De Moivre's Theorem

High School

Definition

Named after Abraham de Moivre, this theorem gives us a relatively simple formula that can help us find the powers and roots of complex numbers. Complex numbers that are in polar form can be raised to certain powers easily. Complex numbers are made of both real and imaginary parts.

Worked examples

\(\left(2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\right)^3 = 2^3\left(\cos\frac{3\pi}{6} + i\sin\frac{3\pi}{6}\right) = 8\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right) = 8i\)

Raise the modulus to the power and multiply the angle by the exponent.

\(\left(\sqrt{3} + i\right)^{10} = \left(2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\right)^{10} = 1024\left(\cos\frac{5\pi}{3} + i\sin\frac{5\pi}{3}\right) = 512 - 512\sqrt{3}\,i\)

Convert to polar form first, apply De Moivre's theorem, then convert back to rectangular form.

\(\left(\cos\frac{2\pi}{5} + i\sin\frac{2\pi}{5}\right)^5 = \cos 2\pi + i\sin 2\pi = 1\)

Finding roots uses the theorem in reverse; this shows one of the fifth roots of unity.

Common mistakes

\((r(\cos\theta + i\sin\theta))^n = r(\cos n\theta + i\sin n\theta)\) → \((r(\cos\theta + i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta)\) You must raise the modulus r to the nth power, not leave it unchanged.
\((3 + 4i)^5\) — apply the theorem directly to rectangular form → Convert \(3 + 4i = 5(\cos 0.927 + i\sin 0.927)\) to polar first, then apply the theorem De Moivre's theorem only works when the complex number is already in polar form.
\((\cos\theta + i\sin\theta)^n = \cos\theta^n + i\sin\theta^n\) → \((\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta\) Multiply the angle by n; do not raise the angle itself to the nth power.

Where you'll use it next

You'll use De Moivre's theorem to find nth roots of complex numbers, solve polynomial equations in the complex plane, and analyze oscillations and waves in engineering and physics courses.

Found in 1 StudyPug lesson

Operations on complex numbers in polar form

Grade 12 Math

Let's find out how to perform some basic operations on complex numbers in polar form! We will briefly introduce the notion of the exponential form of a complex number, then we will focus on multiplication and division on complex numbers in polar form.

12th Grade12th