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<h1 align="center">Euler's Formula for Complex Numbers</h1>
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<p align="center"><i>(There is another "<a href="../geometry/eulers-formula.html">Euler's Formula</a>" about Geometry<b></b>,<br>
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this page is about the one used in Complex Numbers)</i></p>
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<p>First, you may have seen the famous "Euler's Identity":</p>
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<p class="large center"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> + 1 = 0</p>
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<p>It seems absolutely magical that such a neat equation combines:</p>
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<div class="bigul">
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<ul>
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<li><b><i>e</i></b> (<a href="../numbers/e-eulers-number.html">Euler's Number</a>)</li>
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<li><b><i>i</i></b> (the unit <a href="../numbers/imaginary-numbers.html">imaginary number</a>)</li>
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<li><span class="times">π</span> (the famous number <a href="../numbers/pi.html">pi</a> that turns up in many interesting areas)</li>
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<li>1 (the first counting number)</li>
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<li>0 (<a href="../numbers/zero.html">zero</a>)</li>
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</ul>
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</div>
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<p>And also has the basic operations of add, multiply, and an exponent too!</p>
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<p>But if you want to take an interesting trip through mathematics, you will discover how it comes about.</p>
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<p>Interested? Read on!</p>
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<h2>Discovery</h2>
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<p>It was around 1740, and mathematicians were interested in <a href="../numbers/imaginary-numbers.html">imaginary</a> numbers. </p>
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<div class="center80">
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<p>An imaginary number, when squared 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>This is normally impossible (try squaring some numbers, remembering that <a href="../multiplying-negatives.html">multiplying negatives gives a positive</a>, and see if you can get a negative result), but just imagine that you can do it!</p>
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<p>And we can have this special number (called <i><b>i</b></i> for imaginary):</p>
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<p class="center"><span class="large"><b><i>i</i><sup>2</sup> = −1</b></span></p>
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</div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/euler.jpg" alt="Leonhard Euler" height="160" width="150"></p>
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<p> </p>
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<p>Leonhard Euler was enjoying himself one day, playing with imaginary numbers (or so I imagine!), and he took this well known <a href="taylor-series.html">Taylor Series</a> (read about those, they are fascinating):</p>
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<p class="center larger"><i>e</i><sup>x</sup> = 1 + x + <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> + <span class="intbl"><em>x<sup>5</sup></em><strong>5!</strong></span> + ...</p>
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<div style="clear:both"></div>
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<p>And he put <i><b>i</b></i> into it:</p>
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<p class="center larger"><i>e</i><sup>ix</sup> = 1 + ix + <span class="intbl"><em>(ix)<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>(ix)<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>(ix)<sup>4</sup></em><strong>4!</strong></span> + <span class="intbl"><em>(ix)<sup>5</sup></em><strong>5!</strong></span> + ...</p>
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<p> </p>
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<p>And because <b>i<sup>2</sup> = −1</b>, it simplifies to:</p>
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<p class="center larger"><i>e</i><sup>ix</sup> = 1 + ix − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> − <span class="intbl"><em>ix<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> + <span class="intbl"><em>ix<sup>5</sup></em><strong>5!</strong></span> − ...</p>
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<p> </p>
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<p>Now group all the <i><b>i</b></i> terms at the end:</p>
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<p class="center larger"><i>e</i><sup>ix</sup> = ( 1 − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> − ... ) + i( x − <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>5</sup></em><strong>5!</strong></span> − ... )</p>
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<p> </p>
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<p>And here is the miracle ... the two groups are actually the Taylor Series for <b>cos</b> and <b>sin</b>:</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 class="larger" align="center"><b>cos x</b> = 1 − <span class="intbl"><em>x<sup>2</sup></em><strong>2!</strong></span> + <span class="intbl"><em>x<sup>4</sup></em><strong>4!</strong></span> − ...</td>
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</tr>
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<tr>
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<td class="larger" align="center"><b>sin x</b> = x − <span class="intbl"><em>x<sup>3</sup></em><strong>3!</strong></span> + <span class="intbl"><em>x<sup>5</sup></em><strong>5!</strong></span> − ...</td>
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</tr>
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</tbody></table>
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<p>And so it simplifies to:</p>
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<div class="def">
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<p class="large center"><i>e</i><sup><i><b>i</b></i>x</sup> = cos x + <i><b>i</b></i> sin x</p>
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</div>
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<p>He must have been so happy when he discovered this!</p>
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<p>And it is now called <b>Euler's Formula</b>.</p>
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<p> </p>
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<p>Let's give it a try:</p>
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<div class="example">
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<h3>Example: when x = 1.1</h3>
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<div class="so"><i>e</i><sup><i><b>i</b></i>x</sup> = cos x + <i><b>i</b></i> sin x</div>
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<div class="so"><i>e</i><sup><i><b>1.1i</b></i></sup> = cos 1.1 + <i><b>i</b></i> sin 1.1</div>
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<div class="so"><i>e</i><sup><i><b>1.1i</b></i></sup> = 0.45 + 0.89 <i><b>i</b></i> (to 2 decimals)</div>
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<p>Note: we are using <a href="../geometry/radians.html">radians</a>, not degrees.</p>
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</div>
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<p>The answer is a combination of a Real and an Imaginary Number, which together is called a <a href="../numbers/complex-numbers.html">Complex Number</a>.</p>
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<p>We can plot such a number on the <a href="complex-plane.html">complex plane</a> (the real numbers go left-right, and the imaginary numbers go up-down):</p>
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<p class="center"><img src="images/complex-plane-45-89.svg" alt="graph real imaginary 0.45 + 0.89i"><br>
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Here we show the number <b>0.45 + 0.89 <i><b>i</b></i></b><br>
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<br>
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Which is the same as <b><i>e</i><sup><i>1.1i</i></sup></b> </p>
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<p>Let's plot some more!</p>
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<p class="center"><img src="images/complex-plane-many.svg" alt="graph real imaginary many e^ix values"></p>
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<h2>A Circle!</h2>
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<p>Yes, putting Euler's Formula on that graph produces a circle:</p>
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<p class="center"><img src="images/euler-formula-circle.svg" alt="e^ix = cos(x) + i sin(x) on circle"><i><br>
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<b>e</b></i><b><sup><i>i</i>x</sup></b> produces a circle of radius 1<br>
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</p>
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<p> </p>
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<p>And when we include a radius of <b>r</b> we can turn any point (such as <b>3 + 4i</b>) into <i><b>re</b></i><b><sup><i>i</i>x</sup></b> form by finding the correct value of <b>x</b> and <b> r</b>:</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>To turn <b>3 + 4i</b> into <i><b>re</b></i><b><sup><i>i</i>x</sup></b> form we do a <a href="../polar-cartesian-coordinates.html">Cartesian to Polar conversion</a>:</p>
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<ul>
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<li>r = √(3<sup>2</sup> + 4<sup>2</sup>) = √(9+16) = √25 = <b>5</b></li>
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<li>x = tan<sup>-1</sup> ( 4 / 3 ) =<b> 0.927</b> (to 3 decimals)</li>
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</ul>
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<p class="larger"> </p>
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<p class="larger">So <b>3 + 4i</b> can also be <b>5<i>e</i><sup>0.927 <i>i</i></sup></b></p>
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<p class="center"><img src="images/euler-formula-3-4i.svg" alt="3+4i = 5 at 0.927"></p>
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</div>
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<h2>It is Another Form</h2>
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<p>It is basically another way of having a complex number. </p>
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<p>This turns out to very useful, as there are many cases (such as multiplication) where it is easier to use the <i><b>re</b></i><b><sup><i>i</i>x</sup></b> form rather than the <b>a+bi</b> form.</p>
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<h2>Plotting <i>e</i><sup><b>i</b><span class="times">π</span></sup></h2>
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<p>Lastly, when we calculate Euler's Formula for x = <span class="times">π</span> we get:</p>
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<div class="so"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> = cos <span class="times">π</span> + <i><b>i</b></i> sin <span class="times">π</span></div>
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<div class="so"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> = −1<span class="times"></span> + <i><b>i</b></i> × 0 <i> (because cos <span class="times">π</span> = −1 and sin <span class="times">π</span> = 0) </i></div>
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<div class="so"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> = −1</div>
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<p>And here is the point created by <span class="center larger"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup></span> (where our discussion began):</p>
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<p class="center larger"><img src="images/euler-formula-circle-pi.svg" alt="e^ipi = -1 + i on circle"></p>
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<p>And <i><b>e</b></i><b><sup><i>i</i><span class="times">π</span></sup> = −1</b> can be rearranged into:</p>
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<p class="large center"><i>e</i><sup><i><b>i</b></i><span class="times">π</span></sup> + 1 = 0 </p>
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<p class="center"><i>The famous Euler's Identity.</i></p>
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<p> </p>
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<div class="fun">
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<p>Footnote: in fact all these are true: </p>
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<p class="center larger"><img src="images/euler-formula-circle-idents.svg" alt="e^ipi = -1 + i on circle"></p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(8893, 8894, 8895, 8896, 8897, 8898, 8899, 8900, 8901, 8902);</script>
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</div>
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<div class="related">
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<a href="../numbers/e-eulers-number.html">e (Euler's Number)</a>
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<a href="index.html">Algebra
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Index</a> </div>
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