# 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

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.*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

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 $\purpleC{\theta_1+\theta_2}$.

*Want to learn more about complex number polar form? Check out this video.*

## Exponential form

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:

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 $\cos(x)+i\sin(x)$.

Using this definition, we obtain the equivalence of exponential and polar forms: