ضرب الأعداد المركبة و قسمتها

في هذا الدرس سنتعلم كيفية **ضرب وقسمة الأعداد المركبة**، والهدف في النهاية هو كتابة الناتج على الصورة القياسية، بحيث يتكون من **جزء حقيقي** وجزء تخيلي.

ضرب الأعداد المركبة

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="(2+4i)(9-3i)" data-tex="(2+4i)(9-3i)" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="15.138ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 6690.9 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="32" xlink:href="#MJXG-TEX-N-32"></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="mn" transform="translate(2111.4,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2611.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2956.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3345.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(3734.4,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(4456.7,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(5456.9,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(5956.9,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(6301.9,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g></g></svg></div>

فنوزع الضرب كما نفعل مع كثيرات الحدود:

- <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="2 \times 9 = 18" data-tex="2 \times 9 = 18" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.308ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 4556 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="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></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><use data-c="38" xlink:href="#MJXG-TEX-N-38" transform="translate(500,0)"></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="2 \times (-3i) = -6i" data-tex="2 \times (-3i) = -6i" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="16.018ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 7080 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="32" xlink:href="#MJXG-TEX-N-32"></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="mn" transform="translate(2889.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(3389.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(3734.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4401.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(5457,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(6235,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(6735,0)"><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="4i \times 9 = 36i" data-tex="4i \times 9 = 36i" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="11.869ex" height="1.717ex" role="img" focusable="false" viewBox="0 -677 5246 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="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1067.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(2067.4,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(2845.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3901,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(4901,0)"><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="4i \times (-3i) = -12i^2" data-tex="4i \times (-3i) = -12i^2" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.918ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 8361.6 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="mn"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1067.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mo" transform="translate(2067.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(2456.4,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(3234.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mi" transform="translate(3734.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4079.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4746.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(5802,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(6580,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="msup" transform="translate(7580,0)"><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>

ونتذكر 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>

إذن:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="-12i^2=-12(-1)=12" data-tex="-12i^2=-12(-1)=12" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="22.762ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 10060.7 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="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="msup" transform="translate(1778,0)"><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(2837.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(3893.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(4671.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(5671.1,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(6060.1,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(6838.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(7338.1,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(8004.9,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(9060.7,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg></div>

ثم نجمع الحدود المتشابهة:

- الجزء الحقيقي:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="18+12=30" data-tex="18+12=30" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="12.57ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 5556 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><use data-c="38" xlink:href="#MJXG-TEX-N-38" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(1222.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2222.4,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(3500.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(4556,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(500,0)"></use></g></g></g></svg></div>

- الجزء التخيلي:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="-6i+36i=30i" data-tex="-6i+36i=30i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="15.541ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 6869 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="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="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(1278,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1845.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2845.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(3845.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4468.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(5524,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(6524,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

فيكون الناتج النهائي:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="30+30i" data-tex="30+30i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="8.071ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3567.4 748" 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="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(1222.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2222.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="30" xlink:href="#MJXG-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(3222.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

قسمة الأعداد المركبة

عند قسمة عدد مركب على عدد مركب، فإن الهدف هو التخلص من الجزء التخيلي في المقام.

ولتحقيق ذلك نضرب البسط والمقام في **مرافق المقام**.

فإذا كان المقام على الصورة:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="a+bi" data-tex="a+bi" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.714ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2525.4 776" 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="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1751.4,0)"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(2180.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

فإن مرافقه هو:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="a-bi" data-tex="a-bi" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.714ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2525.4 776" 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="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1751.4,0)"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(2180.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

مثال

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="\frac{2i}{3+6i}" data-tex="\frac{2i}{3+6i}" style="display:block;margin:0 auto;vertical-align: -1.738ex;" xmlns="http://www.w3.org/2000/svg" width="6.804ex" height="4.774ex" role="img" focusable="false" viewBox="0 -1342 3007.4 2110" 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="mfrac"><g data-mml-node="mrow" transform="translate(1081.2,676)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></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 data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></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="36" xlink:href="#MJXG-TEX-N-36"></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><rect width="2767.4" height="60" x="120" y="220"></rect></g></g></g></svg></div>

فنضرب البسط والمقام في مرافق المقام:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="3-6i" data-tex="3-6i" 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="33" xlink:href="#MJXG-TEX-N-33"></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="36" xlink:href="#MJXG-TEX-N-36"></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>

فتصبح:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\frac{2i(3-6i)}{(3+6i)(3-6i)}" data-tex="\frac{2i(3-6i)}{(3+6i)(3-6i)}" style="display:block;margin:0 auto;vertical-align: -2.172ex;" xmlns="http://www.w3.org/2000/svg" width="16.133ex" height="5.475ex" role="img" focusable="false" viewBox="0 -1460 7130.9 2420" 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="mfrac"><g data-mml-node="mrow" transform="translate(1470.2,710)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(845,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(1234,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(1956.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(2956.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(3456.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(3801.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g><g data-mml-node="mrow" transform="translate(220,-710)"><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="33" xlink:href="#MJXG-TEX-N-33"></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="mn" transform="translate(2111.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(2611.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2956.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3345.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(3734.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(4456.7,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(5456.9,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(5956.9,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(6301.9,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g></g><rect width="6890.9" height="60" x="120" y="220"></rect></g></g></g></svg></div>

تبسيط البسط

نوزع 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="2i \times 3 = 6i" data-tex="2i \times 3 = 6i" style="display:inline-block;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.738ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 4746 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="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1067.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mn" transform="translate(2067.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(2845.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3901,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(4401,0)"><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="2i \times (-6i) = -12i^2" data-tex="2i \times (-6i) = -12i^2" style="display:inline-block;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.918ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 8361.6 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="mn"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(500,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(1067.2,0)"><use data-c="D7" xlink:href="#MJXG-TEX-N-D7"></use></g><g data-mml-node="mo" transform="translate(2067.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(2456.4,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(3234.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(3734.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4079.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(4746.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(5802,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(6580,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="msup" transform="translate(7580,0)"><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="-12i^2=12" data-tex="-12i^2=12" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="11.07ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 4893.1 965.9" 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="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><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="msup" transform="translate(1778,0)"><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(2837.3,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(3893.1,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g></g></g></svg></div>

إذن يصبح البسط:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="12+6i" data-tex="12+6i" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.94ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 3067.4 748" 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><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(1222.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2222.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(2722.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

تبسيط المقام

يمكن حساب المقام باستخدام فرق بين مربعين:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="(a+b)(a-b)=a^2-b^2" data-tex="(a+b)(a-b)=a^2-b^2" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="23.312ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 10304 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="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(389,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(1140.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(2140.4,0)"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(2569.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2958.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(3347.4,0)"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(4098.7,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(5098.9,0)"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mo" transform="translate(5527.9,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6194.7,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="msup" transform="translate(7250.4,0)"><g data-mml-node="mi"><use data-c="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mn" transform="translate(562,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(8438.2,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(9438.4,0)"><g data-mml-node="mi"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mn" transform="translate(462,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></g></svg></div>

فتصبح:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="3^2-(6i)^2" data-tex="3^2-(6i)^2" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.544ex" height="2.565ex" role="img" focusable="false" viewBox="0 -883.9 4218.6 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="msup"><g data-mml-node="mn"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use></g><g data-mml-node="mn" transform="translate(533,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1158.8,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mo" transform="translate(2159,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(2548,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(3048,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g><g data-mml-node="msup" transform="translate(3393,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g></g></g></g></svg></div>

أي:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="9-36i^2" data-tex="9-36i^2" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.928ex" height="2.185ex" role="img" focusable="false" viewBox="0 -883.9 3504 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="mn"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></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><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="msup" transform="translate(2722.4,0)"><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></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="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="9-36(-1)=9+36=45" data-tex="9-36(-1)=9+36=45" style="display:block;margin:0 auto;vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="25.267ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 11168 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="39" xlink:href="#MJXG-TEX-N-39"></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><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(2722.4,0)"><use data-c="28" xlink:href="#MJXG-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(3111.4,0)"><use data-c="2212" xlink:href="#MJXG-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(3889.4,0)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(4389.4,0)"><use data-c="29" xlink:href="#MJXG-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(5056.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(6112,0)"><use data-c="39" xlink:href="#MJXG-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(6834.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(7834.4,0)"><use data-c="33" xlink:href="#MJXG-TEX-N-33"></use><use data-c="36" xlink:href="#MJXG-TEX-N-36" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(9112.2,0)"><use data-c="3D" xlink:href="#MJXG-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10168,0)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g></g></g></svg></div>

ويمكن أيضًا الوصول إلى النتيجة نفسها بالتوزيع المباشر.

كتابة الناتج بالصورة القياسية

يصبح الناتج:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\frac{12+6i}{45}" data-tex="\frac{12+6i}{45}" style="display:block;margin:0 auto;vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="7.935ex" height="4.638ex" role="img" focusable="false" viewBox="0 -1342 3507.4 2050" 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="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(1222.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2222.4,0)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mi" transform="translate(2722.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g><g data-mml-node="mn" transform="translate(1253.7,-686)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g><rect width="3267.4" height="60" x="120" y="220"></rect></g></g></g></svg></div>

ثم نفصل الجزء الحقيقي عن الجزء التخيلي:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\frac{12}{45}+\frac{6}{45}i" data-tex="\frac{12}{45}+\frac{6}{45}i" style="display:block;margin:0 auto;vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="10.062ex" height="4.638ex" role="img" focusable="false" viewBox="0 -1342 4447.4 2050" 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="mfrac"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="32" xlink:href="#MJXG-TEX-N-32" transform="translate(500,0)"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g><rect width="1200" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1662.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(2662.4,0)"><g data-mml-node="mn" transform="translate(470,676)"><use data-c="36" xlink:href="#MJXG-TEX-N-36"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g><rect width="1200" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(4102.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

وبتبسيط الكسور بقسمة البسط والمقام على 3 نحصل على:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="\frac{4}{15}+\frac{2}{15}i" data-tex="\frac{4}{15}+\frac{2}{15}i" style="display:block;margin:0 auto;vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="10.062ex" height="4.663ex" role="img" focusable="false" viewBox="0 -1353 4447.4 2061" 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="mfrac"><g data-mml-node="mn" transform="translate(470,676)"><use data-c="34" xlink:href="#MJXG-TEX-N-34"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g><rect width="1200" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1662.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mfrac" transform="translate(2662.4,0)"><g data-mml-node="mn" transform="translate(470,676)"><use data-c="32" xlink:href="#MJXG-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="31" xlink:href="#MJXG-TEX-N-31"></use><use data-c="35" xlink:href="#MJXG-TEX-N-35" transform="translate(500,0)"></use></g><rect width="1200" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(4102.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

الخلاصة

- عند ضرب الأعداد المركبة نستخدم التوزيع، مع تذكر 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>

- بعد الضرب نجمع الأجزاء الحقيقية معًا، والأجزاء التخيلية معًا. - عند القسمة نضرب البسط والمقام في **مرافق المقام** للتخلص من الجزء التخيلي. - ثم نكتب الناتج في الصورة القياسية:

<div class="math-display" style="text-align:center;margin:10px 0;direction:ltr;overflow-x:auto;max-width:100%;"><svg aria-label="a+bi" data-tex="a+bi" style="display:block;margin:0 auto;vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="5.714ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 2525.4 776" 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="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(751.2,0)"><use data-c="2B" xlink:href="#MJXG-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(1751.4,0)"><use data-c="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(2180.4,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></div>

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="a" data-tex="a" style="display:inline-block;vertical-align: -0.023ex;" xmlns="http://www.w3.org/2000/svg" width="1.197ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 529 451" 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="1D44E" xlink:href="#MJXG-TEX-I-1D44E"></use></g></g></g></svg></span> 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="bi" data-tex="bi" style="display:inline-block;vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.751ex" height="1.595ex" role="img" focusable="false" viewBox="0 -694 774 705" 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="1D44F" xlink:href="#MJXG-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(429,0)"><use data-c="1D456" xlink:href="#MJXG-TEX-I-1D456"></use></g></g></g></svg></span> هو الجزء التخيلي.