De Moivre's Theorem — Mathematics Knowledge System

Overview

De Moivre's theorem, named after Abraham de Moivre (1667–1754), states that raising a complex number in trigonometric form to an integer power n simply multiplies its argument by n and raises its modulus to the n-th power. This turns an otherwise painful repeated multiplication of complex numbers into a one-line computation, and — run in reverse with fractional exponents — it also produces a clean formula for the n distinct n-th roots of any complex number, evenly spaced around a circle in the complex plane. It is one of the most-used shortcuts in complex analysis and underlies applications from AC circuit phasors to signal processing (roots of unity in the discrete Fourier transform).

Intuition

Since multiplying two numbers in trigonometric form multiplies moduli and adds arguments, multiplying z by itself n times multiplies the modulus by itself n times (giving r^n) and adds the argument to itself n times (giving nθ) — De Moivre's theorem is just the product rule applied repeatedly to the same number. Running the idea backward, an n-th root of z must have modulus r^(1/n) and an argument that becomes θ (mod 360°) once multiplied by n; because adding any multiple of 360° to θ describes the same z, there are exactly n different starting arguments — (θ + 360°k)/n for k = 0, …, n−1 — that all land back on θ after multiplying by n. Geometrically, this places the n roots at equal 360°/n intervals around a circle of radius r^(1/n).

Formal Definition

Definition

For any complex number in trigonometric form z = r(cosθ + i sinθ) and any integer n:

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

De Moivre's theorem

z^{1/n} = r^{1/n}\left(\cos\frac{\theta + 360^\circ k}{n} + i\sin\frac{\theta+360^\circ k}{n}\right), \quad k = 0,1,\dots,n-1

n-th roots (n distinct values)

Worked Examples

Convert 1 + i to trigonometric form: modulus and argument.
$r = \sqrt{1^2+1^2} = \sqrt2, \quad \theta = 45^\circ$
Apply De Moivre's theorem with n = 4.
$(1+i)^4 = (\sqrt2)^4\big(\cos(4\cdot45^\circ) + i\sin(4\cdot45^\circ)\big) = 4(\cos180^\circ + i\sin180^\circ)$
Evaluate cos 180° = −1, sin 180° = 0.
$(1+i)^4 = 4(-1+0i) = -4$

Answer: (1+i)⁴ = −4

Practice Problems

Difficulty 5/10

Use De Moivre's theorem to compute (√3 + i)⁶.

Difficulty 6/10

Find all square roots of z = −4 (write z in trigonometric form first).

Difficulty 7/10

In digital signal processing, the 4th roots of unity (solutions to zⁿ = 1 with n = 4) are used as the basis for a length-4 discrete Fourier transform. Find all four 4th roots of unity using De Moivre's theorem.

Quiz

Summary

De Moivre's theorem: [r(cosθ+isinθ)]ⁿ = rⁿ(cos(nθ)+isin(nθ)) for integer n.
It turns repeated complex multiplication into raising the modulus to a power and multiplying the argument.
Run in reverse, it gives the n distinct n-th roots of a complex number: modulus r^(1/n), arguments (θ+360°k)/n for k=0,…,n−1.
The n-th roots are evenly spaced by 360°/n around a circle — the geometric basis for roots of unity and the discrete Fourier transform.

References

WebsiteWikipedia — De Moivre's formula