R-ATSPE-3

Graph

Polar form!

We do polar form because its easier to do multiplication and division when compared to addition and subtraction in rect form.

A "natural" form of expressing complex number, multiplication, division, raising to powers.

Something something rect form is translations relative to i and j components, i.e. x and y axis respectively (i.e. real and imaginary axis respectively)

Geometry of multiplication of comple numbers

Consider $(2+i)\times (-3+2i)$.

When you plot them out, you notice that the angle made by the product is the sum of the angles of the two complex numbers, and the magnitude of the product ~~is the sum of the magnitudes of the two complex number.~~ and the product of the magnitudes of the two complex numbers.

(plot twist this is polar form)

Something something you can multiply by specific complex numbers which relate to a rotation by a certain angle.

You can think of the multiplication of complex number $z$ with $u$, as dilating by scale factor of $|u|$ and rotating anti-clockwise by an angle related to $u$ (i.e. $Re(u)=r\cos \theta$, $r$ is something special from later

Hence, there is a geometric interpretation of complex number multiplication. When multiplying by a complex number $z$ the things determining the corresponding geometric transfomrations are

the distance $|z|$ from the origin (the 'magnitude' or 'modulus')
the angle $\theta$ from the positive real axis (the 'argument')

This suggest that a way of representing $z$ where $|z| \text{ and }\theta$ are very explicit.

Hence, polar form!

Rewriting

how to rewrite $z=a+bi$ so that $|z|$ and $\theta$ are explicit.

$$\begin{align} |z| &= \sqrt{ a^{2}+b^{2} } \\ \cos \theta &= \frac{a}{|z|} , \text{ so } a=|z|\cos \theta. \\ \text{Similarly, } \sin \theta&=\frac{b}{|z|} , \text{ so } b=|z|\sin \theta \\ z &\implies |z|(\cos \theta+i\sin \theta) \end{align} $$

This is the polar form!!!

Hence, try to rewrite $z=\frac{5}{\sqrt{ 2 }}+\frac{5}{\sqrt{ 2 }}i$. $|z| = \sqrt{ \frac{25}{2}+\frac{25}{2} }=5$ and argument = $\frac{\pi}{4}$

Hence, $z=5\left( \cos\left( \frac{\pi}{4} \right)+i\sin\left( \frac{\pi}{4} \right) \right)$

(Shorthand for this is $z=5cis\left( \frac{\pi}{4} \right)$)

Formal definition of polar form

The polar form of a complex number $z$ is:

$$|z|(\cos \theta + i \sin \theta) $$

Where:

$|z|$ is the magnitude or modulus of $z$ (i.e. the distance from the origin)
$\theta$ is the principal argument (the angle between the vector for $z$ and the positive real axis).
- Note that the principal argument is defined for the domain $-\pi < \theta = \pi$.
- There are "good but rather subtle" reasons as to why there is $<$ and $\leq$ in that order.
- You must adjust a principal argument to be within this range.

Practice

Write the following in polar form: $-3 + 3\sqrt{ 3 }i$
$|-3+3\sqrt{ 3 }i|=6$
Blah blah you get -60 but its +120 cause something something quadrants

General method:

Evaluate $\tan^{-1}\left( \frac{b}{a} \right)$.
If necessary, add or subtract $\pi$ to ensure that:
- $\theta$ describes the complex number
- $-\pi < \theta \leq \pi$

Write the following in Cartesian form: $5cis\left( -\frac{3\pi}{4} \right)$

$$\begin{align} 5cis\left( -\frac{3\pi}{4} \right)&=5\left( -\frac{1}{\sqrt{ 2 }} \right)+5i\left(-\frac{1}{\sqrt{ 2 }}\right) \\ &= -\frac{5}{\sqrt{ 2 }}-\frac{5}{\sqrt{ 2 }}i \end{align} $$