De Moivre's Theorem
Learn De Moivre's theorem for raising complex numbers to powers, with formula, proof outline, and worked examples.
The Formula
De Moivre's theorem is a fundamental result connecting complex numbers and trigonometry. Named after Abraham de Moivre, a French mathematician who published it in 1722, the theorem states that raising a complex number in polar form to any integer power n is equivalent to multiplying the angle by n while keeping the magnitude at 1.
The theorem works because of how complex multiplication operates geometrically. When you multiply two complex numbers in polar form, you multiply their magnitudes and add their angles. Raising to the power n means multiplying the number by itself n times, so the angle gets added n times (becoming nθ) and the unit magnitude raised to any power remains 1.
For a general complex number with magnitude r, the theorem extends to: (r(cos θ + i sin θ))n = rn(cos(nθ) + i sin(nθ)). This makes it vastly simpler to compute high powers of complex numbers compared to repeated multiplication in rectangular form (a + bi).
One of the most powerful applications of De Moivre's theorem is finding the nth roots of a complex number. If you want to find all n values of z1/n, the theorem gives: z1/n = r1/n(cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, 2, ..., n−1. This produces exactly n distinct roots, equally spaced around a circle in the complex plane.
The theorem also provides an elegant way to derive multiple-angle trigonometric identities. By expanding the left side using the binomial theorem and comparing real and imaginary parts with the right side, you can derive formulas for cos(nθ) and sin(nθ) in terms of powers of cos θ and sin θ. For example, expanding n = 3 gives the triple angle formulas.
Variables
| Symbol | Meaning |
|---|---|
| θ | The angle in radians (or degrees) of the complex number |
| n | The exponent (any integer; extends to rationals for roots) |
| i | The imaginary unit, where i² = −1 |
| r | The magnitude (modulus) of the complex number |
| k | Index for nth roots (k = 0, 1, ..., n−1) |
Example 1
Compute (cos 30° + i sin 30°)6.
Apply De Moivre's theorem: cos(6 × 30°) + i sin(6 × 30°)
= cos 180° + i sin 180°
= −1 + 0i = −1
Example 2
Find the three cube roots of 8 (the complex cube roots of 8).
Write 8 in polar form: 8(cos 0 + i sin 0), so r = 8, θ = 0
Cube root magnitude: 81/3 = 2
k = 0: 2(cos 0 + i sin 0) = 2
k = 1: 2(cos 120° + i sin 120°) = 2(−0.5 + 0.866i) = −1 + 1.732i
k = 2: 2(cos 240° + i sin 240°) = 2(−0.5 − 0.866i) = −1 − 1.732i
The three cube roots of 8 are: 2, −1 + √3 i, and −1 − √3 i.
When to Use It
De Moivre's theorem is essential in many areas of mathematics, physics, and engineering.
- Computing powers of complex numbers efficiently
- Finding all nth roots of any complex number
- Deriving multiple-angle trigonometric identities
- Electrical engineering: analyzing AC circuits with phasors
- Signal processing: working with Fourier transforms
- Quantum mechanics: manipulating wave functions in polar form