De Moivre’s Theorem and Powers of Complex Numbers

De Moivre’s theorem states that for any integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ), so a power of a complex number raises the modulus to the power and multiplies the argument by n. In brief: (1 + i)⁸: 1 + i = √2(cos 45° + i sin 45°), so (1 + i)⁸ = (√2)⁸(cos 360° + i sin 360°) = 16. The theorem avoids long expansions and gives multiple-angle formulas for sine and cosine.

Computing (1 + i)⁸ by repeated multiplication takes many steps. In polar form, it takes one. De Moivre’s theorem also leads to formulas such as cos 3θ = 4cos³θ − 3cos θ and to the roots of complex numbers, treated later. This guide states the theorem, proves it by induction, shows how to compute powers of complex numbers, how to derive trigonometric identities and gives a table of powers of 1 + i.

What does de Moivre’s theorem say?

For z = r(cos θ + i sin θ) and every integer n:

zⁿ = rⁿ(cos nθ + i sin nθ).

For r = 1 it takes the form (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. The statement holds for negative n as well, with z⁻ⁿ = 1/zⁿ, and, in a modified form, for fractional exponents.

Why does the theorem hold?

The proof is by induction. For n = 1 the formula is the definition. Assume it holds for n. Then zⁿ⁺¹ = zⁿ · z = rⁿ(cos nθ + i sin nθ) · r(cos θ + i sin θ). By the multiplication rule for complex numbers in polar form, the moduli multiply and the arguments add, so zⁿ⁺¹ = rⁿ⁺¹(cos(n + 1)θ + i sin(n + 1)θ). ∎ The case of negative n follows from the formula for the reciprocal, 1/z = r⁻¹(cos(−θ) + i sin(−θ)).

How do you compute powers of complex numbers?

1. Convert z to polar form (r, θ).
2. Raise r to the nth power and multiply θ by n.
3. Reduce the angle if desired, and convert back to algebraic form if needed.

Example 1: (1 + i)⁸

1 + i = √2(cos π/4 + i sin π/4). Then (1 + i)⁸ = (√2)⁸(cos 2π + i sin 2π) = 16(1 + 0i) = 16. Check: (1 + i)² = 2i, (2i)² = −4, (−4)² = 16 ✓.

Example 2: (−1 + √3 i)⁵

−1 + √3 i = 2(cos 120° + i sin 120°). Then the fifth power is 32(cos 600° + i sin 600°). 600° − 360° = 240°, so 32(cos 240° + i sin 240°) = 32(−1/2 − (√3/2)i) = −16 − 16√3 i.

Example 3: (√3 − i)⁶

√3 − i = 2(cos(−30°) + i sin(−30°)). The sixth power is 64(cos(−180°) + i sin(−180°)) = 64(−1) = −64.

Example 4: a negative exponent

(1 + i)⁻⁴ = (√2)⁻⁴(cos(−180°) + i sin(−180°)) = (1/4)(−1) = −1/4. Check: (1 + i)⁴ = (2i)² = −4, so the reciprocal is −1/4 ✓.

Example 5: a large exponent

i²⁰²⁵: i = cos 90° + i sin 90°, so i²⁰²⁵ = cos(2025 · 90°) + … The angle 182,250° reduces modulo 360°: 182,250 = 360 · 506 + 90, so the angle is 90°, and i²⁰²⁵ = i. This agrees with the cycle of the powers of i: 2025 = 4 · 506 + 1.

Example 6: the sum of a series

1 + z + z² + … + z⁵ for z = cos 60° + i sin 60°. By the geometric sum, (1 − z⁶)/(1 − z). Since z⁶ = cos 360° + i sin 360° = 1, the sum is 0. This shows that the sixth roots of unity add to zero.

How do you derive multiple-angle formulas?

Expand (cos θ + i sin θ)ⁿ with the binomial theorem and compare the real and imaginary parts with cos nθ + i sin nθ.

Example 7: n = 2

(cos θ + i sin θ)² = cos²θ − sin²θ + 2i sin θ cos θ. So cos 2θ = cos²θ − sin²θ and sin 2θ = 2 sin θ cos θ, the double-angle formulas.

Example 8: n = 3

(cos θ + i sin θ)³ = cos³θ + 3i cos²θ sin θ − 3 cos θ sin²θ − i sin³θ. The real part is cos³θ − 3 cos θ sin²θ = cos³θ − 3 cos θ(1 − cos²θ) = 4cos³θ − 3 cos θ. The imaginary part is 3 cos²θ sin θ − sin³θ = 3 sin θ − 4 sin³θ. So cos 3θ = 4cos³θ − 3cos θ and sin 3θ = 3 sin θ − 4 sin³θ. Check: θ = 30°: cos 90° = 0 and 4(√3/2)³ − 3(√3/2) = 4 · 0.6495 − 2.598 = 0 ✓.

Table of powers of 1 + i

Where does the theorem come from?

Abraham de Moivre, a French Huguenot who lived in London, published related results in 1722, in the form of a relation that gives the roots of equations. The form with the complex exponent, (cos x + i sin x)ⁿ = cos nx + i sin nx, is due to Euler in 1748, who also proved it for non-integers via the exponential function. In the modern view, de Moivre’s theorem is a special case of the exponent rule (e^iθ)ⁿ = e^inθ.

Where is it used today?

The theorem is used to compute powers and roots of complex numbers, to derive trigonometric identities and in the Fast Fourier Transform, which relies on powers of a primitive root of unity. In physics, it describes the rotation of phasors: a phasor multiplied by e^iωt rotates with the angular velocity ω, and the nth harmonic rotates n times faster.

Which mistakes are common?

Raising only the modulus

Both parts change: rⁿ and nθ.

Forgetting to convert to polar form first

The theorem needs the form r(cos θ + i sin θ). Write (1 + i) as √2(cos 45° + i sin 45°) before using it.

Using the wrong angle for a number in quadrant II or III

An error in the argument is multiplied by n. Check the quadrant carefully.

Applying the formula with a sum inside

(cos θ + i sin θ)ⁿ equals cos nθ + i sin nθ, but (cos θ)ⁿ + (i sin θ)ⁿ is something else.