Complex Numbers and Polar Coordinates

The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.

Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.

Polar coordinates allow conic sections to be expressed in an elegant way.

Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.

A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Complex numbers can be added and subtracted by adding the real parts and imaginary parts separately. 

Complex numbers can be multiplied using the FOIL algorithm.

Powers of complex numbers can be computed with the the help of the binomial theorem.

The complex conjugate of the number $a+bi$ is $a-bi$. Two complex conjugates of each other multiply to be a real number with geometric significance. 

Division of complex numbers is accomplished by multiplying by the multiplicative inverse. The multiplicative inverse of $z$ is $\frac{\overline{z}}{\abs{z}^2}.$

Complex numbers can be represented in polar coordinates using the formula $a+bi=re^{i\theta}$. This leads to a way to visualize multiplying and dividing complex numbers geometrically.