2.5 Complex Numbers — De Moivre's Theorem

2026 Syllabus Objectives

By the end of these notes, you should be able to:

Understand de Moivre's theorem for positive and negative integer exponents — and understand what it means geometrically (in terms of what multiplication and division of complex numbers do on a diagram).
Prove de Moivre's theorem for a positive integer exponent (for example, using mathematical induction).
Use de Moivre's theorem for positive or negative rational exponents to:
- Express trig ratios of multiple angles (like cos 5θ) in terms of powers of trig ratios of the basic angle θ.
- Express powers of sin θ and cos θ (like sin⁶θ) in terms of multiple angles.
- Sum certain series using the C + iS method.
- Find and use the nth roots of unity.

Before we can understand de Moivre's theorem, you need to be comfortable with writing a complex number in polar (modulus-argument) form.

Any complex number $z = x + iy$ can be written as:

$z = r(\cos\theta + i\sin\theta)$

where:

$r = |z| = \sqrt{x^2 + y^2}$ is the modulus — how far the point is from the origin on an Argand diagram.
$\theta = \arg(z)$ is the argument — the angle the line from the origin to the point makes with the positive real axis.

This is sometimes written in shorthand as $z = r\operatorname{cis}\theta$ .

What happens when you multiply two complex numbers in polar form?

If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$ , then:

$z_1 \cdot z_2 = r_1 r_2 \bigl(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\bigr)$

In plain English: When you multiply two complex numbers, you multiply their moduli and add their arguments.

What happens when you divide?

$\frac{z_1}{z_2} = \frac{r_1}{r_2}\bigl(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\bigr)$

In plain English: When you divide two complex numbers, you divide their moduli and subtract their arguments.

This geometric picture is the foundation for everything that follows.

De Moivre's theorem says:

$\boxed{(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)}$

This works for any integer $n$ (positive, negative, or zero), and also for rational values of $n$ (fractions like $\frac{1}{3}$ ).

More generally, if $z = r(\cos\theta + i\sin\theta)$ , then:

$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$