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327 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Imaginary Numbers</h1>
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<p> </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 style="text-align:center;"><span class="larger">An Imaginary Number, when <a href="../square-root.html">squared</a>, gives
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a <b>negative</b> result.</span></td>
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</tr>
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<tr>
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<td style="text-align:center;"><img src="images/imaginary-squared.svg" alt="imaginary squared => negative"></td>
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</tr>
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</tbody></table>
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<h3>Try</h3>
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<p>Let's try squaring some numbers to see if we can get a negative result:</p>
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<div class="example">
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<ul>
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<div class="bigul">
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<li>2 × 2 = <b>4</b></li>
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<li>(−2) × (−2) = <b>4</b> (because <a href="../multiplying-negatives.html">a negative times a negative gives a positive</a>)</li>
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<li>0 × 0 = <b>0</b></li>
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<li>0.1 × 0.1 = <b>0.01</b></li>
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</div>
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</ul>
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</div>
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<p>No luck! Always <b>positive</b>, or zero.</p>
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<p>It seems like we cannot multiply a number by itself to get a negative answer ...</p>
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<table align="center" width="85%" border="0">
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<tbody>
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<tr>
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<td><img src="../images/style/thought.gif" alt="thought" height="92" width="92"></td>
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<td>
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<p>... but <b>imagine</b> that there is such a number (call it <span class="large">i</span> for imaginary) that could do this:</p>
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<div class="def">
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<div class="center"><span class="large">i × i = <b>−1</b></span><br>
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</div>
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</div>
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<p>Would it be useful, and what could we do with it?</p>
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</td>
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</tr>
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</tbody></table>
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<p>Well, by taking the square root of both sides we get this:</p>
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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><img src="images/imaginary-square-root.svg" alt="equals the square root of -1"></td>
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</tr>
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<tr style="text-align:center;">
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<td>Which means that <span class="large">i</span> is the answer to the square root of −1.</td>
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</tr>
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</tbody></table>
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<p>Which is actually <b>very useful</b> because ...</p>
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<div class="center80">
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<p class="center larger">... by simply <b>accepting</b> that<b> i </b>exists we can solve things<br>
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that need the square root of a negative number.</p>
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</div>
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<p>Let us have a go:</p>
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<div class="example">
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<h3>Example: What is the square root of −9 ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">√(−9)</span><span class="right">= √(9 × −1)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= √(9) × √(−1)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 3 × √(−1)</span></div>
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<div class="row"><span class="left"> </span><span class="right">= 3<b>i</b></span></div>
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</div>
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<p>(see <a href="simplify-square-roots.html">how to simplify square roots</a>)</p>
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</div>
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<p>Hey! that was interesting! The square root of −9 is simply the square root of +9, <b>times <span class="large">i</span></b>.</p>
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<p>In general:</p>
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<div class="center80">
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<p class="center"><span class="large">√(−x) = <b>i</b>√x</span></p>
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</div>
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<p class="center larger">So long as we keep that little "i" there to remind us that we still<br>
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need to multiply by √−1 we are safe to continue with our solution!</p>
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<h2>Using <b>i</b></h2>
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<div class="example">
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<h3>Example: What is (5<b>i</b>)<sup>2</sup> ?</h3>
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<div class="tbl">
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<div class="row"><span class="left">(5<b>i</b>)<sup>2</sup></span><span class="right">= 5<b>i</b> × 5<b>i</b></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 5× 5× <b>i</b> × <b>i</b></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 25 × <b>i</b><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= 25 × −1</span></div>
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<div class="row"><span class="left"> </span><span class="right">= −25</span></div>
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</div>
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</div>
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<p>Interesting! We used an imaginary number (5<b>i</b>) and ended up with a real solution (−25).</p>
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<p>Imaginary numbers can help us solve some equations:</p>
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<div class="example">
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<h3>Example: Solve x<sup>2</sup> + 1 = 0</h3>
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<p>Using <a href="real-numbers.html">Real Numbers</a> there is no solution, but now we <b>can</b> solve it!</p>
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<p>Subtract 1 from both sides:</p>
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<div class="so"> x<sup>2</sup> = −1 </div>
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<p>Take the square root of both sides:</p>
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<div class="so"> x = ± √(−1) </div>
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<div class="so"> x = ± <b>i </b> </div>
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<p class="large">Answer: x = −i or +i</p>
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<p>Check:</p>
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<ul>
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<li>(−i)<sup>2</sup> + 1 = (−i)(−i) + 1 = +i<sup>2</sup> + 1 = −1 + 1 = 0</li>
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<li>(+i)<sup>2</sup> +1 = (+i)(+i) +1 = +i<sup>2</sup> +1 = −1 + 1 = 0</li>
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</ul>
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</div>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/ij.gif" alt="i and j" height="121" width="119"></p>
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<h2>Unit Imaginary Number</h2>
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<p>The square root of minus one <b>√(−1)</b> is the "unit" Imaginary Number, the equivalent of <b>1</b> for Real Numbers.</p>
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<p>In mathematics the symbol for √(−1) is <i><b><span class="large">i</span></b></i> for imaginary.</p>
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<p class="center fun">Can <b>you </b>take the square root of −1?<br>
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Well <b>i</b> can!</p>
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But in electronics they use <span class="large">j</span> (because "i" already means current, and the next letter after i is j).
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<h2>Examples of Imaginary Numbers</h2>
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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">i</td>
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<td class="large" width="80">12.38i</td>
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<td class="large" width="80">−i</td>
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<td class="large" width="80">3i/4</td>
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<td class="large" width="80">0.01i</td>
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<td class="large" width="80"><span class="times">π</span>i</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Imaginary Numbers are not <i>"Imaginary"</i></h2>
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<p>Imaginary Numbers were once thought to be <i>impossible</i>, and so they were called "Imaginary" (to make fun of them).</p>
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<p>But then people researched them more and discovered they were actually <b>useful</b> and <b>important</b> because they filled a gap in mathematics ... but the "imaginary" name has stuck.</p>
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<p>And that is also how the name "<a href="real-numbers.html">Real Numbers</a>" came about (real is not imaginary).</p>
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<h2>Imaginary Numbers are Useful</h2>
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<h3> </h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../algebra/images/complex-plane-vector-add.svg" alt="complex plane vector add"></p>
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<h3>Complex Numbers</h3>
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<p>Imaginary numbers become most useful when combined with real numbers to make <a href="complex-numbers.html">complex numbers</a> like <b>3+5i</b> or <b>6−4i</b></p>
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<div style="clear:both"></div>
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<h3>Spectrum Analyzer</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/spectrum-analyzer.gif" alt="spectrum analyzer" height="88" width="224"></p>
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<p>Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called "Fourier Transforms".</p>
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<p>In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.</p>
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<p>It is part of a subject called "Signal Processing".</p>
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<p> </p>
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<h3>Electricity</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/plug.gif" alt="plug" height="103" width="101"><br>
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<img src="images/sines-thumb.gif" alt="sine waves" height="56" width="120"></p>
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<p>AC (Alternating Current) Electricity changes between positive and negative in a <a href="../sine-graph-exercise.html">sine wave</a>.</p>
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<p>When we combine two AC currents they may not match properly, and it can be <b>very hard</b> to figure out the new current.</p>
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<p>But using complex numbers makes it a lot easier to do the calculations.</p>
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<p>And the result may have "Imaginary" current, but it can still hurt you!</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/mandelbroot-zoom-a.jpg" alt="Mandelbrot Set Zoomed In" height="224" width="300"></p>
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<h3> </h3>
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<h3>Mandelbrot Set</h3>
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<p>The beautiful <a href="mandelbrot.html">Mandelbrot Set</a> (part of it is pictured here) is based on Complex Numbers.</p>
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<div style="clear:both"></div>
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<h3>Quadratic Equation</h3>
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<p class="center"><img src="../algebra/images/quadratic-equation.svg" alt="Quadratic Equation"></p>
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<p class="center">The <a href="../algebra/quadratic-equation.html">Quadratic Equation</a>, which has many uses,<br>
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can give results that include imaginary numbers</p>
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<p>Also Science, Quantum mechanics and Relativity use complex numbers.</p>
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<h2>Interesting Property</h2>
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<p>The Unit Imaginary Number, <span class="large">i</span>, has an interesting property. It "cycles" through 4 different values each time we multiply:</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>
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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 class="larger" align="right">1 × <b>i</b></td>
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<td style="text-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 style="text-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 style="text-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 style="text-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 style="width:30px;"> </td>
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<td><img src="images/i-cycle.svg" alt="i cycle"></td>
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</tr>
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</tbody></table>
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<div style="clear:both"></div>
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<p>So we have this:</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 style="width:100px;"><span class="large">i</span> = <b>√−1</b></td>
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<td style="width:100px;"><span class="large">i<sup>2</sup></span> = <b>−1</b></td>
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<td style="width:100px;"><span class="large">i<sup>3</sup></span> = <b>−√−1</b></td>
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<td style="width:100px;"><span class="large">i<sup>4</sup></span> = <b>+1</b></td>
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</tr>
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<tr style="text-align:center;">
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<td><span class="large">i<sup>5</sup></span> = <b>√−1</b></td>
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<td><span class="large">i<sup>6</sup></span> = <b>−1</b></td>
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<td>...etc</td>
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<td> </td>
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</tr>
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</tbody>
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</table>
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</div>
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<div class="example">
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<h3>Example What is <span class="large">i</span><sup>10</sup> ?</h3>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">i</span><sup>10</sup></span><span class="right">= <span class="large">i</span><sup>4</sup> × <span class="large">i</span><sup>4</sup> × <span class="large">i</span><sup>2</sup></span></div>
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<div class="row"><span class="left"> </span><span class="right">= <span class="large">1</span> × <span class="large">1</span> × <span class="large">−1</span></span></div>
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<div class="row"><span class="left"> </span><span class="right">= <span class="large">−1</span></span></div>
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</div>
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</div>
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<div class="fun">
|
||
<p>And that leads us into another topic, the <a href="../algebra/complex-plane.html">complex plane</a>:</p>
|
||
<p class="center"><img src="images/i-cycle-complex.svg" alt="i cycle on complex plane"></p>
|
||
</div>
|
||
<h2>Conclusion</h2>
|
||
<p style="float:left; margin: 0 20px 5px 0;"><img src="images/imaginary-square-root.svg" alt="i = square root of -1"></p>
|
||
<p>The unit imaginary number, <span class="large">i</span>, equals the square root of minus 1</p>
|
||
<p>Imaginary Numbers are not "imaginary", they really exist and have many uses.</p>
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>
|
||
getQ(428,439,2233,438,2234,2235,2992,2993,3989,3990);
|
||
</script> </div>
|
||
|
||
<div class="related">
|
||
<a href="real-numbers.html">Real Numbers</a>
|
||
<a href="complex-numbers.html">Complex Numbers</a>
|
||
<a href="mandelbrot.html">Mandelbrot Set</a>
|
||
</div>
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