></td></tr>
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محمد ماهير الفرجي Mohammed Mahir Alfarji http://www.blogger.com/profile/11646420026148366824noreply@blogger.com0tag:blogger.com,1999:blog-162104398430917066.post-28171690961043734892011-07-16T10:06:00.000-07:002019-10-09T02:41:09.268-07:00 مجموعات الأعداد، الحسابيات<div dir="ltr" style="text-align: left;" trbidi="on">
<div style="text-align: right;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الرموز
الرياضياتية العربية </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>(</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
المصطلحات الخاصة و المكون البياني</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>)
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>المُستعملة
في هذا الدرس</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>طــ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموعة
الأعداد الصحيحة الطبيعية</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>صـ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموعة
الأعداد الصحيحة</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>كـ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموعة
الأعداد الكسرية</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>حـ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموعة
الأعداد الحقيقية</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red; font-family: "trebuchet ms" , sans-serif;"><b>تمارين</b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"> <span style="color: red;">1تمرين</span></span></span></span></span></div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px;">تحقق من صحة </span><span style="line-height: 15.18px;">العبارات</span><span style="line-height: 20.7px;"> التالية</span></span></span></div>
<div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 15.18px;"> 19 = (13 </span></span><span style="font-size: x-small;"><span style="line-height: 15.18px;">−</span><span style="line-height: 15.18px;"> 10)5 </span></span>−<span style="font-size: x-small;"><span style="line-height: 15.18px;"> 4</span></span></span></div>
<div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 15.18px;"> 16</span></span>−<span style="font-size: x-small;"><span style="line-height: 15.18px;"> = (</span></span><span style="font-size: x-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;">3 </span></span></span><span style="font-size: x-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;"><span style="font-size: x-small;">×</span></span></span></span></span></span></span><span style="font-size: x-small;"><span style="line-height: 15.18px;"><span style="font-size: xx-small;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;"><span style="font-size: x-small;"><span style="line-height: 20.7px;"><span style="font-size: xx-small;"> 2 +</span></span></span></span></span></span> 2 ÷ 4 - 12)5 - 4</span></span></span></span></div>
<div class="MsoNormal" style="color: black; line-height: normal; margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="font-family: "times new roman"; margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: times, "times new roman", serif;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;"><span style="font-size: xx-small;"><br /></span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;">2تمرين</span></span></span></span></div>
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<div class="MsoNormal" style="margin-bottom: 0cm;">
<div class="MsoNormal" style="margin-bottom: 0cm;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;">أحسب وقدم النتيجة على شكل عدد كسري مختزل</span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;">.<span style="font-size: x-small; text-align: right;"><span style="line-height: 15.18px;">= (15 </span></span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: xx-small;">÷</span><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;"><span style="font-size: x-small; text-align: right;"><span style="line-height: 15.18px;"> </span></span></span></span></span></span><img alt="{\frac {2}{15}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe09f7616e54474573ac9f05163ebcfce642df8d" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px; box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; display: inline-block; font-family: sans-serif; font-size: 14px; height: 5.343ex; margin: 0px; padding: 5px; text-align: start; vertical-align: -2.005ex; width: 3.182ex;" /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;"><span style="font-size: x-small; text-align: right;"><span style="line-height: 15.18px;"> </span><span style="line-height: 15.18px;"><span style="font-size: xx-small;"> ÷</span></span></span></span></span></span></span> <span style="font-size: x-small;"><span style="line-height: 15.18px;"><span style="line-height: 20.7px; text-align: justify;"><span style="font-size: xx-small;"><span style="font-size: x-small;">5 </span></span></span> </span></span><span style="font-size: x-small;"><span style="line-height: 15.18px;">− </span><span style="line-height: 15.18px;">2)5 </span></span>+ <img alt="{\frac {7}{11}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbcf5cc810423cbbcb1bcaac7e3d96ae8c82cdde" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px; box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; display: inline-block; font-family: sans-serif; font-size: 14px; height: 5.176ex; margin: 0px; padding: 5px; text-align: start; vertical-align: -1.838ex; width: 3.182ex;" /><br /><span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", times, serif; font-size: 16.52px; line-height: 1; white-space: nowrap;"><span style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;"><br /></span></span></span><span style="font-family: sans-serif; font-size: 14px;"></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;">3</span></span></span></span><span style="font-size: x-small;"><span style="color: red;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa">تمرين</span></span></span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px;">استعمل وحدة طول المستقيم المدرج التالي وارسم خط متصل طوله العدد 2√ و خط متصل طوله</span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px;"> العدد ط</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="color: blue; line-height: 20.7px;"><br /></span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="color: blue;"> النسبة بين محيط الدائرة وقطرها ثابت رياضي (عدد حقيقي لا يتغير مع تغيير قطرالدائرة) نرمز له بالحرف ط)</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="color: blue;"> ( π أو بالحرف اليوناني</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><img alt="http://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Number_line.svg/440px-Number_line.svg.png" src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Number_line.svg/440px-Number_line.svg.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 1px solid rgb(238, 238, 238); box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; padding: 5px;" /></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="color: blue;"><br /></span><span style="color: blue;">بين أن العدد عدد <span style="line-height: 20.7px;">2√ </span><span style="line-height: 20.7px;"> </span> غير كسري٠</span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;">تمرين</span></span></span><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;">4</span></span></span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="color: #252525; font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"> </span></span><span style="line-height: 22.4px; text-align: left;"> احسب الأعداد التالية</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">= </span></span></span><span style="font-size: x-small;"><span style="line-height: 22.4px; text-align: left;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span>5</span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"> + </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-size: x-small;"><span style="font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span></span>2</span></span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">= </span></span></span><span style="font-size: x-small;"><span style="line-height: 22.4px; text-align: left;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span>5</span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"> </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">− </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-size: x-small;"><span style="font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span></span>2</span></span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">= </span></span></span><span style="font-size: x-small;"><span style="line-height: 22.4px; text-align: left;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span>5</span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"> + <span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span>2</span></span></span></span><br /><div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">= </span></span></span><span style="font-size: x-small;"><span style="line-height: 22.4px; text-align: left;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span>5</span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"> </span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA">×</span></span></span></span></span></span> </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span></span>2</span></span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;">= </span></span></span><span style="font-size: x-small;"><span style="line-height: 22.4px; text-align: left;"><span class="lang-grc" lang="grc" style="background: none 0% 0% repeat scroll transparent;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span>5</span></span><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"> </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-size: xx-small;">÷</span> </span></span></span><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;"><span style="font-family: sans-serif; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial;">π</span></span></span></span></span>2</span></span></span></span></div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;">= √2 </span></span></span></span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;">5 </span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;">× √2 2</span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;">= √3 </span></span></span></span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;">5</span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;"> × √2 2</span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;"><span style="line-height: 20.7px;">= </span>√2</span><span style="line-height: 20.7px;"> </span></span><span style="font-size: x-small;"><span style="text-align: justify;"><span style="font-size: xx-small;">÷</span></span><span style="line-height: 20.7px;"> </span><span style="text-align: left;">√6</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;"><span style="line-height: 20.7px;">= </span>√2</span><span style="line-height: 20.7px;"> </span></span><span style="font-size: x-small;"><span style="text-align: justify;"><span style="font-size: xx-small;">÷</span></span><span style="line-height: 20.7px;"> </span><span style="text-align: left;">√8</span></span></span></div>
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<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;"><span style="line-height: 20.7px;">= </span>√2</span><span style="line-height: 20.7px;"> </span></span><span style="font-size: x-small;"><span style="text-align: justify;"><span style="font-size: xx-small;">÷</span></span><span style="line-height: 20.7px;"> </span><span style="text-align: left;">√3</span></span></span></div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span><br /><div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;">= √2 </span></span></span></span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;">5 +</span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;"> √2 2</span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="font-family: sans-serif; font-size: 14px; line-height: 22.4px;"><span class="lang-grc" lang="grc" style="background-image: none; font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif;">= √2 </span></span></span></span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;">5 -</span><span style="font-family: "arial unicode ms", "dejavu sans", athena, gentium, "palatino linotype", "lucida sans unicode", "lucida grande", code2000, sans-serif; font-size: 14px;"> √2 2</span></div>
</div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"> <span style="color: red;"></span></span></span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 20.7px; text-align: justify;"><span lang="ar-sa"><span style="color: red;">5تمرين</span></span></span></span></span><br /><div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 22.4px;">اكتب الأعداد التالية على شكل قوة</span></span></span></div>
<div style="line-height: 18.48px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"> </span></span><span lang="AR-SA" style="color: black;"> <sup>6</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"> </span></span></span></span></span></span></span></span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>6</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"> </span></span></span></span></span></span></span><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: xx-small;">÷ </span><sup style="font-family: times, "times new roman", serif;">2</sup><span style="font-family: times, "times new roman", serif;">5 </span><span style="font-family: times, "times new roman", serif;"> </span><br /><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"> </span></span></span></span></span></span></span><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span></span><span style="font-family: times, "times new roman", serif; font-size: xx-small;">÷</span><span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span></span><sup style="font-family: times, "times new roman", serif;">6</sup><span style="font-family: times, "times new roman", serif;">5 </span><span style="font-family: times, "times new roman", serif;"> </span></div>
<div style="line-height: 18.48px;">
<div style="line-height: 18.48px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>0</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"> </span></span></span></span></span><span style="font-size: xx-small;">÷</span><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span></span><span style="font-family: times, "times new roman", serif;">5 </span><span style="font-family: times, "times new roman", serif; font-size: 13.3333px;"> </span></div>
<div style="line-height: 18.48px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><sup>2</sup>5</span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"> </span></span></span></span></span></span></span><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span style="color: black;"><span lang="AR-SA"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA">+</span><span lang="AR-SA"> </span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><sup>2</sup>5 </span></span></span></span></span></span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA">+</span></span></span></span></span> </span></span></span></span></span><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><sup>2</sup>5 </span></span></span></span></span></span></span></span></span></span>+</span></span><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span><span style="font-size: x-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA"><sup>2</sup>5 </span></span></span></span></span></span></span></span></span></span></span><span style="font-size: xx-small;"><span style="line-height: 25.2px;"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;"> </span></span></span></span></span></span></span></div>
<div style="line-height: 18.48px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><br /></span></span></div>
</div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="line-height: 18.48px;">ما هي الكتابة العلمية للعدد 3000؟</span></span></span></div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;">ما هي الكتابة العلمية للعدد </span><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;">7345,719</span></span></span></span></span></span></span><span style="text-align: left;"> ؟</span></span></span></div>
<div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;">ما هي الكتابة العلمية للعدد </span></span><span style="font-size: x-small;"><span style="text-align: left;">−</span><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span dir="RTL" lang="AR-SA" style="line-height: 20.7px;"><span lang="AR-SA" style="line-height: 20.7px;"><span dir="RTL"><span lang="AR-SA" style="color: black;">7345,719</span></span></span></span></span></span></span><span style="text-align: left;"> ؟</span></span></span></div>
<div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;"><br /></span></span></span></div>
<div>
<span style="font-family: times, "times new roman", serif;"><span style="font-size: x-small;"><span style="text-align: left;"><br /></span></span></span></div>
<span style="color: red; font-family: trebuchet ms, sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">الدرس</span></span></b></span><br />
<span style="font-family: trebuchet ms, sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نظام
العد،الخانات</b></span></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نظام
العد هو طريقة التعامل مع رسوم الأرقام
للتعبير عن قيمتها وكيفية تطبيق العمليات
الحسابية عليها٠</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>علم
الخانات مكننا من كتابة أي عدد و القيام
بكل العمليات الحسابية٠ ففي النظام العشري
مثلا تُستعمل رموز الأرقام من </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>0
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>إلى
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>9 </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
خاناته و تُعبِّر خاناته عن مضاعفات قوى
العدد عشرة ٠</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">أمثلة</span></b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أمثلة
لتفكيك عدد</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>خمسة
وستون وتسع مائة وألف</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1965 =
5 + 60 + 900 + 1000</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>سبعة
وخمسون</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50 + 7
= 57</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">العمليات
الحسابية</span></b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>علم
الخانات مكننا من القيام بكل العمليات
الحسابية</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ففي
عملية الجمع مثلا بين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>57</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>فالعميلة
تتم كالتالي </b></span></span></span></span>
</div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>63 = 60
+ 3 = 50 + 10 + 3 = 50 + 13 = 50 + 7+ 6 = 57 + 6</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وكذلك
في عملية الطرح مثلا</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>= 57 -
63 </b></span></span></span></span>
</div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مجموعات
الأعداد</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><br /></span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">1-
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مجموعة
الأعداد</span></b><b><span style="color: red;"> الطبيعية ط</span></b><b> </b></span></span></span><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /><span style="color: red;"><br /></span></b><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">2- </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مجموعة الأعداد الصحيحة ص</span></b></span></span></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><br /></span></b></span></span></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">3-
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مجموعة
الأعداد النسبية ك </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القسمة
هي توزيع بالتساوي</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يُرمز
إلى القسمة بالعلامة </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>÷.</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القسمة
على الصفر هي عملية غير معرفة</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وسبب
ذلك هو أنه إذا ضُرب الصفر في عدد ما، فإن
النتيجة تساوي دائما الصفر</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
التعبير أ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>÷
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب ،
يسمى أ مقسوما أو بسطا، ويسمى ب مقسوما
عليه أو مقاما</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يُشار
أيضاً إلى عملية القسمة بواسطة خط أفقي
يأتي فوقه المقسوم ويأتي تحته المقسوم
عليه، على الصُّورةِ أ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.</b></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><br /></span></b></span></span></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">أمثلة</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">:</span><br />20
÷ 4 = 5</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يسمى
خارج القسمة أو ناتج القسمة</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.<br /><br />7
÷ 3 =</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
المثل الثاني نتيجة القسمة غير منتهية٬
في هذه الحالة يُكتب خارج القسمة على
الصُّورةِ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>7\4
<br /><br />2 ÷ 5 = 0,4</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
المثل الثالث نتيجة القسمة منتهية٬ في
هذه الحالة يُكتب خارج القسمة </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>0,4
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أو
على الصُّورةِ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2\5</b></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>العدد
الكسري أو العدد الجذري هو عدد يُعبِّر
على خارج قسمة لعددين صحيحين بشرط أن
المقام لا يساوي صفر</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يُكتب
العدد النسبي على الصُّورةِ أ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب
حيثُ أ ، ب عددانِ صحيحانِ ، ب لا يساوي
صفرا</b></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموعة
الأعداد النسبية هي المجموعة التي تشتمل
على جميع الأعداد النسبية، ونستخدم الرمز
ك للدلالة عليها</b></span></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><br /></span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">يمكن
كتابة أي عدد نسبي بعدد غير منتهي من
الأشكال، ويعتبر الشكل أبسط ما يكون عندما
لا يكون للبسط </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">(</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">الصورة</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">)
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">والمقام
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">(</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">المخرج</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">)
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">أي
قواسم مشتركة</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><br /></span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"> </span></b></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مثل</span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3\5
= 6\10 = 9\15 = 12\20 = 15\25 =.... </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">تساوي
عددان كسريان</span> </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أ،
ب، ج، ح أعداد طبيعية غير منعدمة </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>(</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تخالف
صفر</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>)</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يكون
عددان كسريان، أ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب
و ج</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ح
متساويان فقط وفقط إذا كان أ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>×
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ح </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>=
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>×
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ج</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">العمليات
الحسابية </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">عملية
الجمع و الطرح في المجموعة ك</span> </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مجموع
عددين كسريين لهما نفس المقام هو عدد كسري
بسطه هو مجموع البسطين ومقامه هو نفس
المقام</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>طرح
عددين كسريين لهما نفس المقام هو عدد كسري
بسطه هو طرح البسطين ومقامه هو نفس المقام</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">أمثلة
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">:</span>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\5
+ 2\5 = 6\5</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\5
− 2\5 = 2\5 </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لحساب
مجموع عددين كسريين مختلفين في المقام
نوحد مقاميهما و نطبق قاعدة حساب مجموع
عددين كسريين لهما نفس المقام</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لحساب
طرح عددين كسريين مختلفين في المقام نوحد
مقاميهما و نطبق قاعدة حساب طرح عددين
كسريين لهما نفس المقام </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أمثلة
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\5
+ 2\7 = 28\ 35 + 10\35 = 38\35</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\3
− 2\5 = 20\15 −6\15 = 14\15</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>عملية
الضرب في المجموعة ك</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لحساب
جداء عددين كسريين نضرب البسط في البسط
و المقام في المقام</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مثل</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
<br />4\5 × 3 = 12\5</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\5
× 3\2 = 12\10 = 6\5</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مقلوب
عدد كسري </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مقلوب
العدد الكسري أ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب
، أ لا يساوي صفر، ب لا يساوي صفر، هو نظيره
ب</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>\</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أ
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">أمثلة
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">:
</span> </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مقلوب
العدد الكسري </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4\5
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
العدد الكسري </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5\4
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مقلوب
العدد الكسري </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
العدد الكسري </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1\2
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>عملية
القسمة في المجموعة ك </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>خارج
عدد كسري على عدد كسري هو جداء العدد
الكسري الأول في مقلوب الكسر الثاني</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مثل
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>: </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2\3
÷ 5\4 = 2\3 × 4\5 = 8\15 </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تمرين
تطبيقي </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أحسب
وقدم النتيجة على شكل عدد كسري مختزل </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">4</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مجموعة
الاعداد الحقيقية ح</span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نشأت
فكرة الأعداد الحقيقية بسبب وجود أطوال
لا يمكن التعبير عن قياسها باستعمال أعداد
نسبية، لهذا يتم إنشاء مجموعة الأعداد
الحقيقية </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الجذر
التربيعي</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">5×5
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">هي
عملية تربيع للعدد </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">5</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">،
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>2</sup></span><span style="color: black;">5
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">تُقرأ
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">5
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">تربيع
أو</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">5
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">أس
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">2
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">العدد
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">25
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">هو
مربع العدد </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">5
</span></b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>العدد
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5 </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
الجذر التربيعي للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>25
</b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الجذر
التربيعي للعدد الموجب س يعبر على العدد
الموجب الذي مربعه يساوي س و يشار إليه
بـِ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
√ </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أمثلة
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>: </b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الجذر
التربيعي للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>٠ </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الجذر
التربيعي للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>٠ </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الجذر
التربيعي للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>0</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>٫</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>04
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>0</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>٫</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>٠ </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.74cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مستقيم
الأعداد</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">عدد
حقيقي هو قيمة كمية ما تمثَّل عادة على
مستقيم متصل</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">.
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">فالعدد
الحقيقي يمثله نقطه على خط الاعداد، كما
أن كل نقطه على خط الاعداد تمثل عدد حقيقي
</span></b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm; margin-left: 0.74cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><a href="https://blogger.googleusercontent.com/img/proxy/AVvXsEglH0nad-VvSxh5GQ5_SCWNYb4dn7lPcXGRIIDgHLqX6Wm-Ar9Ra8LNLErVDWBjWooFnxVSDdfXMZwBmn86Yi5WkV1RqjoY4_c3WatUieWsXWIb_lWc3BE9rqf0XoFa1GMqbSYIJLrRzqdzoGshd1-Jya4jbLxcqfE3aKQV2XCcSnV7V448DCVQsOUFxOgsmXLQ0j1nU-EKwUrgE3qxchQ3vZg=" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img align="BOTTOM" border="0" name="images1" src="https://blogger.googleusercontent.com/img/proxy/AVvXsEglH0nad-VvSxh5GQ5_SCWNYb4dn7lPcXGRIIDgHLqX6Wm-Ar9Ra8LNLErVDWBjWooFnxVSDdfXMZwBmn86Yi5WkV1RqjoY4_c3WatUieWsXWIb_lWc3BE9rqf0XoFa1GMqbSYIJLrRzqdzoGshd1-Jya4jbLxcqfE3aKQV2XCcSnV7V448DCVQsOUFxOgsmXLQ0j1nU-EKwUrgE3qxchQ3vZg=" /></a><span style="color: red;"> </span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>خاصية</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
حالة ضرب جذر تربيعي لعدد في جذر تربيعي
لعدد آخر يتم ضرب العددين في بعضهما تحت
إشارة الجذر التربيع</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><span style="font-style: normal;">وكذلك
في حالة قسمة جذر تربيعي لعدد على جذر
تربيعي لعدد آخر يتم قسمة العددين تحت
إشارة الجذر</span></span><span style="color: black;"> </span><span style="color: black;"><span style="font-style: normal;">التربيعي</span></span></b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="font-style: normal; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">أمثلة
</span></span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">:</span></span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">5-</span></span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">الرفع</span></span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">الضرب
المتكرر أو الرفع أو الترقية هو تكرار
ضرب العدد في نفسه عدة مرات مثل</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">:
3×3×3 </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">أو
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">1×1×1×1×1
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">ولكنها
يتم اختصار هذه العملية في صيغة بسيطة
فمثلا </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">3×3×3×3
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">تُكتب
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup><span style="font-style: normal;">4</span></sup><span style="font-style: normal;">3
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">و
تُقرأ </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">3
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">أس
أربعة، وتسمى </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">3
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">بالأساس
أو المبنى و </span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">4
</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="font-style: normal;">بالأُس
أو القُوّة </span></b></span></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><span style="font-style: normal;">الأساس
</span></span>هو العدد الذي يتم تكراره في عملية
الضرب المتكرر</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الأس
هي قوة العدد أو عدد مرات تكراره </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>فعلى سبيل
المثال </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>2</sup></span><span style="color: black;">3</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أساسها
يساوى </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لأن
الثلاثة هي العدد الذي تم تكريره</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>3</sup></span><span style="color: black;">6</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أساسها
يساوى </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6
(</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>العدد
الذي تم تكريره</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>)</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><span style="font-style: normal;">،</span></span>
<span style="color: black;"><span style="font-style: normal;">و</span></span>
أسها يساوى </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لأن
الأساس الذي يساوى </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>قد
تم تكريرها ثلاثة مرات</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تُقرأ العملية
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>3</sup></span><span style="color: black;">6</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>كما
يلي </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
6 </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أس
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3 </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أو
القوة الثلاثة للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">ملاحظة</span>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لا
داعى لكتابة الواحد إذا كان الواحد أسا
لعدد ما لأن أي عدد مرفوع له أس واحد يساوي
نفس العدد</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثل </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">6</span>
= <span style="color: black;"><sup>1</sup></span><span style="color: black;">6</span>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>متطابقات
وخصائص</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>عند
ضرب عددين أو أكثر ذى أساسات متساوية فإن
الناتج يكون نفس الأساس مرفوع له مجموع
الأسس</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثل</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>2</sup></span><span style="color: black;">5×
</span><span style="color: black;"><sup>7</sup></span><span style="color: black;">5=
</span><span style="color: black;"><sup>9</sup></span><span style="color: black;">5</span>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>عند
قسمة عددين أو أكثر ذى أساسات متساوية
فإن الناتج يكون نفس الأساس مرفوع له حاصل
طرح الأسس</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثل</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;"><sup>9</sup></span><span style="color: black;">5÷
</span><span style="color: black;"><sup>7</sup></span><span style="color: black;">5=
</span><span style="color: black;"><sup>2</sup></span><span style="color: black;">5</span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>إذا
كان هناك عدد مرفوع لأس والكل مرفوع لأس
آخر فإن الناتج يكون نفس العدد مرفوع له
حاصل ضرب الأسين</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثل </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>(</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>3</sup>
)<sup>4</sup> = </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>12</sup></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ضرب
عدد مرفوع لأس بعدد مرفوع لنفس الأس فإن
الناتج يكون ضرب العددين مرفوع لنفس الأس</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثل </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>3</sup>
<span style="color: black;">×</span> </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>حـ</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>3</sup>
= (</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ب
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">×</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ح</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>)<sup>3</sup>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الكتابة
العلمية</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الاصطلاح
العلمي أو الكتابة العلمية هو كتابة عدد
على شكل جداء عددين</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>,
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>الأول
هو عدد عشري قيمته المطلقة محصورة بين
واحد وعشرة</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أما
الآخر فهو العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>10
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مرفوع
لأس معلوم</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 0.71cm; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يتم استخدام
هذه الكتابة في اختصار الأعداد الكبيرة</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وهي
تكتب على الشكل التالي</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ج</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.10<sup>
</sup></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>ص</sup>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>حيث</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القيمة
المطلقة للعدد ج أكبر أو يساوي </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وأصغر
قطعا من </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>10</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
وص عدد صحيح٠ </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أمثلة</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">300
= 3 × </span><span style="color: black;"><sup>2</sup></span><span style="color: black;">10</span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: black;"><span style="background: #ffffff;">−</span></span>
<span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">7345,79
= </span><span style="color: black;"><span style="background: #ffffff;">−</span></span><span style="color: black;">
7,34579 × </span><span style="color: black;"><sup>3</sup></span><span style="color: black;">10</span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: black;">7345,79</span><span style="color: red;">
</span><span style="color: black;">× </span><span style="color: black;"><sup>5</sup></span><span style="color: black;">10
= 7,34579 × </span><span style="color: black;"><sup>3</sup></span><span style="color: black;">10
</span>× <sup>3</sup>10 <span style="color: black;">= 7,34579 × </span><span style="color: black;"><sup>8</sup></span><span style="color: black;">10
</span></b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">6-
</span></span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"><span style="font-style: normal;">نظرية
الترتيب</span></span></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن
س٬ ص٬ عددين حقيقيين</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 0.53cm; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
تكافئ ص <span style="background: #ffffff;">−</span> س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>>
0 </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 0.53cm; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 0.53cm; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن س٬ ص٬ ك٬
ثلاثة أعداد حقيقية</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
تكافئ س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>+
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ك </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>+
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ك</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن
س٬ ص٬ ك٬ ثلاثة أعداد حقيقية بحيث ك عدد
موجب</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
تكافئ ك س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ك
ص</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن
س٬ ص٬ ك٬ ثلاثة أعداد حقيقية بحيث ك عدد
سالب</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
تكافئ ك س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ك
ص</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن
س٬ ص٬ عددين حقيقيين موجبين</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>إذا
كان س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
فإن س</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>2</sup>
< </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>2</sup>
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>إذا
كان س </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
فإن س√ </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص√٠
</b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 115%; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن س٬ ص٬
عددين حقيقيين موجبين٬ س<span style="color: black;"> لا
يساوي صفر،</span>ص لا يساوي صفر </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص
تكافئ س</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>-1</sup>
> </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ص</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><sup>-1</sup></b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.5cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تمرين
تطبيقي</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="line-height: 0.53cm; margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتكن س٬ ص٬
عددين حقيقيين بحيث ص </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>س
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>قارن
العددين س <span style="background: #ffffff;">−</span> </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>10
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و ص
<span style="background: #ffffff;">−</span> </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>10
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>قارن
العددين <span style="background: #ffffff;">−</span> س و <span style="background: #ffffff;">−</span>
ص </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;"> </span></b></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><span style="color: red;">الحسابيات</span></span></span></span></b></span><br />
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><b><br /></b></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><b>ليكن أ، ب عددين من مجموعة الأعداد الصحيحة ص، نقول إن ب يَقسِم أ أو ب قاسم لِـ أ أو أ مُضاعف لِـ ب و نكتب ب</b><b><b><span style="font-family: "trebuchet ms" , sans-serif;">\</span></b>أ إذا وجد ك في ص حيث أ = ك ب</b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><br /></b></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;">مثال</span></span></span></span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b>مجموعة مضاعفات العدد 7</b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">عدد
أولي، القاسم المشترك الأكبر لعددين</span></b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>العدد
الأولي هو عدد طبيعي أكبر قطعاً من </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
لا يقبل القسمة إلا على نفسه وعلى الواحد
فقط</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>يُدعى
كل عدد طبيعي أكبر قطعاً من </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وغير
أولي عددا مؤلفا</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مثال</b></span></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5 </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
عدد أولي لأنه لا يقبل القسمة إلا على </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وعلى
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
بينما </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
عدد مؤلف لأنه قابل للقسمة على </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
وعلى </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وعلى
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>وعلى
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><span style="color: red;"><b>اختبار
أولية عدد ما</b></span></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القسمة
المتكررة</b></span></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تتمثل
هاته الطريقة في قسمة العدد المراد تحديد
أوليته على جميع الأعداد الصحيحة الأكبر
من الواحد والأصغر من جذره التربيعي،
واللائي يكن في نفس الوقت أعدادا أولية</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>إذا
لم تنتج إحدى هذه القسمات باقيا، فإن
العدد المراد تحديد أوليته ليس بالأولي</b></span></span></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
سبيل المثال، بالنسبة للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>37</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
فإنه يكفي النظر إلى الأعداد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ولا
ينبغي النظر إلى العددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>4
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>6
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لأنهما
عددان غير أوليين</b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مبرهنة
التحليل إلى جداء أعداد أولية</b></span></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>في
الرياضيات التحليل إلى العوامل أو تحليل
العدد الصحيح هو عملية تفكيكه إلى جداء
عوامله الأولية، أي كتابة هذا العدد على
شكل جداء أعداد أولية، بحيث يكون حاصل
ضربها مساوٍ للعدد الأصلي</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مثلا</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>:
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تحليل
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5×3×3</b></span></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مبرهنة</b></span></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>المبرهنة
الأساسية في الحسابيات أو ما يعرف بمبرهنة
التحليل إلى جداء أعداد أولية هي مبرهنة
رياضية تنص على أن كل عدد صحيح طبيعي غير
منعدم يمكن كتابته على شكل جداء أعداد
أولية، وهذه الكتابة وحيدة</b></span></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: #ff6600;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>بعض
خوارزميات التحليل</b></span></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هناك
طرق عديدة تستعمل لتحليل الأعداد الصحيحة،
خصوصا عندما يكون العدد كبيرا </b></span></span></span></span>
</div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">القسمات
المتتابعة</span></b></span></span></span></span></div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تتم
بقسمة العدد على التوالي على الأعداد
الأولية قسمات تامة والتوقف عند الوصول
إلى خارج مساو للعدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
أو لعدد أولي </b></span></span></span></span>
</div>
<div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div dir="LTR" id="Section1">
<div align="RIGHT" style="margin-left: 0.26cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مثال</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><br /></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>لتحليل
العدد الصحيح </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>180</b></span></span></span></span></div>
</div>
<dl><dd><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">
</span></span><br />
<table border="0" cellpadding="2" cellspacing="0" style="width: 220px;">
<colgroup><col width="105"></col></colgroup><colgroup><col width="107"></col>
</colgroup><tbody>
<tr>
<td bgcolor="#f0f0f0" width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>العدد
وناتج القسمة </b></span></span></span></span>
</div>
</td>
<td bgcolor="#f0f0f0" width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>عدد
أولي مقسوم عليه</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>180</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>90</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>2</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>15</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>3</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: blue;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5</b></span></span></span></span></span></div>
</td>
</tr>
<tr>
<td width="105"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>1</b></span></span></span></span></div>
</td>
<td width="107"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
</td>
</tr>
</tbody></table>
</dd><dd><table border="0" cellpadding="2" cellspacing="0" style="width: 11px;">
<colgroup><col width="7"></col>
</colgroup><tbody>
<tr>
<td width="7"><div align="RIGHT">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
</td>
</tr>
</tbody></table>
</dd></dl>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm; page-break-before: auto;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>أي أن </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>180
= 5×3×3×2×2</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.53cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm; page-break-before: auto;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القاسم
المشترك الأكبر لعددين</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القاسم المشترك
الأكبر لعددين، هو أكبر عدد يقسم في نفس
الوقت العددين معاً بدون أي باقي قسمة</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>فمثلاً القاسم
المشرك الأكبر للعددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>48
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>60
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>12. </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>طريقة
حساب القاسم المشترك الأكبر لعددين</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>استعمال
التعميل إلى جداء أعداد أولية</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm; margin-left: 0.01cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نكتب العددان
على شكل جداء عوامل أولية</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ونبحث
عن قاسمهما المشترك الأكبر</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>.</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>فمثلاً
القاسم المشترك الأكبر للعددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تحليل
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5×3×3
. </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>تحليل
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5×5×2
. </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القاسم
المشترك الأكبر للعددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>استعمال
خوارزمية اقليدس</b></span></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نقسم
العدد الأكبر على الأصغر ثم نأخذ باقي
القسمة مع العدد الأصغر الناتج ونعيد
العملية مع هذين العددين الجديدين حتى
نحصل على باقي هو الصفر فيكون العدد الأصغر
هو القاسم المشترك الأكبر </b></span></span></span></span>
</div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: #ff3300;">مثلا</span></b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: #ff3300;">:</span>
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>القاسم
المشترك الأكبر للعددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نقسم
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ثم
نأخذ باقي القسمة مع العدد الأصغر ونعيد
العملية، أي </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>مع
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>ونعيد
العملية</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>نقسم
العدد </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>على
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>،
باقي هذه القسمة هو صفر إذن العدد الأصغر
أي </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>5
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>هو
القاسم المشترك الأكبر للعددين </b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>45
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>و
</b></span></span><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>50</b></span></span></span></span></div>
<div align="RIGHT" dir="RTL" style="margin-bottom: 0cm;">
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="color: red;">الأعداد الأولية فيما بينها</span></b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b>يكون عددان صحيحان أوليين فيما بينهما عندما يكون القاسم المشترك الأكبر بينهما مساويا للعدد 1.</b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;">خاصية</span></span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b>العددان الصحيحان ا و ب أوليان فيما بينهما إذا وفقط إذا وجد عددان صحيحان خ و ج بحيث أ خ + ب ج = 1.</b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;">المضاعف المشترك الأصغر</span></span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">المضاعف المشترك الأصغر لعددين صحيحين هو أصغر عدد صحيح موجب مضاعف لكلا هذين العددين، وهذا يعني أنه من الممكن قسمة المضاعف المشترك الأصغر على العددين بدون باقي قسمة.</span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span></div>
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;">علاقة تكافؤ</span><br /><br /><span style="color: red;">علاقة انعكاسية</span><br />تكون العلاقة ~ علاقة انعكاسية على المجموعة أ عندما يرتبط كل عنصر س من أ مع نفسه في العلاقة ~ ٱي من أجل كل عنصر س من أ ، </span></b></span><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">فإن </b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;"><span style="font-family: "trebuchet ms" , sans-serif;">س ~ س.</span></b><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><br /><span style="color: red;">علاقة تماثلية</span><br />تكون العلاقة ~ علاقة تماثلية على المجموعة أ : إذا كان من أجل كل عنصرين س ، ص من المجموعة أ</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;">،</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"> </span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">س ~ ص فإن ص ~ س .<br /><br /><span style="color: red;">علاقة تعدي</span><br />تكون العلاقة ~ علاقة تعدي على المجموعة أ : إذا كان من أجل كل ثلاثة عناصر س ، ص ، ل من المجموعة أ</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;">،</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"> </span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">س ~ ص و ص ~ ل فإن س ~ ل .<br /><br /><span style="color: red;">علاقة تكافؤ</span><br />تكون العلاقة ع علاقة تكافؤ على المجموعة أ إذا وفقط إذا كانت علاقة انعكاسية وتماثلية وتعدي معاً.<br /><br />علاقة تكافؤ ~ هي علاقة تقسم مجموعة ما، إلى عدد من المجموعات الجزئية حيث يصير كل عنصر من المجموعة الأصلية عنصرا من مجموعة جزئية واحدة بالتحديد. يعتبر عنصران من المجموعة متكافئين إذا وفقط إذا انتميا إلى نفس المجموعة الجزئية. ننتج إذن أصناف تكافؤ و نُكوِّن مجموعة جديدة (مجموعة أصناف التكافؤ) تسمى مجموعة خارج القسمة</span></b></span><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">.</b><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">ونرمز </span></b></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b style="text-align: -webkit-right;"><span style="font-family: "trebuchet ms" , sans-serif;">لمجموعة</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"> أصناف تكافؤ</span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"> </span></b></span><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">المجموعة أ</b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">، </b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;"> </b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;"><span style="font-family: "trebuchet ms" , sans-serif;">بِـ أ\ ~</span></b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">.</b><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><br />ﺼﻨﻑ ﺘﻜﺎﻓﺅ، مجموعة خارج القسمة</span><br />س عنصر من المجموعة أ، ﺼﻨﻑ ﺘﻜﺎﻓﺅ العنصر س ونرمز له بِـ [س] هو المجموعة الجزية من المجموعة أ، [س] = {ص من أ، بحيث ص ~ س}<br /><br />ﻜل ﻋﻨﺼﺭ ﻤﻥ ﺼﻨﻑ ﺍﻟﺘﻜﺎﻓﺅ ﻫﻭ ﻤﻤﺜل ﻟﻬﺫﺍ ﺍﻟﺼﻨف<br /><br />مجموعة خارج القسمة هيي مجموعة ﻤﻤﺜلي ﺍﻟأﺼﻨاف</span></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><b>حسابيات نمطية</b></span></span><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;">،</span></span></b> الموافقة بترديد بعدد صحيح ن</b></span></span></span>
<br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b>ن عدد طبيعي، </b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b>نقول إن عددين صحيحين </b></span></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><span style="font-family: "trebuchet ms" , sans-serif;"><b style="text-align: -webkit-right;"><span style="font-family: "trebuchet ms" , sans-serif;">ا </span></b><b>و ب متوافقان بترديد العدد ن يعني أن <span style="font-family: "trebuchet ms" , sans-serif;">الطرح</span> </b></span></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><span style="font-family: "trebuchet ms" , sans-serif;"><b style="text-align: -webkit-right;"><span style="font-family: "trebuchet ms" , sans-serif;">ا</span></b><b> </b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="background: #ffffff;">−</span></b></span></span></span> ب </b></span></span><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">ينتمي للمجموعة ن ص</b><span style="text-align: left; text-decoration-line: underline;">،</span><b style="font-family: "trebuchet ms", sans-serif; text-align: left;"> </b><b style="font-family: "trebuchet ms", sans-serif; text-align: left;">أي </b><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b><span style="font-family: "trebuchet ms" , sans-serif;">عدد مُضاعف لِلعدد ن</span></b></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">.</span></b> </b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b>ونرمز</b><b style="text-align: -webkit-right;"><span style="font-family: "trebuchet ms" , sans-serif;">ا </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;">≡ ب</span> </b></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">[</span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b>ن</b></span>]</span></b> </b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b>و نقرأ </b><b style="text-align: -webkit-right;"><span style="font-family: "trebuchet ms" , sans-serif;">ا </span></b><b>يوافق ب بترديد ن</b></span><span style="font-family: "trebuchet ms" , sans-serif; text-align: left;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">.</span></b></b></span></b></span></div>
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<span style="font-family: "trebuchet ms" , sans-serif;"><br /></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">أمثلة </span></b></span></span></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b> <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">7</span></span></span> ≡ <span style="font-family: "trebuchet ms" , sans-serif;">3</span></span></b></span></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">[</span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">4</span></span></b></span>]</span></b></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b> <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">7</span></span></span> ≡ <span style="font-family: "trebuchet ms" , sans-serif;">3</span></span></b></span></span></b></b></span></span><b><b><b> </b><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">[</span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">4</span></span></b></span>]</span></b></b></span> </b></b><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">7</span></span></span> ≡ </span></b></span></span></b></b></span></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="background: #ffffff;">− 3</span></b></span></span></span></b></span></span></span> </b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">[<span style="font-family: "trebuchet ms" , sans-serif;">10</span></span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">]</span></b></b></span></span></b></b></span> <br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b> <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">7</span></span></span> ≡ <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">9</span></span></span></span></b></span></span><b> </b><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">[<span style="font-family: "trebuchet ms" , sans-serif;">2</span></span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">] </span></b></b></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><br /></span></b></b></span></span>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="color: red;"><b><br /><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> ص\ ن ص</span></span></b></span></span><br />
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">ن عدد طبيعي أكبر قطعا من 1، الموافقة بترديد بعدد صحيح ن علاقة تكافؤ في مجموعة الأعداد الصحيحة ص، </span></b></b></span><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">ننتج إذن</span></b> مجموعة أصناف تكافؤ (مجموعة خارج القسمة بترديد العدد ن) التي نرمز لها بِـ ص\ ن ص.</span></b></b></span></span><br />
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<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">ص\ <span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">ن</span></b></b></span> ص = </span></b></b></span></span><b><span style="font-family: "trebuchet ms" , sans-serif;">{<span style="font-family: "trebuchet ms" , sans-serif;">0 </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> 1</span></span> </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">، </span></b></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">2</span> </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> <span style="font-family: "trebuchet ms" , sans-serif;">3</span></span></span> </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">، <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">4</span></span></span></b></span></span></b></span></span></b></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> ....</span></span> </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">،</span></b></span></span></b></span></span></b></span></span></b></span></span></b></span></span></b><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"> ن</span></b></b></span><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"> </span></b></span></span></b></span></span></b></span></span></b></span></span></b></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="background: #ffffff;">− 1</span></b></span></span></span></b></span></span></span></b></span></span></b></b></span></span></b></span></span></b></span></span></b></span></span></b></span></span></b></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;">}</span></b></div>
<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b><span style="color: red;">مثال</span></b></span></span></span></span></div>
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<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">ص\ <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">2</span></span> ص = </span></b></b></span></span><b><span style="font-family: "trebuchet ms" , sans-serif;">{<span style="font-family: "trebuchet ms" , sans-serif;">0 </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> 1</span></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;">}</span></b><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;"> </span></b></b></span></div>
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<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><b><span style="font-family: "trebuchet ms" , sans-serif;">ص\ <span style="font-family: "trebuchet ms" , sans-serif;">7</span> ص = </span></b></b></span><b><span style="font-family: "trebuchet ms" , sans-serif;">{<span style="font-family: "trebuchet ms" , sans-serif;">0 </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> 1</span></span> </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">، </span></b></span></span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">2</span> </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> <span style="font-family: "trebuchet ms" , sans-serif;">3</span></span></span> </span></b></span></span></b></span></span></b></span><b><b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">، <span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;">4</span></span></span></b></span></span></b></b></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> </span>،<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"> <span style="font-family: "trebuchet ms" , sans-serif;">5</span></span></span> </span></b><b><span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "trebuchet ms" , sans-serif;"><b><span style="font-family: "trebuchet ms" , sans-serif;">، 6</span></b></span></span></b></span></span></b></span></span></b></span></span></b></span>}</span></b><br />
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<span style="font-family: "trebuchet ms" , sans-serif;"><span style="font-family: "simplified arabic";"><span style="font-size: small;"><b>alfarjimohammed@gmail.com</b></span></span></span><br />
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