Overview

Complex numbers provide a clean language for rotation and roots of unity. Contest problems often use $z\overline{z}=|z|^2$ and polar form.

Key Ideas

• $z=re^{i\theta}$ with $r=|z|$ and argument $\theta$.
• $n$th roots of unity satisfy $z^n=1$ and are evenly spaced on the unit circle.
• Multiplication rotates and scales points in the plane.

Worked Example

Compute $(1+i)^4$.

Since $1+i=\sqrt{2}e^{i\pi/4}$, we have $(1+i)^4=(\sqrt{2})^4 e^{i\pi}=4(-1)=-4$.

Practice Problems