الأعداد المركبة: جمعها وطرحها
الأعداد التخيلية والأعداد المركبة
الرياضيات — الأعداد المركبة
— الحل: تعريف الوحدة التخيلية i.
القاعدة العامة — لأي عدد موجب a
— الجزء الحقيقي = 1، معامل التخيلي = 5.
الطرح
الجمع
| المفهوم | القاعدة |
|---|---|
| الوحدة التخيلية | i = √−1 ، i² = −1 |
| جذر عدد سالب | √−a = i√a |
| العدد المركب | z = a + bi |
| الجمع والطرح | حقيقي مع حقيقي، تخيلي مع تخيلي |
| الضرب | توزيع عادي ثم استبدال i² بـ −1 |
— العدد المركب: جزء حقيقي a وجزء تخيلي bi على شكل a + bi.
— الجمع والطرح: نعمل على الأجزاء المتشابهة فقط — لا نجمع حقيقياً مع تخيلي.
— الضرب: قواعد الجبر العادي مع استبدال i² بـ −1 دائماً.
— ضرب عدد مركب في مرافقه ينتج عدداً حقيقياً بحتاً.
الأعداد المركبة — جمع وطرح — مسائل محلولة
الرياضيات — الأعداد المركبة
| العملية | القاعدة |
|---|---|
| الجمع | (a+bi)+(c+di) = (a+c)+(b+d)i |
| الطرح | (a+bi)−(c+di) = (a−c)+(b−d)i |
| تنبيه الطرح | وزّع الإشارة السالبة على القوسين أولاً |
— الطرح: وزّع الإشارة السالبة على القوسين قبل الجمع لتجنب الأخطاء.
— الناتج: دائماً على شكل a + bi حيث a و b أعداد حقيقية.
اختبار: الأعداد التخيلية والأعداد المركبة
1 / 10في هذا الدرس سنتعرف على **الأعداد التخيلية**، والتي ظهرت بسبب الجذر التربيعي للأعداد السالبة، لأننا لا نستطيع إيجاد عدد حقيقي إذا ضربناه في نفسه يعطينا عددًا سالبًا.
لماذا ظهرت الأعداد التخيلية؟
نعرف أن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{9} = 3" data-tex="\sqrt{9} = 3" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.209ex" height="2.442ex" role="img" focusable="false" viewBox="0 -997.3 3186.6 1079.3" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,137.2)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="877.2"></rect></g><g data-mml-node="mo" transform="translate(1630.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2686.6,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g></g></g></svg></div>
لكن إذا كان لدينا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{-9}" data-tex="\sqrt{-9}" style="display:block;margin:0 auto;vertical-align: -0.21ex;" xmlns="http://www.w3.org/2000/svg" width="4.821ex" height="2.398ex" role="img" focusable="false" viewBox="0 -967.3 2131 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="1278" height="60" x="853" y="847.3"></rect></g></g></g></svg></div>
فلا يوجد عدد حقيقي مربعه يساوي **سالب 9**.
لذلك استُخدم الرمز **<span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span>**، وهو اختصار لكلمة **Imaginary**، وعُرّف بحيث:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="i^2 = -1" data-tex="i^2 = -1" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.677ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 3393.1 965.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1059.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2115.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2893.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></div>
تبسيط الجذور السالبة
إذا كان لدينا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{-9}" data-tex="\sqrt{-9}" style="display:block;margin:0 auto;vertical-align: -0.21ex;" xmlns="http://www.w3.org/2000/svg" width="4.821ex" height="2.398ex" role="img" focusable="false" viewBox="0 -967.3 2131 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="1278" height="60" x="853" y="847.3"></rect></g></g></g></svg></div>
فنكتبها على الصورة:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{-1 \times 9}" data-tex="\sqrt{-1 \times 9}" style="display:block;margin:0 auto;vertical-align: -0.21ex;" xmlns="http://www.w3.org/2000/svg" width="8.718ex" height="2.398ex" role="img" focusable="false" viewBox="0 -967.3 3853.4 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1500.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(2500.4,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="3000.4" height="60" x="853" y="847.3"></rect></g></g></g></svg></div>
ثم نفصل الجذر إلى:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{-1} \times \sqrt{9}" data-tex="\sqrt{-1} \times \sqrt{9}" style="display:block;margin:0 auto;vertical-align: -0.21ex;" xmlns="http://www.w3.org/2000/svg" width="10.648ex" height="2.466ex" role="img" focusable="false" viewBox="0 -997.3 4706.4 1090" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="1278" height="60" x="853" y="847.3"></rect></g><g data-mml-node="mo" transform="translate(2353.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="msqrt" transform="translate(3353.4,0)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,137.2)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="877.2"></rect></g></g></g></svg></div>
وبما أن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{-1}=i" data-tex="\sqrt{-1}=i" style="display:block;margin:0 auto;vertical-align: -0.21ex;" xmlns="http://www.w3.org/2000/svg" width="8.619ex" height="2.398ex" role="img" focusable="false" viewBox="0 -967.3 3809.6 1060" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="1278" height="60" x="853" y="847.3"></rect></g><g data-mml-node="mo" transform="translate(2408.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3464.6,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
و:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\sqrt{9}=3" data-tex="\sqrt{9}=3" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.209ex" height="2.442ex" role="img" focusable="false" viewBox="0 -997.3 3186.6 1079.3" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msqrt"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g></g><g data-mml-node="mo" transform="translate(0,137.2)"><use data-c="221A" xlink:href="#MJXG-TEX-N-221A"></use></g><rect width="500" height="60" x="853" y="877.2"></rect></g><g data-mml-node="mo" transform="translate(1630.8,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2686.6,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g></g></g></svg></div>
فإن الناتج يصبح:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="3i" data-tex="3i" style="display:block;margin:0 auto;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.912ex" height="1.554ex" role="img" focusable="false" viewBox="0 -665 845 687" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
لماذا نحتفظ بالرمز <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span>؟
نترك الرمز <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span> في الناتج، لأن العمليات الحسابية اللاحقة قد تعيدنا إلى الأعداد الحقيقية.
فعلى سبيل المثال:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="i^2=-1" data-tex="i^2=-1" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.677ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 3393.1 965.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1059.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2115.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2893.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></div>
وبذلك قد يختفي الجزء التخيلي بعد إجراء بعض العمليات.
مثال على الضرب
إذا كان لدينا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="(1+i)(1-i)" data-tex="(1+i)(1-i)" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="12.875ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 5690.9 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(1111.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2111.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2456.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2845.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(3234.4,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(3956.7,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(4956.9,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(5301.9,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g></g></svg></div>
فنوزع الضرب:
- <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="1 \times 1 = 1" data-tex="1 \times 1 = 1" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="9.176ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 4056 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2500.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3556,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="1 \times (-i) = -i" data-tex="1 \times (-i) = -i" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="13.756ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6080 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mo" transform="translate(1722.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(2111.4,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(2889.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(3234.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3901.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4957,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(5735,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i \times 1 = i" data-tex="i \times 1 = i" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="8.475ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3746 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(567.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1567.4,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2345.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3401,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span> - <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i \times (-i) = -i^2" data-tex="i \times (-i) = -i^2" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="14.393ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 6361.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(567.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mo" transform="translate(1567.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(1956.4,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(2734.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(3079.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3746.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(4802,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(5580,0)"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,363) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></g></svg></span>
وبما أن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="i^2=-1" data-tex="i^2=-1" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.677ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 3393.1 965.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1059.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2115.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2893.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></div>
فإن:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="-i^2 = -(-1)=1" data-tex="-i^2 = -(-1)=1" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="17.106ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 7560.7 1133.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(778,0)"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1837.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2893.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(3671.1,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(4060.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(4838.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5338.1,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6004.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(7060.7,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></div>
فتصبح النتيجة:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="1-i+i+1=2" data-tex="1-i+i+1=2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="16.269ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 7190.9 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1722.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2289.7,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(3289.9,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(3857.1,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(4857.3,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(5635.1,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6690.9,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></svg></div>
ونلاحظ أن الناتج **عدد حقيقي**.
الأعداد المركبة
يدخلنا هذا إلى مفهوم **الأعداد المركبة**، وهي الأعداد التي تتكون من جزأين:
- جزء حقيقي. - وجزء تخيلي يحتوي على <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span>.
فمثلًا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="1+5i" data-tex="1+5i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.809ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2567.4 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
العدد **1** هو الجزء الحقيقي، و**<span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="5i" data-tex="5i" style="display:inline-block;vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.912ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 845 688" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span>** هو الجزء التخيلي.
جمع وطرح الأعداد المركبة
في عمليات الجمع والطرح نتعامل مع كل جزء على حدة:
- نجمع أو نطرح الأجزاء الحقيقية معًا. - ثم نجمع أو نطرح الأجزاء التخيلية معًا.
### مثال على الطرح
إذا كان لدينا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="(5+i)-(3+3i)" data-tex="(5+i)-(3+3i)" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="16.772ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7413.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(1111.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2111.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2456.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3067.7,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(4067.9,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(4456.9,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(5179.1,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6179.3,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(6679.3,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7024.3,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g></g></svg></div>
فنطرح الأجزاء الحقيقية:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="5-3=2" data-tex="5-3=2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="9.176ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 4056 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(2500.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3556,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></svg></div>
ثم نطرح الأجزاء التخيلية:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="i-3i=-2i" data-tex="i-3i=-2i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="12.147ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 5369 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(567.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1567.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(2067.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2690.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(3746,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(4524,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(5024,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
فيكون الناتج:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="2-2i" data-tex="2-2i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.809ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 2567.4 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
### مثال على الجمع
إذا كان لدينا:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="(5+i)+(3+3i)" data-tex="(5+i)+(3+3i)" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="16.772ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7413.3 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(389,0)"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(1111.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2111.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2456.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3067.7,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mo" transform="translate(4067.9,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(4456.9,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(5179.1,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(6179.3,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(6679.3,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(7024.3,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g></g></svg></div>
نجمع الأجزاء الحقيقية:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="5+3=8" data-tex="5+3=8" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="9.176ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 4056 748" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="35" xlink:href="#MJXG-TEX-N-35"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(2500.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3556,0)"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g></g></g></svg></div>
ثم نجمع الأجزاء التخيلية:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="i+3i=4i" data-tex="i+3i=4i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.387ex" height="1.717ex" role="img" focusable="false" viewBox="0 -677 4591 759" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(567.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1567.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(2067.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2690.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3746,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(4246,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
فيكون الناتج:
<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="8+4i" data-tex="8+4i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.809ex" height="1.717ex" role="img" focusable="false" viewBox="0 -677 2567.4 759" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><use data-c="38" xlink:href="#MJXG-TEX-N-38"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>
الخلاصة
- ظهرت الأعداد التخيلية بسبب الجذر التربيعي للأعداد السالبة. - يرمز للعدد التخيلي بالرمز **<span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span>**، حيث <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i^2=-1" data-tex="i^2=-1" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.677ex" height="2.072ex" role="img" focusable="false" viewBox="0 -833.9 3393.1 915.9" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mn" transform="translate(378,363) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1059.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2115.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2893.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g></g></g></svg></span>. - يمكن تبسيط الجذور السالبة بإخراج <span class="math-inline" dir="ltr" style="display:inline-block;direction:ltr;unicode-bidi:isolate;vertical-align:baseline;white-space:nowrap;margin:0 2px;"><svg aria-label="i" data-tex="i" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="0.781ex" height="1.52ex" role="img" focusable="false" viewBox="0 -661 345 672" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span> خارج الجذر. - العدد المركب يتكون من **جزء حقيقي** و**جزء تخيلي**. - عند جمع أو طرح الأعداد المركبة، نجمع أو نطرح الأجزاء الحقيقية معًا، ثم الأجزاء التخيلية معًا.
عن هذا الدرس
في هذا الدرس سنتعرف على الأعداد التخيلية، والتي ظهرت بسبب الجذر التربيعي للأعداد السالبة، لأننا لا نستطيع إيجاد عدد حقيقي إذا ضربناه في نفسه يعطينا عددًا… هذا الدرس ضمن الفصل الأول لمنهج الثانوية ثاني ثانوي في أكاديمية موسى.