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add complex number polar form division
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add complex number polar form division to Multiplying complex number
polar forms, creating Multiplying and dividing complex number polar
forms
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@agentydragon
agentydragon committed Dec 30, 2011
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220 changes: 220 additions & 0 deletions exercises/multiplying_and_dividing_complex_number_polar_forms.html
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<!--
This exercise (and other polar form ones) are written for
the trigonometric complex number form (z = r(cos phi + i sin phi)).
It might be cool to sometimes mix this up with z = r e^{i phi}.
-->
<!DOCTYPE html>
<html data-require="math math-format">
<head>
<meta charset="UTF-8" />
<title>Multiplying and dividing complex number polar forms</title>
<script src="../khan-exercise.js"></script>
<script>
var currentAngle = 0, currentRadius = 1;

function polarForm(radius, angle) {
var angleRep = KhanUtil.piFraction(angle * Math.PI / 12);
return radius + " \\cdot (cos(" + angleRep + ") + i \\cdot sin(" + angleRep + "))";
}

function redrawFormula() {
var angleRep = KhanUtil.piFraction(currentAngle * Math.PI / 12);
var equation = polarForm(currentRadius, currentAngle);
equation = KhanUtil.cleanMath(equation);
jQuery("#number-label").html("<code>"+equation+"</code>").tmpl();
jQuery("#current-radius").html(currentRadius);
jQuery("#current-angle").html("<code>"+angleRep+"</code>").tmpl();
}

function updateComplex(deltaAngle, deltaRadius) {
currentAngle += deltaAngle;
currentRadius += deltaRadius;

if (currentRadius < 1) currentRadius = 1;
if (currentRadius > 15) currentRadius = 15;

while (currentAngle < 0) currentAngle += 24;
while (currentAngle > 23) currentAngle -= 24;

jQuery("#angle input").val(currentAngle);
jQuery("#radius input").val(currentRadius);

redrawFormula();
}
</script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="multiply-cplx">
<div class="vars" data-ensure="ANSWER_RADIUS < 15">
<var id="A_RADIUS">randRange(1, 7)</var>
<var id="A_ANGLE">randRange(0, 23)</var> <!-- the angle is in 1/12's of pi -->
<var id="A_ANGLE_REP">piFraction(A_ANGLE * PI / 12)</var>
<var id="A_REP">polarForm(A_RADIUS, A_ANGLE)</var>

<var id="B_RADIUS">randRange(1, 7)</var>
<var id="B_ANGLE">randRange(0, 23)</var>
<var id="B_ANGLE_REP">piFraction(B_ANGLE * PI / 12)</var>
<var id="B_REP">polarForm(B_RADIUS, B_ANGLE)</var>

<var id="ANSWER_RADIUS">(A_RADIUS * B_RADIUS)</var>
<var id="ANSWER_ANGLE">(A_ANGLE + B_ANGLE) % 24</var>
<var>
(function() {
currentAngle = 0;
currentRadius = 1;
})()
</var>
</div>
<p class="question">Multiply the following complex numbers:</p>
<p><code>[<var>A_REP</var>] \cdot [<var>B_REP</var>]</code></p>

<!-- TODO: factor the answer type out... -->
<div class="solution" data-type="multiple">
<table> <!-- with apologies to any self-respecting webdesigner, I just need the buttons to line up... -->
<tr>
<td style="width: 100px">
Radius: <span id="current-radius">1</span>
</td>
<td>
<input type="button" class="simple-button action-gradient mini-button" value="+" onclick="updateComplex( 0, 1 )" />
<input type="button" class="simple-button action-gradient mini-button" style="margin-left: 5px;" value="-" onclick="updateComplex( 0, -1 )" />
</td>
</tr>
<tr>
<td>
Angle: <span id="current-angle"><code>0 \pi</code></span>
</td>
<td>
<input type="button" class="simple-button action-gradient mini-button" value="+" onclick="updateComplex( 1, 0 )" />
<input type="button" class="simple-button action-gradient mini-button" style="margin-left: 5px;" value="-" onclick="updateComplex( -1, 0 )" />
</td>
</tr>
</table>

<p id="number-label" style="margin: 8px 0px 2px 0px"><code><var>polarForm(1, 0)</var></code></p>

<span class="sol" id="angle" style="display: none"><var>ANSWER_ANGLE</var></span>
<span class="sol" id="radius" style="display: none"><var>ANSWER_RADIUS</var></span>
</div>

<div class="hints">
<p>
Multiplying complex numbers in polar forms can be done by multiplying the radii
and adding the angles.
</p>
<p>
The first number (<code><var>A_REP</var></code>) has angle <code><var>A_ANGLE_REP</var></code> and radius <var>A_RADIUS</var>.
The second number (<code><var>B_REP</var></code>) has angle <code><var>B_ANGLE_REP</var></code> and radius <var>B_RADIUS</var>.
</p>
<p>
The radius of the result will be <code><var>A_RADIUS</var> \cdot <var>B_RADIUS</var></code>, which is <var>ANSWER_RADIUS</var>.
</p>
<div data-if="A_ANGLE + B_ANGLE > 12 * 2" data-unwrap>
<p>
The sum of the angles is <code><var>A_ANGLE_REP</var> + <var>B_ANGLE_REP</var> = <var>piFraction((A_ANGLE + B_ANGLE) * PI / 12)</var></code>.
</p>
<p>
That is more than <code>2 \pi</code>. A complex number goes a full circle if its angle is increased by <code>2 \pi</code>, so it
goes back to itself. Because of that, angles of complex numbers are convient to keep between 0 and <code>2 \pi</code>.
</p>
<p>
After taking <code>2 \pi</code> from the sum, the angle becomes <code><var>piFraction(ANSWER_ANGLE * PI / 12)</var></code>.
<!-- The A and B angles are both between 0 and 2pi, so the maximum angle here is 4pi, so it's safe not to handle further 2pi multiples. -->
</p>
</div>
<p data-else>
The angle of the result is <code><var>A_ANGLE_REP</var> + <var>B_ANGLE_REP</var> = <var>piFraction(ANSWER_ANGLE * PI / 12)</var></code>.
</p>
</div>
</div>

<div id="divide-cplx">
<div class="vars">

<var id="ANSWER_RADIUS">randRange(1, 10)</var>
<var id="ANSWER_ANGLE">randRange(0, 23)</var>

<var id="B_RADIUS">randRange(1, 7)</var>
<var id="B_ANGLE">randRange(0, 23)</var>
<var id="B_ANGLE_REP">piFraction(B_ANGLE * PI / 12)</var>
<var id="B_REP">polarForm(B_RADIUS, B_ANGLE)</var>

<var id="A_RADIUS">ANSWER_RADIUS * B_RADIUS</var>
<var id="A_ANGLE">(ANSWER_ANGLE + B_ANGLE) % 24</var>
<var id="A_ANGLE_REP">piFraction(A_ANGLE * PI / 12)</var>
<var id="A_REP">polarForm(A_RADIUS, A_ANGLE)</var>

<var>
(function() {
currentAngle = 0;
currentRadius = 1;
})()
</var>
</div>
<p class="question">Divide the following complex numbers:</p>
<p><code>\frac{<var>A_REP</var>}{<var>B_REP</var>}</code></p>

<div class="solution" data-type="multiple">
<table>
<tr>
<td style="width: 100px">
Radius: <span id="current-radius">1</span>
</td>
<td>
<input type="button" class="simple-button action-gradient mini-button" value="+" onclick="updateComplex( 0, 1 )" />
<input type="button" class="simple-button action-gradient mini-button" style="margin-left: 5px;" value="-" onclick="updateComplex( 0, -1 )" />
</td>
</tr>
<tr>
<td>
Angle: <span id="current-angle"><code>0 \pi</code></span>
</td>
<td>
<input type="button" class="simple-button action-gradient mini-button" value="+" onclick="updateComplex( 1, 0 )" />
<input type="button" class="simple-button action-gradient mini-button" style="margin-left: 5px;" value="-" onclick="updateComplex( -1, 0 )" />
</td>
</tr>
</table>

<p id="number-label" style="margin: 8px 0px 2px 0px"><code>1 \cdot (cos(0 \pi) + i \cdot sin(0 \pi))</code></p>

<span class="sol" id="angle" style="display: none"><var>ANSWER_ANGLE</var></span>
<span class="sol" id="radius" style="display: none"><var>ANSWER_RADIUS</var></span>
</div>

<div class="hints">
<p>
Dividing complex numbers in polar forms can be done by dividing the radii
and subtracting the angles.
</p>
<p>
The first number (<code><var>A_REP</var></code>) has angle <code><var>A_ANGLE_REP</var></code> and radius <var>A_RADIUS</var>.
The second number (<code><var>B_REP</var></code>) has angle <code><var>B_ANGLE_REP</var></code> and radius <var>B_RADIUS</var>.
</p>
<p>
The radius of the result will be <code>\frac{<var>A_RADIUS</var>}{<var>B_RADIUS</var>}</code>, which is <var>ANSWER_RADIUS</var>.
</p>
<div data-if="A_ANGLE - B_ANGLE < 0" data-unwrap>
<p>
The difference of the angles is <code><var>A_ANGLE_REP</var> - <var>B_ANGLE_REP</var> = <var>piFraction((A_ANGLE - B_ANGLE) * PI / 12)</var></code>.
</p>
<p>
That is a negative number. A complex number goes a full circle if its angle is increased by <code>2 \pi</code>, so it
goes back to itself. Because of that, angles of complex numbers are convient to keep between 0 and <code>2 \pi</code>.
</p>
<p>
After adding <code>2 \pi</code> to the difference, the angle becomes <code><var>piFraction(ANSWER_ANGLE * PI / 12)</var></code>.
</p>
</div>
<p data-else>
The angle of the result is <code><var>A_ANGLE_REP</var> - <var>B_ANGLE_REP</var> = <var>piFraction(ANSWER_ANGLE * PI / 12)</var></code>.
</p>
</div>

</div>
</div>
</div>
</body>
</html>
131 changes: 0 additions & 131 deletions exercises/multiplying_complex_number_polar_forms.html
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