# Complex number forms review

Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.

## What are the different complex number forms?

Rectangular$a+bi$
Polar$r(\cos\theta+i\sin\theta)$
Exponential$r\cdot e^{i\theta}$

## Rectangular form

$\blueD a+\greenD bi$
The rectangular form of a complex number gives its $\blueD{\text{real}}$ part and its $\greenD{\text{imaginary}}$ part. It treats the number as the sum of those two parts.
As such, it is really useful for adding and subtracting complex numbers.
We can also plot a complex number given in rectangular form in the complex plane. The real and imaginary parts determine the real and imaginary coordinates of the number.

## Polar form

$\goldD r(\cos\purpleC\theta+i\sin\purpleC\theta)$
Polar form emphasizes the graphical attributes of complex numbers: $\goldD{\text{absolute value}}$ (the distance of the number from the origin in the complex plane) and $\purpleC{\text{angle}}$ (the angle that the number forms with the positive Real axis). These are also called $\goldD{\text{modulus}}$ and $\purpleC{\text{argument}}$.
Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:
This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values $\goldD{r_1}$ and $\goldD{r_2}$ and angles $\purpleC{\theta_1}$ and $\purpleC{\theta_2}$ will have an absolute value $\goldD{r_1r_2}$ and angle $\purpleC{\theta_1+\theta_2}$.
$\goldD r\cdot e^{i\purpleC\theta}$
Exponential form uses the same attributes as polar form, $\goldD{\text{absolute value}}$ and $\purpleC{\text{angle}}$. It only displays them in a different way that is more compact. For example, the multiplicative property can now be written as follows:
$(\goldD{r_1}\cdot e^{i\purple{\theta_1}})\cdot(\goldD{r_2}\cdot e^{i\purple{\theta_2}})=\goldD{r_1}\goldD{r_2}\cdot e^{i(\purple{\theta_1+\theta_2})}$
This form stems from Euler's expansion of the exponential function $e^z$ to any complex number $z$. The reasoning behind it is quite advanced, but its meaning is simple: for any real number $x$, we define $e^{ix}$ to be $\cos(x)+i\sin(x)$.
$\goldD r\cdot e^{i\purpleC\theta}=\goldD r(\cos\purpleC\theta+i\sin\purpleC\theta)$