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542 lines
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<div id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Complex Number Multiplication</h1>
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<p>A <a href="../numbers/complex-numbers.html">Complex Number</a> is a combination of a Real Number and an Imaginary Number:</p>
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<div class="def">
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<p>A <a href="../numbers/real-numbers.html">Real Number</a> is the type of number we use every day.</p>
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<p>Examples: 12.38, ½, 0, −2000</p>
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</div>
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<div class="def">
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<p>An <a href="../numbers/imaginary-numbers.html">Imaginary Number</a>, when <a href="../square-root.html">squared</a> gives a negative result:</p>
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<p class="center"><img src="../numbers/images/imaginary-squared.svg" alt="imaginary squared gives negative"></p>
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<p>The "unit" imaginary number when squared equals −1</p>
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<p class="center"><span class="large">i<sup>2</sup> = <b>−1</b></span></p>
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<p>Examples: 5<b>i</b>, −3.6<b>i</b>, <b>i</b>/2, 500<b>i</b></p>
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</div>
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<div class="example">
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<h3>Examples of Complex Numbers:</h3>
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<div class="simple">
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<table style="margin:auto">
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<tbody>
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<tr>
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<td class="center">3.6 + 4<b>i</b></td>
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<td><br></td>
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<td class="center">(real part is 3.6, imaginary part is 4<b>i</b>)</td></tr>
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<tr>
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<td class="center">−0.02 + 1.2<b>i</b> </td>
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<td><br></td>
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<td class="center">(real part is −0.02, imaginary part is 1.2<b></b><b>i</b>)</td></tr>
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<tr>
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<td class="center">25 − 0.3<b>i</b></td>
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<td><br></td>
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<td class="center">(real part is 25, imaginary part is −0.3<b>i</b>)</td></tr>
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</tbody></table>
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</div>
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<p>Either part can be zero:</p>
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<div class="simple">
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<table style="margin:auto">
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<tbody>
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<tr>
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<td class="center">0 + 2<b>i</b></td>
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<td><br>
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</td>
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<td class="center">(no real part, imaginary part is 2<b>i</b>)</td>
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<td><br></td>
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<td>same as 2<b>i</b></td></tr>
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<tr>
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<td class="center">4 + 0<b>i</b></td>
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<td><br></td>
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<td class="center">(real part is 4, no imaginary part)</td>
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<td><br></td>
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<td>same as 4<b></b></td></tr>
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</tbody></table>
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</div>
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</div>
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<h2>Multiplying </h2>
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<p>To multiply complex numbers:</p>
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<p class="center"><b>Each part of the first complex number </b> gets multiplied by<br>
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<b>each part of the second complex number</b></p>
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<p>Just use "FOIL", which stands for "<b>F</b>irsts, <b>O</b>uters, <b>I</b>nners, <b>L</b>asts" (see <a href="polynomials-multiplying.html">Binomial Multiplication</a> for more details):</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/foil-complex.svg" alt="FOIL: Firsts, Outers, Inners, Lasts"></td>
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<td>
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<ul>
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<li>Firsts: <b>a × c</b></li>
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<li>Outers: <b>a × di</b></li>
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<li>Inners: <b>bi × c</b></li>
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<li>Lasts: <b>bi × di</b></li>
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</ul></td>
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</tr>
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<tr align="center">
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<td colspan="2">
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<p class="larger">(a+b<b>i</b>)(c+d<b>i</b>) = ac + ad<b>i</b> + bc<b>i</b> + bd<b>i<sup>2</sup></b></p></td>
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</tr>
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</tbody></table>
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<p>Like this:</p>
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<div class="example">
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<h3>Example: (3 + 2<b>i</b>)(1 + 7<b>i</b><b></b>)</h3>
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<div class="tbl">
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<div class="row"><span class="left">(3 + 2<b>i</b>)(1 + 7<b>i</b><b></b>) </span><span class="right">= 3×1 + 3×7<b>i</b><b></b> + 2<b>i</b>×1+ 2<b>i</b><b></b>×7<b>i</b><b></b></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 3 + 21<b>i</b> + 2<b>i</b> + 14<b>i</b><b></b><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 3 + 21<b>i</b> + 2<b>i</b><b></b> − 14 <i>(because </i><b>i</b><i><sup>2</sup> = −1)</i></span></div>
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<div class="row"><span class="left"> </span><span class="right">= −11 + 23<b>i</b><b></b></span></div>
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</div>
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</div>
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<p>Here is another example:</p>
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<div class="example">
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<h3>Example: (1 + <b>i</b>)<sup>2</sup></h3>
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<div class="tbl">
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<div class="row"><span class="left">(1 + <b>i</b>)<sup>2</sup></span><span class="right">= (1 + <b>i</b>)(1 + <b>i</b>)</span></div>
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<div class="row"><span class="left"></span><span class="right">= 1×1 + 1×<b>i</b> + 1×<b>i</b> + <b>i</b><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 1 + 2<b>i</b> − 1 <i>(because <b>i</b><sup>2</sup> = −1)</i></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 0 + 2<b>i</b></span></div>
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</div> </div>
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<h3>But There is a Quicker Way!</h3>
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<p>Use this rule:</p>
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<p class="center"><span class="larger">(a+b<b>i</b>)(c+d<b>i</b>) = (ac−bd) + (ad+bc)<b>i</b></span></p>
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<div class="example">
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<h3>Example:</h3>
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<div class="tbl">
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<div class="row"><span class="left">(3 + 2<b>i</b>)(1 + 7<b>i</b>) =</span> <span class="right">(3×1 − 2×7) + (3×7 + 2×1)<b>i</b> </span></div>
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<div class="row"><span class="left">=</span> <span class="right">−11 + 23<b>i</b></span></div>
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</div>
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</div>
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<h3>Why Does That Rule Work?</h3>
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<p>It is just the "FOIL" method after a little work:</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td>(a+b<b>i</b>)(c+d<b>i</b>)</td>
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<td align="center" width="20">=</td>
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<td>ac + ad<b>i</b> + bc<b>i</b> + bd<b>i</b><sup>2</sup></td>
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<td width="20"> </td>
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<td>FOIL method</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center">=</td>
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<td>ac + ad<b>i</b> + bc<b>i</b> − bd</td>
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<td width="20"> </td>
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<td>(because <b>i</b><sup>2</sup>=−1)</td>
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</tr>
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<tr>
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<td> </td>
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<td align="center">=</td>
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<td>(ac − bd) + (ad + bc)<b>i</b></td>
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<td width="20"> </td>
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<td>(gathering like terms)</td>
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</tr>
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</tbody></table>
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<p class="center">And there you have the <span class="larger">(ac − bd) + (ad + bc)<b>i</b></span> pattern.</p>
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<p>This rule is certainly faster, but if you forget it, just remember the FOIL method.</p>
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<p> </p>
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<p class="center"><b>Now let's see what multiplication looks like on the Complex Plane.</b></p>
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<h2>The Complex Plane</h2>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><span class="center larger">This is the <a href="complex-plane.html">complex plane</a>:</span></td>
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<td>
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<p class="center"><img src="images/complex-plane.svg" alt="complex plane"></p>
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<p class="center">It is a <b>plane</b> for <b>complex</b> numbers!</p></td>
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</tr>
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</tbody></table>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><span class="center">We can plot a complex number like <b>3 + 4i</b></span> :
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<p>It is placed</p>
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<ul>
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<li>3 units along (the real axis),</li>
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<li>and 4 units up (the imaginary axis).</li>
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</ul></td>
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<td width="30" valign="top"> </td>
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<td><br>
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<img src="images/complex-plane-3-4i.svg" alt="complex plane 3+4i"></td>
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</tr>
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</tbody></table>
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<h2>Multiplying By i</h2>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td>
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<p>Let's multiply it by <b>i</b>:</p>
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<div class="so"> (3 + 4<b>i</b>) x <b>i</b> = 3<b>i</b> + 4<b>i</b><sup>2</sup></div>
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<p>Which simplifies to (because <b>i</b><sup>2</sup> = −1):</p>
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<div class="so">−4 + 3<b>i</b></div></td>
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<td width="30" valign="top"> </td>
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<td valign="bottom"><img src="images/complex-plane-vector-by-i.gif" alt="complex plane vector 3+4i by i = -4+31" height="120" width="164"></td>
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</tr>
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</tbody></table>
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<p>And here is the cool thing ... it's the same as <b>rotating by a right angle</b> (90° or <span class="times">π</span>/2)</p>
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<p>Was that just a weird coincidence?</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td>
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<p>Let's try multiplying by <b>i</b> again:</p>
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<div class="so">(−4 + 3<b>i</b>) x <b>i</b> = −4<b>i</b> + 3<b>i</b><sup>2</sup> = −3 − 4<b>i</b></div>
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<p>and <i>again:</i></p>
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<div class="so"><b><i></i></b>(−3 − 4<b>i</b>) x <b>i</b> = −3<b>i</b> − 4<b>i</b><sup>2</sup> = 4 − 3<b>i</b></div>
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<p>and <i>again:</i></p>
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<div class="so"><b><i></i></b>(4 − 3<b>i</b>) x <b>i</b> = 4<b>i</b> − 3<b>i</b><sup>2</sup> = 3<b><i> +</i></b> 4<b>i</b></div></td>
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<td width="30" valign="top"> </td>
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<td valign="bottom"><img src="images/complex-plane-vector-by-i4.svg" alt="complex plane vector by i 4 times is full rotation"></td>
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</tr>
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</tbody></table>
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<p>Well, isn't that stunning? Each time it rotates by a right angle, until it ends up where it started.</p>
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<p>Let's try it on the number 1:</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td class="larger" align="right">1 × <b>i</b></td>
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<td align="right"> </td>
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<td class="larger" align="left">= <b>i</b></td>
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</tr>
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<tr>
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<td class="larger" align="right"><b>i</b> × <b>i</b></td>
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<td align="right"> </td>
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<td class="larger" align="left">= −1</td>
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</tr>
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<tr>
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<td class="larger" align="right">−1 × <b>i</b></td>
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<td align="right"> </td>
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<td class="larger" align="left">= −<b>i</b></td>
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</tr>
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<tr>
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<td class="larger" align="right">−<b>i</b> × <b>i</b></td>
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<td align="right"> </td>
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<td class="larger" align="left">= 1</td>
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</tr>
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<tr>
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<td colspan="3" align="right">Back to 1 again!</td>
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</tr>
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</tbody></table></td>
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<td width="30"> </td>
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<td><span class="center"><img src="../numbers/images/i-cycle-complex.svg" alt="i cycle on complex plane"></span></td>
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</tr>
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</tbody></table>
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<p>Each time a right angle rotation.</p>
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<p><i>Choose your own complex number and try that for yourself, it is good practice.</i></p>
|
||
<p>Let's look more closely at angles now.</p>
|
||
<h2>Polar Form</h2>
|
||
<table align="center" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td>Our friendly complex number <b>3 + 4i</b> :</td>
|
||
<td valign="top"> </td>
|
||
<td><img src="images/complex-plane-3-4i-vector.svg" alt="complex plane 3+4i vector"></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>Here it is again, but</p>
|
||
<p class="center"><b>in polar form:</b><br>
|
||
(distance and angle)</p></td>
|
||
<td valign="top"> </td>
|
||
<td><img src="images/complex-plane-3-4i-polar.svg" alt="complex plane 3+4i polar"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>So the complex number <b>3 + 4i</b> can also be shown as distance (5) and angle (0.927 radians).</p>
|
||
<p>How do we do the conversions?</p>
|
||
<div class="example">
|
||
<h3>Example: the number <span class="center"><b>3 + 4i</b></span></h3>
|
||
<p>We can do a <a href="../polar-cartesian-coordinates.html">Cartesian to Polar conversion</a>:</p>
|
||
<ul>
|
||
<li><b>r = √(x<sup>2</sup> + y<sup>2</sup>)</b> = √(3<sup>2</sup> + 4<sup>2</sup>) = √25 = <b>5</b></li>
|
||
<li><b><i>θ</i> = tan<sup>-1</sup> (y/x)</b> = tan<sup>-1</sup> (4/3) =<b> 0.927</b> (to 3 decimals)</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>We can also take Polar coordinates and convert them to Cartesian coordinates:</p>
|
||
<ul>
|
||
<li><b>x = r × cos( <i>θ</i> )</b> = 5 × cos( 0.927 ) = 5 × 0.6002... = <b>3</b> (close enough)</li>
|
||
<li><b>y = r × sin(<i> θ</i> )</b> = 5 × sin( 0.927 ) = 5 × 0.7998... = <b>4</b> (close enough)</li>
|
||
</ul>
|
||
</div>
|
||
<p>In fact, a common way to write a complex number in Polar form is</p>
|
||
<table align="center" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td><span class="center">x + <b>i</b>y</span></td>
|
||
<td align="center" width="20">=</td>
|
||
<td><span class="center">r cos <i>θ</i> + <b>i</b> r sin <i>θ</i></span></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td align="center">=</td>
|
||
<td><span class="center"> r(cos <i>θ</i> + <b>i</b> sin <i>θ</i>)</span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>And "cos <i>θ</i> + <b>i</b> sin <i>θ</i>" is often shortened to "cis <i>θ</i>", so:</p>
|
||
<p class="center larger"><b>x + iy</b> = <b>r cis <i>θ</i></b></p>
|
||
<p class="words"><b>cis</b> is just shorthand for <b>cos <i>θ</i> + i sin <i>θ</i></b></p>
|
||
So we can write:
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<p class="center larger"><b>3 + 4i</b> = <b>5 cis 0.927</b></p>
|
||
<p>In some subjects, like electronics, "cis" is used a lot!</p>
|
||
<h2>Now For Some More Multiplication</h2>
|
||
<p>Let's try another multiplication:</p>
|
||
<div class="example">
|
||
<h3>Example: Multiply 1+i by 3+i</h3>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">(1+<b>i</b>) (3+<b>i</b>) =</span><span class="right">1(3+<b>i</b>) + <b>i</b>(3+<b>i</b>)</span></div>
|
||
<div class="row"><span class="left"> =</span><span class="right">3 + <b>i</b> + 3<b>i</b> + <b>i</b><sup>2</sup></span></div>
|
||
<div class="row"><span class="left"> =</span><span class="right">3 + 4<b>i</b> − 1</span></div>
|
||
<div class="row"><span class="left"> =</span><span class="right">2 + 4<b>i</b><b></b></span></div>
|
||
</div>
|
||
<p>And here is the result on the Complex Plane:</p>
|
||
<p class="center"><img src="images/complex-plane-1-i-3-i.gif" alt="complex plane 1+i, 3+i, 2+4i" height="176" width="166"></p>
|
||
</div>
|
||
<p>But it is more interesting to see those numbers in Polar Form:</p>
|
||
<div class="example">
|
||
<h3>Example: (continued)<span class="center"><b></b></span></h3>
|
||
<p>Convert <b>1+i</b> to Polar:</p>
|
||
<ul>
|
||
<li><b>r = </b> √(1<sup>2</sup> + 1<sup>2</sup>) = <b>√2</b></li>
|
||
<li><b><i>θ</i> = </b>tan<sup>-1</sup> (1/1) =<b> 0.785</b> (to 3 decimals)</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>Convert <b>3+i</b> to Polar:</p>
|
||
<ul>
|
||
<li><b>r = </b> √(3<sup>2</sup> + 1<sup>2</sup>) = <b>√10</b></li>
|
||
<li><b><i>θ</i> = </b>tan<sup>-1</sup> (1/3) =<b> 0.322</b> (to 3 decimals)</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>Convert <b>2+4i</b> to Polar:</p>
|
||
<ul>
|
||
<li><b>r = </b> √(2<sup>2</sup> + 4<sup>2</sup>) = <b>√20</b></li>
|
||
<li><b><i>θ</i> = </b>tan<sup>-1</sup> (4/2) =<b> 1.107</b> (to 3 decimals)</li>
|
||
</ul>
|
||
<p> </p>
|
||
<p>Have a look at the <b>r</b> values for a minute. Are they related somehow? <br>
|
||
And what about the <b><i>θ</i></b> values?</p>
|
||
<p>Here is that multiplication in one line (using "cis"):</p>
|
||
<p class="center larger">(√2 cis 0.785) <b>×</b> (√10 cis 0.322) <b>=</b> √20 cis 1.107</p>
|
||
<p>This is the interesting thing:</p>
|
||
<ul>
|
||
<li><b>√2 x √10 = √20</b><b></b></li>
|
||
<li><b> 0.785 + 0.322 = 1.107</b></li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center">The magnitudes get multiplied. <br>
|
||
And the angles get added.</p>
|
||
</div>
|
||
<div class="center80">
|
||
<p>When multiplying in Polar Form: <b>multiply the magnitudes, add the angles.</b></p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="fun">
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/complex-plane-i.gif" alt="complex plane i is right angle" height="108" width="111"></p>
|
||
<p>And that is why multiplying by <b>i</b> rotates by a right angle:</p>
|
||
<div class="so"><b>i</b> has a magnitude of 1 and forms a right angle on the complex plane</div>
|
||
</div>
|
||
<h2>Squaring</h2>
|
||
<p>To square a complex number, multiply it by itself:</p>
|
||
<ul>
|
||
<li>multiply the magnitudes: magnitude × magnitude = magnitude<sup>2</sup></li>
|
||
<li>add the angles: angle + angle = 2 , so we double them.</li>
|
||
</ul>
|
||
<p>Result: square the magnitudes, double the angle.</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/complex-plane-vector-1-21-squared.gif" alt="complex plane vector 1+2i squared is -3+4i" height="121" width="162"></p>
|
||
<h3>Example: Let us square 1 +<b><i></i></b> 2<b>i</b>:</h3>
|
||
<div class="so"><b><i></i></b>(1 +<b><i></i></b> 2<b>i</b>)(1 +<b><i></i></b> 2<b>i</b>) = 1 + 4<b>i</b> + 4<b>i</b><sup>2</sup> = −3<b><i> +</i></b> 4<b>i</b></div>
|
||
<p>On the diagram the angle looks to be (and is!) doubled.</p>
|
||
<p>Also:</p>
|
||
<ul>
|
||
<li>The magnitude of (1+2<b>i</b>) = √(1<sup>2</sup> + 2<sup>2</sup>) = √5</li>
|
||
<li>The magnitude of (−3+4<b>i</b>) = √(3<sup>2</sup> + 4<sup>2</sup>) = √25 = 5</li>
|
||
</ul>
|
||
<p>So the magnitude got squared, too.</p>
|
||
</div>
|
||
<p>In general, a complex number like:</p>
|
||
<p class="center larger">r(cos <i>θ</i> + <b>i</b> sin <i>θ</i>)</p>
|
||
<p>When <b>squared</b> becomes:</p>
|
||
<p class="center larger">r<sup>2</sup>(cos 2<i>θ</i> + <b>i</b> sin 2<i>θ</i>)</p>
|
||
<p class="center"><i>(the magnitude <b>r</b> gets squared and the angle <b>θ</b> gets doubled.)</i></p>
|
||
<p>Or in the shorter "cis" notation:</p>
|
||
<p class="def center large">(r cis <i>θ</i>)<sup>2</sup> = r<sup>2</sup> cis 2<i>θ</i></p>
|
||
<p class="center"> </p>
|
||
<p style="float:left; margin: 0 10px 15px 0;"><img src="images/de-moivre.jpg" alt="de moivre" height="116" width="100"></p>
|
||
<h2>De Moivre's Formula</h2>
|
||
<p>And the mathematician <i>Abraham de Moivre</i> found it works for any integer exponent <b>n</b>:</p>
|
||
<p class="center larger">[ r(cos <i>θ</i> + <b>i</b> sin <i>θ</i>) ]<sup>n</sup> = r<sup>n</sup>(cos n<i>θ</i> + <b>i</b> sin n<i>θ</i>)</p>
|
||
<p class="center"><i>(the magnitude becomes <b>r<sup>n</sup></b> the angle becomes <b>nθ</b>.)</i></p>
|
||
<p>Or in the shorter "cis" notation:</p>
|
||
<p class="center def large">(r cis <i>θ</i>)<sup>n</sup> = r<sup>n</sup> cis n<i>θ</i></p>
|
||
<p class="center"> </p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/complex-plane-0-8i.gif" alt="complex plane 0-8i" height="198" width="100"></p>
|
||
<h3>Example: What is (1+<b>i</b>)<sup>6</sup></h3>
|
||
<p>Convert <b>1+<b>i</b></b> to Polar:</p>
|
||
<ul>
|
||
<li><b>r = </b> √(1<sup>2</sup> + 1<sup>2</sup>) = <b>√2</b></li>
|
||
<li><b><i>θ</i> = </b>tan<sup>-1</sup> (1/1) =<b> <span class="times">π</span>/4</b></li>
|
||
</ul>
|
||
<p>In "cis" notation: 1+<b>i</b> = <span class="center">√2 cis <span class="times">π</span>/4</span></p>
|
||
<p>Now, with an exponent of 6, <b>r</b> becomes <b>r<sup>6</sup></b>, <i>θ</i> becomes <i>6θ</i><i></i>:</p>
|
||
<p class="center">(√2 cis <span class="times">π</span>/4)<sup>6</sup> = (√2)<sup>6</sup> cis 6<span class="times">π</span>/4 = 8 cis 3<span class="times">π</span>/2</p>
|
||
<p>The magnitude is now 8, and the angle is <span class="center">3<span class="times">π</span>/2</span> (=270°)</p>
|
||
<p>Which is also <b>0−8<b>i</b></b> (see diagram)</p>
|
||
</div>
|
||
<p> </p>
|
||
<h2>Summary</h2>
|
||
<ul class="larger">
|
||
<li>Use "FOIL" to multiply complex numbers,</li>
|
||
<li>Or use the formula:
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="center larger">(a+b<b>i</b>)(c+d<b>i</b>) = (ac−bd) + (ad+bc)<b>i</b></div></li>
|
||
<li>Or use polar form and then multiply the magnitudes and add the angles.</li>
|
||
<li>De Moivre's Formula can be used for integer exponents:
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
<div class="center larger">[ r(cos <i>θ</i> + <b>i</b> sin <i>θ</i>) ]<sup>n</sup> = r<sup>n</sup>(cos n<i>θ</i> + <b>i</b> sin n<i>θ</i>)</div></li>
|
||
<li>Polar form <b>r cos <i>θ</i> + i r sin <i>θ</i></b> is often shortened to<b> r cis <i>θ</i></b></li>
|
||
</ul>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>getQ(276, 8002, 8003, 8888, 8889, 8891, 277, 8004, 8005, 8890, 8892);</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="../numbers/complex-numbers.html">Complex Numbers</a>
|
||
<a href="index.html">Algebra Index</a>
|
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