mirror of
https://github.com/peter-tanner/legacy_site.git
synced 2024-11-30 06:10:16 +08:00
169 lines
7.6 KiB
HTML
169 lines
7.6 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="UTF-8" />
|
|
<title>📝 Maths!</title>
|
|
<meta name="description" content="year 12 WACE specialist ATAR stuff" />
|
|
<meta name="viewport" content="width=device-width" />
|
|
<title>MathJax example</title>
|
|
<script>
|
|
MathJax = {
|
|
tex: { macros: { cis: "\\mathop{\\rm{cis}}\\nolimits" } },
|
|
chtml: { displayAlign: "center", scale: 1.1 },
|
|
};
|
|
</script>
|
|
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
|
|
<script
|
|
id="MathJax-script"
|
|
async
|
|
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"
|
|
></script>
|
|
<link rel="stylesheet" href="style.css" />
|
|
<link
|
|
rel="stylesheet"
|
|
href="https://stackpath.bootstrapcdn.com/bootstrap/4.5.2/css/bootstrap.min.css"
|
|
integrity="sha384-JcKb8q3iqJ61gNV9KGb8thSsNjpSL0n8PARn9HuZOnIxN0hoP+VmmDGMN5t9UJ0Z"
|
|
crossorigin="anonymous"
|
|
/>
|
|
</head>
|
|
|
|
<body>
|
|
<center>
|
|
<h1>A little rant on cis notation</h1>
|
|
</center>
|
|
|
|
<a
|
|
class="link"
|
|
style="left: 1%; top: 1%"
|
|
href="https://peter-tanner.github.io/legacy_site/maths"
|
|
>🔗 Back to MATHS home page</a
|
|
><br />
|
|
<a
|
|
class="link"
|
|
style="left: 1%; top: 1%"
|
|
href="https://peter-tanner.github.io"
|
|
>🔗 Back to home page</a
|
|
><br />
|
|
|
|
Warning: This page requires javascript to render the math.
|
|
|
|
<hr />
|
|
<br />
|
|
This is a rant on why I think \(e^{\pi\theta}\) should have been the
|
|
standard in WA specialist. All opinions are my own, and yes, this is an
|
|
incredibly nitpicky topic. And no, I don't expect any changes but if it
|
|
occurs I'll be pleasantly surprised.<br />
|
|
|
|
To many readers, this wouldn't be a debate because you'd be there going,
|
|
"what the heck is \(\cis\)?"<br />
|
|
This is the first downside of \(\cis\) notation: it is less known and
|
|
sparsely used compared to Euler's formula. \(\cis\) is the mathematical
|
|
equivalent of the imperial system.<br />
|
|
To answer the question, \(\cis\) is an abbreviation for "\(\cos\) plus
|
|
\(i\sin\)". This is actually one big thing I like about \(\cis\) notation,
|
|
in that it's an abbreviation which is easy to remember. In comparison,
|
|
euler's formula doesn't really make that much sense, you just have to accept
|
|
that it represents "\(\cos\) plus \(i\sin\)".<br />
|
|
This pro becomes less significant when you realize that a complex number can
|
|
be expressed as a vector, and we all know that from the unit circle that
|
|
\(\cos\) goes on the \(x\) axis, and \(\sin\) on the \(y\) axis. Likewise,
|
|
\(\cos\) goes on the real axis and \(\sin\) on the imaginary axis.
|
|
<div class="card">
|
|
<div class="card-body">
|
|
<center>\(\cis\) notation of a complex number, \(z\)</center>
|
|
\[\boxed{\|z\|\cdot \cis(\theta) =
|
|
\|z\|\cdot\left[\cos(\theta)+i\sin(\theta)\right]}\]<br />
|
|
<center>Euler's formula</center>
|
|
\[\boxed{\|z\|\cdot e^{i\theta} =
|
|
\|z\|\cdot\left[\cos(\theta)+i\sin(\theta)\right]}\]
|
|
</div>
|
|
</div>
|
|
My real issue with \(\cis\) notation is that it's not as obvious what
|
|
certain operations do. You can't use previous knowledge acquired from index
|
|
laws. By using \(\cis\), you are restricting yourself and you lose a lot of
|
|
the complexity (Haha get it? complex? okay...) that is possible with the
|
|
polar form.<br />
|
|
<div class="card">
|
|
<div class="card-body">
|
|
<center>
|
|
<b>Question 1</b>: Express \(i^i\) in the form \(\alpha+\beta i\)
|
|
</center>
|
|
For any method, the first step is to turn this into polar form.
|
|
\begin{align} \text{Define }z&=i\\ \implies&{\|z\| = 1}\\
|
|
\implies&{\theta = \frac{\pi}{2}} \end{align} Now we want to evaluate
|
|
\(z^z\)<br />
|
|
Let's try get somewhere with \(\cis\). \begin{align} z &=
|
|
1\cdot\cis\left(\frac{\pi}{2}\right)\\ &=
|
|
\cis\left(\frac{\pi}{2}\right)\\ z^z &=
|
|
\cis\left(\frac{\pi}{2}\right)^{\cis\left(\frac{\pi}{2}\right)}\\ &=
|
|
\cis\left(\frac{\pi}{2}\right)^i\\ &= \cis\left(i\frac{\pi}{2}\right)\\
|
|
&= ? \end{align} Now what? We have an \(i\) in the phase.<br />
|
|
It would be much easier if this was a complex exponential! \begin{align}
|
|
z &= 1\cdot e^{i\frac{\pi}{2}}\\ &= e^{i\frac{\pi}{2}}\\ z^z &=
|
|
{\left[e^{i\frac{\pi}{2}}\right]}^{e^{i\frac{\pi}{2}}}\\ &=
|
|
{\left[e^{i\frac{\pi}{2}}\right]}^{i}\\ &= {\left[e^{i\cdot
|
|
i\frac{\pi}{2}}\right]}\\ &= e^{-\frac{\pi}{2}} \end{align} Wow! And
|
|
it's neatly packaged as an exponent!
|
|
</div>
|
|
</div>
|
|
I understand the use of \(\cis\) to represent a complex number as a polar
|
|
coordinate if your curriculum doesn't understand the concept of calculus
|
|
when the concept of complex numbers is being taught<br />
|
|
But \(e\) (Euler's number) is a concept taught in year 12 methods. And year
|
|
12 methods is a prerequisite for year 12 specialist. So students should be
|
|
familiar with \(e\) and calculus, so why isn't it being used?<br /><br />
|
|
My argument isn't really that strong, I recognise that. Just wanted to go on
|
|
a 3am MathRant™ :)<br /><br />
|
|
<div class="card">
|
|
<div class="card-body">
|
|
<center>
|
|
<b>Bonus Math Tip</b>: Derive the double angle identities.
|
|
</center>
|
|
No need for Euler's formula here, use \(\cis\) for this trick if you
|
|
want.<br />
|
|
Not sure why you'd need to know this - sure, in methods these identities
|
|
aren't given on the formula sheet but they are in specialist. Methods
|
|
seems to stick to the basic identities such as the 2As and pythagorean,
|
|
but you're expected to remember them. I always just put these identities
|
|
on my notes.<br />
|
|
Use this trick to derive any set of angle identities (triple, quadruple,
|
|
etc.). Have fun expanding brackets though.<br /><br />
|
|
Let the first angle be \(A\) and the second angle \(B\). \begin{align}
|
|
e^{\pi A}\cdot e^{\pi B} &=
|
|
\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(B)+i\sin(B)\right]\\ e^{\pi
|
|
(A+B)} &= \cos(A)\cos(B) + i\cos(A)\sin(B) + i\sin(A)\cos(B) +
|
|
i^2\sin(A)\sin(B)\\ \cos(A+B) + i\sin(A+B) &= \left[\cos(A)\cos(B) -
|
|
\sin(A)\sin(B)\right] + i\cdot\left[\cos(A)\sin(B) + \sin(A)\cos(B)
|
|
\right] \end{align} Consider the real and complex components of this
|
|
equation to get the two identities. \begin{cases} \text{Real:}&\cos(A+B)
|
|
= \cos(A)\cos(B) - \sin(A)\sin(B)\\ \text{Imaginary:}&\sin(A+B) =
|
|
\cos(A)\sin(B) + \sin(A)\cos(B) \end{cases} And you can repeat this all
|
|
again for negative second angle. \begin{align} e^{\pi A}\cdot e^{\pi
|
|
(-B)} &=
|
|
\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(-B)+i\sin(-B)\right]\\ &=
|
|
\left[\cos(A)+i\sin(A)\right]\cdot\left[\cos(B)-i\sin(B)\right]\\ e^{\pi
|
|
(A-B)} &=
|
|
\cos(A)\cos(B)-i\cos(A)\sin(B)+i\sin(A)\cos(B)-i^2\sin(A)\sin(B)\\
|
|
\cos(A-B) + i\sin(A-B) &= \left[\cos(A)\cos(B) + \sin(A)\sin(B)\right] +
|
|
i\cdot\left[\sin(A)\cos(B)-\cos(A)\sin(B)\right] \end{align}
|
|
\begin{cases} \text{Real:}&\cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B)\\
|
|
\text{Imaginary:}&\sin(A-B) = \sin(A)\cos(B)-\cos(A)\sin(B) \end{cases}
|
|
</div>
|
|
</div>
|
|
|
|
<br />
|
|
<a
|
|
class="link"
|
|
style="left: 1%; bottom: 1%"
|
|
href="https://peter-tanner.github.io/legacy_site/maths"
|
|
>🔗 Back to MATHS home page</a
|
|
><br />
|
|
<a
|
|
class="link"
|
|
style="left: 1%; bottom: 1%"
|
|
href="https://peter-tanner.github.io"
|
|
>🔗 Back to home page</a
|
|
><br />
|
|
</body>
|
|
</html>
|