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<title>Complex Plane</title>
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<meta name="description" content="A plane for complex numbers!" />
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Complex Plane</h1>
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<table border="0" align="center">
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<tr>
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<td><img src="images/complex-plane.jpg" width="250" height="116" alt="complex plane (flying type)" /></td>
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<td>No, <b>not</b> that complex plane ...</td>
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</tr>
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<tr>
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<td><span class="center larger">... <b>this</b> complex plane:</span></td>
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<td><p class="center"><img src="images/complex-plane.svg" alt="complex plane (math type)" /></p>
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<p class="center">A <b>plane</b> for <b>complex</b> numbers!</p></td>
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</tr>
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</table>
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<p class="center"><i>(Also called an "Argand Diagram")</i></p>
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<h2>Real and Imaginary make Complex</h2>
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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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<p>When we square a Real Number we get a positive (or zero) result:</p>
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<p class="center"><b>2<sup>2</sup> = 2 × 2 = 4</b><br>
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<b>1<sup>2</sup> = 1 × 1 = 1</b><br>
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<b>0<sup>2</sup> = 0 × 0 = 0</b></p>
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<p>What can we square to get −1?</p>
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<p class="center"><span class="large"><b>?</b></span><b><sup>2</sup> = −1</b></p>
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<p>Squaring −1 does not work because <a href="../multiplying-negatives.html">multiplying negatives gives a positive</a>: (−1) × (−1) = +1, and no other Real Number works either.</p>
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<p>So it seems that mathematics is incomplete ...</p>
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<p class="center">... but we can fill the gap by <i><b>imagining</b></i> there is a number that, when multiplied by itself, gives −1<br>
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(call it <b>i</b> for imaginary):</p>
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<p class="center large"><b>i<sup>2</sup> = −1</b></p>
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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 is negative" />.</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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<p>And together:</p>
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<div class="def">
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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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<p>Examples: 3.6 + 4<b>i</b>, −0.02 + 1.2<b>i</b>, 25 − 0.3<b>i</b>, 0 + 2<b>i</b></p>
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</div>
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<h2>Putting a Complex Number on a Plane</h2>
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<p>You may be familiar with the <a href="../number-line.html">number line</a>:</p>
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<p class="center"><img src="../numbers/images/number-line.svg" max-width="100%" alt="number line -10 to +10" /></p>
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<p>But where do we put a complex number like <span class="large">3+4<b>i</b></span> ?</p>
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<p>Let's have the real number line go left-right as usual, and have the <b>imaginary number line go up-and-down</b>:</p>
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<table border="0" align="center">
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<tr>
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<td><p>We can then plot a complex number like <b>3 + 4i</b> :</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><span class="center"><img src="images/complex-plane-3-4i.svg" alt="complex plane 3+4i" /></span></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td><p>And here is <span class="center"><b>4 - 2i</b></span> :</p>
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<ul>
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<li>4 units along (the real axis),</li>
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<li>and 2 units down (the imaginary axis).</li>
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</ul></td>
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<td valign="top"> </td>
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<td><img src="images/complex-plane-4-2i.svg" alt="complex plane 4-2i" /></td>
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</tr>
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</table><p> </p>
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<div class="def">
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<p>And that is the <b>complex plane</b>:</p>
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<ul>
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<li><b>complex</b> because it is a combination of real and imaginary,</li>
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<li><b>plane</b> because it is like a <a href="../geometry/plane.html">geometric plane</a> (2 dimensional).</li>
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</ul>
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</div>
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<h2>Whole New World</h2>
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<p>Now let's bring the <b>idea of a plane</b> (<a href="../data/cartesian-coordinates.html">Cartesian coordinates</a>, <a href="../polar-cartesian-coordinates.html">Polar coordinates</a>, <a href="vectors.html">Vectors</a> etc) to complex numbers.</p>
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<p>It will open up a whole new world of numbers that are more complete and elegant, as you will see.</p>
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<h2>Complex Number as a Vector</h2>
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<p>We can think of a complex number as a <a href="vectors.html">vector</a>.</p>
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<p class="center"><img src="images/vector.gif" width="142" height="81" alt="vector" /><br>
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This is a vector. <br>
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It has magnitude (length) and direction.</p>
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<table border="0" align="center">
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<tr>
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<td><span class="center">And here is the complex number <b>3 + 4i</b></span>
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<p class="center"><b>as a Vector</b>:</p></td>
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<td width="30" valign="top"> </td>
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<td><img src="images/complex-plane-3-4i-vector.svg" alt="complex plane 3+4i vector" /></td>
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</tr>
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</table>
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<h2>Adding</h2>
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<p>You can add complex numbers as vectors, too:</p>
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<table border="0" align="center">
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<tr>
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<td><p>To add the complex numbers <b>3 + 5i</b> and <b>4 − 3i</b> :</p>
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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>separately, like this:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">(3 + 5<b>i</b>) + (4 − 3<b>i</b>) =</span></span><span class="right">(3 + 4)</span><span class="right">+ (5 − 3)<b>i</b></span></div>
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<div class="row"><span class="left">=</span><span class="right">7</span><span class="right">+ 2<b>i</b></span></div>
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</div>
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</td>
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<td valign="top"> </td>
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<td><img src="images/complex-plane-vector-add.svg" alt="complex plane vector addition" /></td>
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</tr>
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</table>
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<h2>Polar Form</h2>
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<table border="0" align="center">
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<tr>
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<td align="right">Let's use <b>3 + 4i </b> again:</td>
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<td width="30" valign="top"> </td>
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<td><img src="images/complex-plane-3-4i-vector.svg" alt="complex plane 3+4i vector" /></td>
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</tr>
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<tr>
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<td align="right"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="right"><p>Here it is <b>in polar form:</b></p></td>
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<td valign="top"> </td>
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<td><img src="images/complex-plane-3-4i-polar.svg" alt="complex plane 3-4i is polar 5 at 0.927" /></td>
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</tr>
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</table>
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<p>So the complex number <b>3 + 4i</b> can also be shown as distance (5) and angle (0.927 radians).</p>
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<p>Let's see how to convert from one form to the other using <a href="../polar-cartesian-coordinates.html">Cartesian to Polar conversion</a>:</p>
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<div class="example">
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<h3>Example: the number <span class="center"><b>3 + 4i</b></span></h3>
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<p>From <b>3 + 4i</b> :</p>
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<ul>
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<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>
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<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>
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</ul>
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<p>And we get distance (5) and angle (0.927 radians)</p>
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<p>Back again:</p>
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<ul>
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<li><b>x = r × cos( <i>θ</i> )</b> = 5 × cos( 0.927 ) = 5 × 0.6002... = <b>3</b> (close enough)</li>
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<li><b>y = r × sin(<i> θ</i> )</b> = 5 × sin( 0.927 ) = 5 × 0.7998... = <b>4</b> (close enough)</li>
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</ul>
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<p>And distance 5 and angle 0.927 becomes 3 and 4 again</p></div>
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<p>In fact a common way to write a complex number in Polar form is</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="center">x + <b>i</b>y =</span></span><span class="right"><span class="center">r cos <i>θ</i> + <b>i</b> r sin <i>θ</i></span></span></div>
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<div class="row"><span class="left">=</span><span class="right"><span class="center"> r(cos <i>θ</i> + <b>i</b> sin <i>θ</i>)</span></span></div>
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</div>
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<p>And "cos <i>θ</i> + <b>i</b> sin <i>θ</i>" is often shortened to "cis <i>θ</i>", so:</p>
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<p class="center larger"><b>x + iy</b> = <b>r cis <i>θ</i></b></p>
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<p class="words"><b>cis</b> is just shorthand for <b>cos <i>θ</i> + i sin <i>θ</i></b></p>
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So we can write:
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<p class="center larger"><b>3 + 4i</b> = <b>5 cis 0.927</b></p>
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<p>In some subjects, like electronics, "cis" is used a lot!</p>
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<h2>Summary</h2>
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<ul class="larger">
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<li>The complex plane is a plane with:
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<ul>
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<li>real numbers running left-right and</li>
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<li>imaginary numbers running up-down.</li>
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</ul></li>
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<li>To convert from Cartesian to Polar Form:
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<ul>
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<li><b>r = √(x<sup>2</sup> + y<sup>2</sup>)</b></li>
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<li><b><i>θ</i> = tan<sup>-1</sup> ( y / x )</b></li>
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</ul></li>
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<li>To convert from Polar to Cartesian Form:
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<ul>
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<li><b>x = r × cos( <i>θ</i> )</b></li>
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<li><b>y = r × sin(<i> θ</i> )</b></li>
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</ul></li>
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<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>
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</ul>
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<p>Next ... learn about <span class="center"><a href="complex-number-multiply.html">Complex Number Multiplication</a></span>.</p>
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<p> </p>
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<div class="questions">
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<script>getQ(7994, 7995, 7996, 7997, 7998, 7999, 8000, 8001, 8002, 8003);</script>
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</div>
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<div class="related">
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<a href="../numbers/complex-numbers.html">Complex Numbers</a>
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<a href="index.html">Algebra Index</a>
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Copyright © 2020 MathsIsFun.com
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