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176 lines
7.9 KiB
HTML
176 lines
7.9 KiB
HTML
<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="UTF-8" />
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<title>📝 Maths!</title>
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<meta name="description" content="year 12 WACE specialist ATAR stuff" />
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<meta name="viewport" content="width=device-width" />
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<title>MathJax example</title>
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<script>
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MathJax = {
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tex: { macros: { cis: "\\mathop{\\rm{cis}}\\nolimits" } },
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chtml: { displayAlign: "center", scale: 1.1 },
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};
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</script>
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<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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<script
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id="MathJax-script"
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async
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src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
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></script>
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<link rel="stylesheet" href="style.css" />
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<link
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rel="stylesheet"
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href="https://stackpath.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css"
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integrity="sha384-JcKb8q3iqJ61gNV9KGb8thSsNjpSL0n8PARn9HuZOnIxN0hoP+VmmDGMN5t9UJ0Z"
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crossorigin="anonymous"
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/>
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<meta name="robots" content="noindex, nofollow" />
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</head>
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<body>
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<b
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>PLEASE DO NOT USE ANY CONTENT FROM HERE! ALL UNMAINTAINED AND THERE'S
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PROBABLY NOTHING USEFUL HERE! CONTENT IS NOW UN-INDEXED TO PREVENT
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CONFUSION, SO YOU CAN ONLY ACCESS THIS PART OF THE SITE THROUGH LINKS.
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CONTENT IS ONLY KEPT HERE FOR HISTORY OF THE SITE -2023</b
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>
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<center>
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<h1>A little rant on cis notation</h1>
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</center>
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<a
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class="link"
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style="left: 1%; top: 1%"
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href="https://www.petertanner.dev/legacy_site/maths"
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>🔗 Back to MATHS home page</a
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><br />
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<a
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class="link"
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style="left: 1%; top: 1%"
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href="https://www.petertanner.dev/legacy_site"
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>🔗 Back to home page</a
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><br />
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Warning: This page requires javascript to render the math.
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<hr />
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<br />
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This is a rant on why I think \(e^{\pi\theta}\) should have been the
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standard in WA specialist. All opinions are my own, and yes, this is an
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incredibly nitpicky topic. And no, I don't expect any changes but if it
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occurs I'll be pleasantly surprised.<br />
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To many readers, this wouldn't be a debate because you'd be there going,
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"what the heck is \(\cis\)?"<br />
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This is the first downside of \(\cis\) notation: it is less known and
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sparsely used compared to Euler's formula. \(\cis\) is the mathematical
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equivalent of the imperial system.<br />
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To answer the question, \(\cis\) is an abbreviation for "\(\cos\) plus
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\(i\sin\)". This is actually one big thing I like about \(\cis\) notation,
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in that it's an abbreviation which is easy to remember. In comparison,
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euler's formula doesn't really make that much sense, you just have to accept
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that it represents "\(\cos\) plus \(i\sin\)".<br />
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This pro becomes less significant when you realize that a complex number can
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be expressed as a vector, and we all know that from the unit circle that
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\(\cos\) goes on the \(x\) axis, and \(\sin\) on the \(y\) axis. Likewise,
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\(\cos\) goes on the real axis and \(\sin\) on the imaginary axis.
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<div class="card">
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<div class="card-body">
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<center>\(\cis\) notation of a complex number, \(z\)</center>
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\[\boxed{\|z\|\cdot \cis(\theta) =
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\|z\|\cdot\left[\cos(\theta)+i\sin(\theta)\right]}\]<br />
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<center>Euler's formula</center>
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\[\boxed{\|z\|\cdot e^{i\theta} =
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\|z\|\cdot\left[\cos(\theta)+i\sin(\theta)\right]}\]
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</div>
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</div>
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My real issue with \(\cis\) notation is that it's not as obvious what
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certain operations do. You can't use previous knowledge acquired from index
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laws. By using \(\cis\), you are restricting yourself and you lose a lot of
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the complexity (Haha get it? complex? okay...) that is possible with the
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polar form.<br />
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<div class="card">
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<div class="card-body">
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<center>
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<b>Question 1</b>: Express \(i^i\) in the form \(\alpha+\beta i\)
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</center>
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For any method, the first step is to turn this into polar form.
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\begin{align} \text{Define }z&=i\\ \implies&{\|z\| = 1}\\
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\implies&{\theta = \frac{\pi}{2}} \end{align} Now we want to evaluate
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\(z^z\)<br />
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Let's try get somewhere with \(\cis\). \begin{align} z &=
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1\cdot\cis\left(\frac{\pi}{2}\right)\\ &=
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\cis\left(\frac{\pi}{2}\right)\\ z^z &=
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\cis\left(\frac{\pi}{2}\right)^{\cis\left(\frac{\pi}{2}\right)}\\ &=
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\cis\left(\frac{\pi}{2}\right)^i\\ &= \cis\left(i\frac{\pi}{2}\right)\\
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&= ? \end{align} Now what? We have an \(i\) in the phase.<br />
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It would be much easier if this was a complex exponential! \begin{align}
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z &= 1\cdot e^{i\frac{\pi}{2}}\\ &= e^{i\frac{\pi}{2}}\\ z^z &=
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{\left[e^{i\frac{\pi}{2}}\right]}^{e^{i\frac{\pi}{2}}}\\ &=
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{\left[e^{i\frac{\pi}{2}}\right]}^{i}\\ &= {\left[e^{i\cdot
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i\frac{\pi}{2}}\right]}\\ &= e^{-\frac{\pi}{2}} \end{align} Wow! And
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it's neatly packaged as an exponent!
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</div>
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</div>
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I understand the use of \(\cis\) to represent a complex number as a polar
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coordinate if your curriculum doesn't understand the concept of calculus
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when the concept of complex numbers is being taught<br />
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But \(e\) (Euler's number) is a concept taught in year 12 methods. And year
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12 methods is a prerequisite for year 12 specialist. So students should be
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familiar with \(e\) and calculus, so why isn't it being used?<br /><br />
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My argument isn't really that strong, I recognise that. Just wanted to go on
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a 3am MathRant™ :)<br /><br />
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<div class="card">
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<div class="card-body">
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<center>
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<b>Bonus Math Tip</b>: Derive the double angle identities.
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</center>
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No need for Euler's formula here, use \(\cis\) for this trick if you
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want.<br />
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Not sure why you'd need to know this - sure, in methods these identities
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aren't given on the formula sheet but they are in specialist. Methods
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seems to stick to the basic identities such as the 2As and pythagorean,
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but you're expected to remember them. I always just put these identities
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on my notes.<br />
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Use this trick to derive any set of angle identities (triple, quadruple,
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etc.). Have fun expanding brackets though.<br /><br />
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Let the first angle be \(A\) and the second angle \(B\). \begin{align}
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e^{\pi A}\cdot e^{\pi B} &=
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\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(B)+i\sin(B)\right]\\ e^{\pi
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(A+B)} &= \cos(A)\cos(B) + i\cos(A)\sin(B) + i\sin(A)\cos(B) +
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i^2\sin(A)\sin(B)\\ \cos(A+B) + i\sin(A+B) &= \left[\cos(A)\cos(B) -
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\sin(A)\sin(B)\right] + i\cdot\left[\cos(A)\sin(B) + \sin(A)\cos(B)
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\right] \end{align} Consider the real and complex components of this
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equation to get the two identities. \begin{cases} \text{Real:}&\cos(A+B)
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= \cos(A)\cos(B) - \sin(A)\sin(B)\\ \text{Imaginary:}&\sin(A+B) =
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\cos(A)\sin(B) + \sin(A)\cos(B) \end{cases} And you can repeat this all
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again for negative second angle. \begin{align} e^{\pi A}\cdot e^{\pi
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(-B)} &=
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\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(-B)+i\sin(-B)\right]\\ &=
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\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(B)-i\sin(B)\right]\\ e^{\pi
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(A-B)} &=
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\cos(A)\cos(B)-i\cos(A)\sin(B)+i\sin(A)\cos(B)-i^2\sin(A)\sin(B)\\
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\cos(A-B) + i\sin(A-B) &= \left[\cos(A)\cos(B) + \sin(A)\sin(B)\right] +
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i\cdot\left[\sin(A)\cos(B)-\cos(A)\sin(B)\right] \end{align}
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\begin{cases} \text{Real:}&\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)\\
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\text{Imaginary:}&\sin(A-B) = \sin(A)\cos(B)-\cos(A)\sin(B) \end{cases}
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</div>
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</div>
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<br />
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<a
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class="link"
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style="left: 1%; bottom: 1%"
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href="https://www.petertanner.dev/legacy_site/maths"
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>🔗 Back to MATHS home page</a
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><br />
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<a
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class="link"
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style="left: 1%; bottom: 1%"
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href="https://www.petertanner.dev/legacy_site"
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>🔗 Back to home page</a
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><br />
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</body>
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</html>
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