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## Precalculus

### Course: Precalculus>Unit 3

Lesson 6: Polar form of complex numbers

# 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$r\left(\mathrm{cos}\left(\theta \right)+i\mathrm{sin}\left(\theta \right)\right)$
Exponential$r\cdot {e}^{i\theta }$

## Rectangular form

$a+bi$
The rectangular form of a complex number is a sum of two terms: the number's $\text{real}$ part and the number's $\text{imaginary}$ part multiplied by $i$.
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

$r\left(\mathrm{cos}\left(\theta \right)+i\cdot \mathrm{sin}\left(\theta \right)\right)$
Polar form emphasizes the graphical attributes of complex numbers: $\text{absolute value}$ (the distance of the number from the origin in the complex plane) and $\text{angle}$ (the angle that the number forms with the positive Real axis). These are also called $\text{modulus}$ and $\text{argument}$.
Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:
$r\left(\mathrm{cos}\left(\theta \right)+i\cdot \mathrm{sin}\left(\theta \right)\right)=\stackrel{a}{\stackrel{⏞}{r\mathrm{cos}\left(\theta \right)}}+\stackrel{b}{\stackrel{⏞}{r\mathrm{sin}\left(\theta \right)}}\cdot i$
This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values ${r}_{1}$ and ${r}_{2}$ and angles ${\theta }_{1}$ and ${\theta }_{2}$ will have an absolute value ${r}_{1}{r}_{2}$ and angle ${\theta }_{1}+{\theta }_{2}$.
$r\cdot {e}^{i\theta }$
Exponential form uses the same attributes as polar form, $\text{absolute value}$ and $\text{angle}$. It only displays them in a different way that is more compact. For example, the multiplicative property can now be written as follows:
$\left({r}_{1}\cdot {e}^{i{\theta }_{1}}\right)\cdot \left({r}_{2}\cdot {e}^{i{\theta }_{2}}\right)={r}_{1}{r}_{2}\cdot {e}^{i\left({\theta }_{1}+{\theta }_{2}\right)}$
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 $\mathrm{cos}\left(x\right)+i\mathrm{sin}\left(x\right)$.
$r\cdot {e}^{i\theta }=r\left(\mathrm{cos}\left(\theta \right)+i\mathrm{sin}\left(\theta \right)\right)$