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<!DOCTYPE html>
<html data-require="math">
<head>
<meta charset="UTF-8">
<title>Imaginary unit powers</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="less-than-4" data-weight="2">
<div class="vars">
<var id="EXP">randRange(1, 4)</var>
<var id="SOLUTION">['1', 'i', '-1', '-i'][EXP % 4]</var>
</div>
<p class="question"><code>\large{i ^ {<var>EXP</var>} = ?}</code></p>
<div class="solution"><code><var>SOLUTION</var></code></div>
<ul class="choices" data-category="true">
<li><code>1</code></li>
<li><code>i</code></li>
<li><code>-1</code></li>
<li><code>-i</code></li>
</ul>
<div class="hints">
<p data-if="EXP === 1">
Anything to the first power is the number itself.
</p>
<p data-if="EXP >= 2">
The most important property of the imaginary unit <code>i</code> is
that <code>\blue{i ^ 2} = \pink{-1}</code>.
</p>
<p data-if="EXP === 3"><code>
\qquad \begin{eqnarray}
i ^ 3 &amp;=&amp; (\pink{i ^ 2}) \cdot i \\
&amp;=&amp; (\blue{-1}) \cdot i \\
&amp;=&amp; -i
\end{eqnarray}
</code></p>
<p data-if="EXP === 4"><code>
\qquad \begin{eqnarray}
i ^ 4 &amp;=&amp; (\pink{i ^ 2}) ^ 2 \\
&amp;=&amp; (\blue{-1}) ^ 2 \\
&amp;=&amp; 1
\end{eqnarray}
</code></p>
<p><code>\qquad i ^ <var>EXP</var> = <var>SOLUTION</var></code></p>
</div>
</div>
<div id="more-than-4" data-weight="5">
<div class="vars">
<var id="EXP">4 + randRange(1, 30)</var>
<var id="SOLUTION">['1', 'i', '-1', '-i'][EXP % 4]</var>
<var id="WHOLES">floor(EXP / 4)</var>
<var id="REMAINDER">EXP % 4</var>
<var id="REMAINDER_TEXT">i18n._("remainder")</var>
</div>
<p class="question"><code>\large{i ^ {<var>EXP</var>} = ?}</code></p>
<div class="solution"><code><var>SOLUTION</var></code></div>
<ul class="choices" data-category="true">
<li><code>1</code></li>
<li><code>i</code></li>
<li><code>-1</code></li>
<li><code>-i</code></li>
</ul>
<div class="hints">
<p>
The most important property of the imaginary unit <code>i</code> is:<br>
<code>\qquad \blue{i ^ 2} = \pink{-1}</code>
</p>
<p>
Therefore:<br>
<code>\qquad i ^ 4 = (\blue{i ^ 2}) ^ 2 = (\pink{-1}) ^ 2 = 1</code>
</p>
<p>So, we can simplify the expression by rewriting it in terms of <code>i^4</code>.</p>
<div>
<p>
Because <code><var>EXP</var> \div \blue{4} = \green{<var>WHOLES</var>}
<span data-if="REMAINDER">\text{ <var>REMAINDER_TEXT</var> } \red{<var>REMAINDER</var>}</span></code>,
</p>
<p><code>
\qquad \begin{eqnarray}
i ^ {<var>EXP</var>} &amp;=&amp; (i^\blue{4})^\green{<var>WHOLES</var>}
<span data-if="REMAINDER"> \cdot i^\red{<var>REMAINDER</var>}</span> \\
&amp;=&amp; (1)^\green{<var>WHOLES</var>}
<span data-if="REMAINDER">\cdot i^\red{<var>REMAINDER</var>}</span> \\
&amp;=&amp; <span data-if="REMAINDER">i^\red{<var>REMAINDER</var>}</span>
<span data-else="">1</span>
\end{eqnarray}
</code></p>
</div>
<p data-if="REMAINDER === 1">
Anything to the first power is the number itself.<br>
<code>\qquad i ^ \red{1} = i</code>
</p>
<p data-if="REMAINDER >= 2">
As stated above, <code>\blue{i ^ 2} = \pink{-1}</code>.
</p>
<p data-if="REMAINDER === 3">
<code>\qquad \begin{eqnarray}
i ^ \red{3} &amp;=&amp; (\blue{i ^ 2}) \cdot i \\
&amp;=&amp; (\pink{-1}) \cdot i \\
&amp;=&amp; -i
\end{eqnarray}
</code>
</p>
<p>
<code>i ^ {<var>EXP</var>} = <span data-if="REMAINDER">i ^ {<var>REMAINDER</var>} =</span> <var>SOLUTION</var></code>
</p>
</div>
</div>
</div>
</div>
</body>
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