Complex Numbers

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.