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441 lines
21 KiB
HTML
441 lines
21 KiB
HTML
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Complex Numbers</h1>
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<p class="center"><img src="images/complex-example.svg" alt="complex example 7 + 3i" height="58" width="157"> <i><br>
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A Complex Number</i></p>
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<div class="def">
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<p class="center">A Complex Number is a combination of a<br>
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<b>Real Number</b> and an <b>Imaginary Number</b></p>
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</div>
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<p class="larger"> </p>
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<div class="words">
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<p class="large"><a href="real-numbers.html">Real Numbers</a> are numbers like:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td class="large" width="80">1</td>
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<td class="large" width="80">12.38</td>
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<td class="large" width="80">−0.8625</td>
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<td class="large" width="80">3/4</td>
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<td class="large" width="80">√2</td>
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<td class="large" width="80">1998</td>
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</tr>
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</tbody></table>
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</div>
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<p>Nearly any number you can think of is a Real Number!</p>
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</div>
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<div class="words">
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<p class="large"><a href="imaginary-numbers.html">Imaginary Numbers</a> when <b>squared</b> give a <b>negative</b> result.</p>
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<p>Normally this doesn't happen, because:</p>
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<ul>
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<li>when we <a href="../square-root.html">square</a> a positive number we get a positive result, and</li>
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<li>when we square a negative number we also get a positive result (because <a href="../multiplying-negatives.html">a negative times a negative gives a positive</a>), for example <b>−2 × −2 = +4</b></li>
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</ul>
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<p>But just imagine such numbers exist, because we want them.</p>
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</div>
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<p>Let's talk some more about imaginary numbers ...</p>
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<div class="def">
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<p>The "unit" imaginary number (like <span class="large">1</span> for Real Numbers) is <span class="large">i</span>, which is the square root of −1</p>
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<div class="center"><img src="images/imaginary-square-root.svg" alt="equals the square root of -1" height="44" width="135"></div>
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<p>Because when we square <span class="large">i</span> we get <span class="large">−1</span></p>
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<p class="large center">i<sup>2</sup> = −1</p>
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<p>Examples of Imaginary Numbers:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td class="large" width="80">3i</td>
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<td class="large" width="80">1.04i</td>
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<td class="large" width="80">−2.8i</td>
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<td class="large" width="80">3i/4</td>
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<td class="large" width="80">(√2)i</td>
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<td class="large" width="80">1998i</td>
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</tr>
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</tbody></table>
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</div>
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<p>And we keep that little "i" there to remind us we need to multiply by √−1</p></div>
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<h2>Complex Numbers</h2>
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<p>When we combine a Real Number and an Imaginary Number we get a <b>Complex Number</b>:</p>
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<p class="center"><img src="images/complex-number.svg" alt="Complex Number" height="85" width="297"></p>
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<h3>Examples:</h3>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<td class="large" width="100">1 + i</td>
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<td class="large" width="100">39 + 3i</td>
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<td class="large" width="100">0.8 − 2.2i</td>
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<td class="large" width="100">−2 + <font face="Times New Roman, Times, serif" size="+2">π</font>i</td>
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<td class="large" width="100">√2 + i/2</td>
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</tr>
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</tbody></table>
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</div>
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<p class="center"> </p>
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<div class="center80">
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<h3>Can a Number be a Combination of Two Numbers?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../images/fractions/pizza-3-8.svg" alt="pie 3/8" height="120" width="120"></p>
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<p>Can we make up a number from two other numbers? Sure we can!</p>
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<p>We do it with <a href="../fractions.html">fractions</a> all the time. The fraction <span class="frac-large"><sup>3</sup>/<sub>8</sub></span> is a number made up of a 3 and an 8. We know it means "3 of 8 equal parts".</p>
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<p>Well, a Complex Number is just <b>two numbers added together</b> (a Real and an Imaginary Number).</p>
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</div>
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<h2>Either Part Can Be Zero</h2>
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<p>So, a Complex Number has a real part and an imaginary part.</p>
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<p>But either part can be <b>0</b>, so all Real Numbers and Imaginary Numbers are also Complex Numbers.</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th width="120">Complex Number</th>
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<th width="120">Real Part</th>
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<th width="120">Imaginary Part</th>
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<th> </th>
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</tr>
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<tr style="text-align:center;">
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<td style="width:120px;">3 + 2<span class="large">i</span></td>
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<td style="width:120px;">3</td>
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<td style="width:120px;">2</td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:120px;">5</td>
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<td style="width:120px;">5</td>
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<td style="width:120px;"><b>0</b></td>
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<td>Purely Real</td>
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</tr>
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<tr style="text-align:center;">
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<td style="width:120px;">−6i</td>
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<td style="width:120px;"><b>0</b></td>
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<td style="width:120px;">−6</td>
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<td>Purely Imaginary</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Complicated?</h2>
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<div class="words">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/building-complex.jpg" alt="building complex" height="151" width="150"></p>
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<p>Complex does <b>not</b> mean complicated.</p>
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<p>It means the two types of numbers, real and imaginary, together form a <b>complex</b>, just like a building complex (buildings joined together).</p>
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</div>
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<h2>A Visual Explanation</h2>
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<p>You know how the number line goes <b>left-right</b>?</p>
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<p>Well let's have the imaginary numbers go <b>up-down</b>:</p>
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<p class="center"><img src="../algebra/images/complex-plane.svg" alt="complex plane" height="172" width="169"><br>
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<br>
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And we get the <a href="../algebra/complex-plane.html">Complex Plane</a></p>
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<p>A complex number can now be shown as a point:</p>
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<p class="center"><img src="../algebra/images/complex-plane-3-4i.svg" alt="complex plane 3+4i" height="179" width="179"><br>
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The complex number 3 + 4<b>i</b></p>
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<h2>Adding</h2>
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<p>To add two complex numbers we add each part separately:</p>
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<p class="center larger">(a+b<b>i</b>) + (c+d<b>i</b>) = (a+c) + (b+d)<b>i</b></p>
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<div class="example">
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<h3>Example: add the complex numbers <b>3 + 2<i>i</i></b> and <b>1 + 7<i>i</i></b></h3>
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<ul>
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<li>add the real numbers, and</li>
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<li>add the imaginary numbers:</li>
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</ul>
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<p class="center">(3 + 2<span class="large">i</span>) + (1 + 7<span class="large">i</span>)<br>
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= 3 + 1 + (2 + 7)<i><b>i</b></i><br>
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= 4 + 9<span class="large">i</span></p>
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</div>
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<p>Let's try another:</p>
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<div class="example">
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<h3>Example: add the complex numbers <b>3 + 5<i>i</i></b> and <b>4 − 3<i>i</i></b></h3>
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<p class="center">(3 + 5<i><b>i</b></i>) + (4 − 3<i><b>i</b></i>)<br>
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= 3 + 4 + (5 − 3)<i><b>i</b></i><br>
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= 7 + 2<i><b>i</b></i></p>
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<p>On the complex plane it is:</p>
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<p class="center"><img src="../algebra/images/complex-plane-vector-add.svg" alt="complex plane vector addition" height="187" width="236"></p>
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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="../algebra/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="../algebra/images/foil-complex.svg" alt="foil" height="133" width="238"></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 × d<i>i</i></b></li>
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<li>Inners: <b>b<i>i</i> × c</b></li>
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<li>Lasts: <b>b<i>i</i> × d<i>i</i></b></li>
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</ul></td>
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</tr>
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<tr style="text-align:center;">
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<td colspan="2">
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<p class="larger">(a+b<i><b>i</b></i>)(c+d<i><b>i</b></i>) = ac + ad<b><i>i</i></b> + bc<b><i>i</i></b> + bd<b><i>i</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<span class="large">i</span>)(1 + 7<span class="large">i</span>)</h3>
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<div class="tbl">
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<div class="row"><span class="left">(3 + 2<span class="large">i</span>)(1 + 7<span class="large">i</span>) </span><span class="rtlt">= 3×1 + 3×7<span class="large">i</span> + 2<span class="large">i</span>×1+ 2<span class="large">i</span>×7<span class="large">i</span></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= 3 + 21<span class="large">i</span> + 2<span class="large">i</span> + 14<span class="large">i</span><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= 3 + 21<span class="large">i</span> + 2<span class="large">i</span> − 14</span> (because i<sup>2</sup> = −1) </div>
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<div class="row"><span class="left"> </span><span class="rtlt">= −11 + 23<span class="large">i</span></span></div>
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</div>
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</div>
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<p>And this:</p>
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<div class="example">
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<h3>Example: (1 + <span class="large">i</span>)<sup>2</sup></h3>
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<div class="tbl">
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<div class="row"><span class="left"> (1 + <span class="large">i</span>)(1 + <span class="large">i</span>)</span><span class="rtlt">= 1×1 + 1×<span class="large">i</span> + 1×<span class="large">i</span> + <span class="large">i</span><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="rtlt">= 1 + 2<span class="large">i</span> − 1 </span> (because i<sup>2</sup> = −1)</div>
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<div class="row"><span class="left"> </span><span class="rtlt">= 0 + 2<span class="large">i</span></span></div>
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</div>
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</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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<p>Example: (3 + 2<span class="large">i</span>)(1 + 7<span class="large">i</span>) = (3×1 − 2×7) + (3×7 + 2×1)<span class="large">i</span> = −11 + 23<span class="large">i</span></p>
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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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<div class="tbl">
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<div class="row"><span class="left">(a+b<b>i</b>)(c+d<b>i</b>) =</span><span class="right">ac + ad<b>i</b> + bc<b>i</b> + bd<b>i</b><sup>2</sup></span> FOIL method</div>
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<div class="row"><span class="left"> =</span><span class="right">ac + ad<b>i</b> + bc<b>i</b> − bd</span> (because <b>i</b><sup>2 </sup>= −1)</div>
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<div class="row"><span class="left"> =</span><span class="right">(ac − bd) + (ad + bc)<b>i</b></span> (gathering like terms)</div>
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</div>
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<p>And there we have the <span class="larger">(ac − bd) + (ad + bc)<b>i</b> </span> pattern.</p>
|
||
<p>This rule is certainly faster, but if you forget it, just remember the FOIL method.</p>
|
||
|
||
<h3>Let us try <span class="large">i<sup>2</sup></span></h3>
|
||
<p>Just for fun, let's use the method to calculate<span class="large"> i<sup>2</sup></span></p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: i<sup>2</sup></h3>
|
||
<p>We can write <span class="large">i</span> with a real and imaginary part as <span class="large">0 + i</span></p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">i<sup>2</sup></span><span class="rtlt">= (0 + i)<sup>2</sup></span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= (0 + i)(0 + i)</span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= (0×0 − 1×1) + (0×1 + 1×0)<b>i</b></span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt">= −1 + 0<b>i</b></span></div>
|
||
<div class="row"><span class="left"> </span><span class="rtlt"><span class="larger">= <b>−1</b></span></span></div>
|
||
</div>
|
||
</div>
|
||
<p>And that agrees nicely with the definition that<span class="large"> i<sup>2 </sup>= −1</span></p>
|
||
<p>So it all works wonderfully!</p>
|
||
<p>Learn more at <a href="../algebra/complex-number-multiply.html">Complex Number Multiplication</a>.</p>
|
||
|
||
|
||
<h2>Conjugates</h2>
|
||
|
||
<p>We will need to know about conjugates in a minute!</p>
|
||
<p>A <a href="../algebra/conjugate.html">conjugate</a> is where we <b>change the sign in the middle</b> like this:</p>
|
||
<p class="center"><img src="images/complex-conjugate.svg" alt="Complex Conjugate" height="92" width="264"></p>
|
||
<p>A conjugate is often written with a bar over it:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example:</h3>
|
||
<p class="center"><span style="border-top:1px solid black;">5 − 3<b>i</b></span> = 5 + 3<b>i</b></p>
|
||
</div>
|
||
|
||
|
||
<h2>Dividing</h2>
|
||
|
||
<p>The conjugate is used to help complex division.</p>
|
||
<p>The trick is to <b>multiply both top and bottom</b> by the<b> conjugate of the bottom</b>.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Do this Division:</h3>
|
||
<p class="center larger"><span class="intbl"><em>2 + 3<b>i</b></em><strong><span class="larger">4 − 5<b>i</b></span></strong></span></p>
|
||
<p>Multiply top and bottom by the conjugate of <span class="larger">4 − 5<b>i</b></span> :<span class="larger"></span></p>
|
||
<p class="center large"><span class="intbl"><em>2 + 3<b>i</b></em><strong>4 − 5<b>i</b></strong></span>×<span class="intbl"><em>4 + 5<b>i</b></em><strong>4 + 5<b>i</b></strong></span> = <span class="intbl"><em>8 + 10<b>i</b> + 12<b>i</b> + 15<b>i</b><sup>2</sup></em><strong>16 + 20<b>i</b> − 20<b>i</b> − 25<b>i</b><sup>2</sup></strong></span></p>
|
||
<p>Now remember that <span class="larger">i<sup>2</sup> = −1</span>, so:</p>
|
||
<p class="center large">= <span class="intbl"><em>8 + 10<b>i</b> + 12<b>i</b> − 15</em><strong>16 + 20<b>i</b> − 20<b>i</b> + 25</strong></span></p>
|
||
<p>Add Like Terms (and notice how on the bottom <span class="larger">20<b>i</b> − 20<b>i</b></span> cancels out!):</p>
|
||
<p class="center large">= <span class="intbl"><em>−7 + 22<b>i</b></em><strong>41</strong></span></p>
|
||
<p>Lastly we should put the answer back into <span class="larger">a + b<b>i</b></span> form:</p>
|
||
<p class="center larger">= <span class="intbl"><em>−7 </em><strong><span class="larger">41</span></strong></span> + <span class="intbl"><em>22</em><strong><span class="larger">41</span></strong></span><b>i</b></p>
|
||
<p>DONE!</p>
|
||
</div>
|
||
<p>Yes, there is a bit of calculation to do. But it <b>can</b> be done.</p>
|
||
|
||
|
||
<h2>Multiplying By the Conjugate</h2>
|
||
|
||
<p>There is a faster way though.</p>
|
||
<p>In the previous example, what happened on the bottom was interesting:</p>
|
||
<p class="center large">(4 − 5<b>i</b>)(4 + 5<b>i</b>) = 16 + 20<b>i</b> − 20<b>i</b> − 25<b>i</b><sup>2</sup></p>
|
||
<p>The middle terms (20<b>i</b> − 20<b>i</b>) cancel out:</p>
|
||
<p class="center large">(4 − 5<b>i</b>)(4 + 5<b>i</b>) = 16 − 25<b>i</b><sup>2</sup></p>
|
||
<p>Also <span class="larger"><b>i</b><sup>2</sup> = −1</span> :</p>
|
||
<p class="center large">(4 − 5<b>i</b>)(4 + 5<b>i</b>) = 16 + 25</p>
|
||
<p>And 16 and 25 are (magically) squares of the 4 and 5:</p>
|
||
<p class="center large">(4 − 5<b>i</b>)(4 + 5<b>i</b>) = 4<sup>2</sup> + 5<sup>2</sup></p>
|
||
<p>Which is really quite a simple result. The general rule is:</p>
|
||
<div class="def">
|
||
<p class="center"><span class="larger">(a + b<b>i</b>)(a − b<b>i</b>) = a<sup>2</sup></span><span class="larger"> + b<sup>2</sup></span></p>
|
||
</div>
|
||
<p>That can save us time when we do division, like this:</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: Let's try this again</h3>
|
||
<p class="center larger"><span class="intbl"><em>2 + 3<b>i</b></em><strong><span class="larger">4 − 5<b>i</b></span></strong></span></p>
|
||
<p>Multiply top and bottom by the conjugate of <span class="larger">4 − 5<b>i</b></span> :<span class="larger"></span></p>
|
||
<p class="center large"><span class="intbl"><em>2 + 3<b>i</b></em><strong>4 − 5<b>i</b></strong></span> × <span class="intbl"><em>4 + 5<b>i</b></em><strong>4 + 5<b>i</b></strong></span> = <span class="intbl"><em>8 + 10<b>i</b> + 12<b>i</b> + 15<b>i</b><sup>2</sup></em><strong>16 + 25</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>−7 + 22<b>i</b></em><strong>41</strong></span></p>
|
||
<p>And then back into <span class="larger">a + b<b>i</b></span> form:</p>
|
||
<p class="center larger">= <span class="intbl"><em>−7 </em><strong><span class="larger">41</span></strong></span> + <span class="intbl"><em>22</em><strong><span class="larger">41</span></strong></span><b>i</b></p>
|
||
<p>DONE!</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Notation</h2>
|
||
|
||
<p>We often use <b>z</b> for a complex number. And <b>Re()</b> for the real part and <b>Im()</b> for the imaginary part, like this:</p>
|
||
<p class="center"><img src="images/complex-re-im.svg" alt="Complex Number Re() and Im()" height="85" width="180"></p>
|
||
<p>Which looks like this on the complex plane:</p>
|
||
<p class="center"><img src="../algebra/images/complex-plane-z-example.svg" alt="complex plane z example" height="187" width="236"></p>
|
||
<p> </p>
|
||
|
||
|
||
<h2>The Mandelbrot Set</h2>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/mandelbrot-set.jpg" alt="Mandelbrot Set" height="225" width="300"></td>
|
||
<td>
|
||
<p>The beautiful <a href="mandelbrot.html">Mandelbrot Set</a> (pictured here) is based on Complex Numbers.</p>
|
||
<p>It is a plot of what happens when we take the simple equation <b>z</b><sup>2</sup>+<b>c</b> (both complex numbers) and feed the result back into <b>z</b> time and time again.</p>
|
||
<p>The color shows how fast <b>z</b><sup>2</sup>+<b>c</b> grows, and black means it stays within a certain range.</p></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">
|
||
<p>Here is an image made by zooming into the Mandelbrot set</p></td>
|
||
<td><img src="images/mandelbroot-zoom-a.jpg" alt="Mandelbrot Set Zoomed In" height="224" width="300"></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;">And here is the center of the previous one zoomed in even further:</td>
|
||
<td><img src="images/mandelbroot-zoom-b.jpg" alt="Mandelbrot Set Zoomed In More" height="224" width="300"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
<div class="questions">440, 1070, 273, 1071, 1072, 443, 3991, 271, 3992, 3993</div>
|
||
|
||
<div class="related">
|
||
<a href="../sets/number-types.html">Common Number Sets</a>
|
||
<a href="../algebra/complex-plane.html">Complex Plane</a>
|
||
<a href="mandelbrot.html">Mandelbrot Set</a>
|
||
<a href="complex-number-calculator.html">Complex Number Calculator</a>
|
||
</div>
|
||
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