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

What are the different complex number forms?

Exponentialreiθr\cdot e^{i\theta}

Rectangular form

a+bi\blueD a+\greenD bi
The rectangular form of a complex number gives its real\blueD{\text{real}} part and its imaginary\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.
Want to learn more about complex number rectangular form? Check out this video about the complex plane and this video about adding and subtracting complex numbers.

Polar form

r(cosθ+isinθ)\goldD r(\cos\purpleC\theta+i\sin\purpleC\theta)
Polar form emphasizes the graphical attributes of complex numbers: absolute value\goldD{\text{absolute value}} (the distance of the number from the origin in the complex plane) and angle\purpleC{\text{angle}} (the angle that the number forms with the positive Real axis). These are also called modulus\goldD{\text{modulus}} and argument\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 r1\goldD{r_1} and r2\goldD{r_2} and angles θ1\purpleC{\theta_1} and θ2\purpleC{\theta_2} will have an absolute value r1r2\goldD{r_1r_2} and angle θ1+θ2\purpleC{\theta_1+\theta_2}.
Want to learn more about complex number polar form? Check out this video.

Exponential form

reiθ\goldD r\cdot e^{i\purpleC\theta}
Exponential form uses the same attributes as polar form, absolute value\goldD{\text{absolute value}} and angle\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:
(r1eiθ1)(r2eiθ2)=r1r2ei(θ1+θ2)(\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 eze^z to any complex number zz. The reasoning behind it is quite advanced, but its meaning is simple: for any real number xx, we define eixe^{ix} to be cos(x)+isin(x)\cos(x)+i\sin(x).
Using this definition, we obtain the equivalence of exponential and polar forms:
reiθ=r(cosθ+isinθ)\goldD r\cdot e^{i\purpleC\theta}=\goldD r(\cos\purpleC\theta+i\sin\purpleC\theta)