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.

Core Skills

Switch Between Forms

Use $z=a+bi$ for algebra and $z=re^{i\theta}$ for powers and rotations.

Use Conjugates and Modulus

Apply $z\overline{z}=|z|^2$ to eliminate imaginary parts or compute lengths.

Use De Moivre

Compute powers and roots quickly using $r^n e^{in\theta}$.

Worked Example

Compute $(1+i)^4$.

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

More Examples

Example 1: Roots of Unity

Find all solutions to $z^3=1$.

$z=e^{2\pi i k/3}$ for $k=0,1,2$.

Example 2: Modulus

If $z=3-4i$, compute $|z|$.

$5$.

Example 3: Rotation

If $z$ is multiplied by $e^{i\pi/3}$, what happens geometrically?

It rotates by $60^\circ$ and keeps the same magnitude.

Strategy Checklist

• Choose rectangular or polar form based on the task.
• Use conjugates to eliminate denominators.
• Reduce angles modulo $2\pi$.

Practice Problems