De Moivre's Theorem and Roots

Any complex number z = a + bi can be written in polar form as z = r(cos θ + i sin θ), where r = |z| = √(a² + b²) is the modulus and θ = arg(z) = arctan(b/a) is the argument. Using Euler's formula, this is also written z = re^{iθ}. The angle θ is defined modulo 2π, meaning θ and θ + 2kπ represent the same point for any integer k. This periodicity is crucial for finding multiple roots.

If z = r(cos θ + i sin θ) and n is any integer, then z^n = r^n(cos(nθ) + i sin(nθ)). The proof follows by induction using the angle-addition formulas for sine and cosine, or immediately from Euler's formula: z^n = (re^{iθ})^n = r^n e^{inθ} = r^n(cos(nθ) + i sin(nθ)). For example, if z = 2(cos 30° + i sin 30°), then z^4 = 2^4(cos 120° + i sin 120°) = 16(−1/2 + i√3/2) = −8 + 8i√3. Powers simply scale the modulus and rotate the angle.

To find the nth roots of a complex number w = R(cos φ + i sin φ), we solve z^n = w. Writing z = r(cos θ + i sin θ), we need r^n = R and nθ = φ + 2kπ for integer k. This gives r = R^{1/n} (the positive real nth root of R) and θ_k = (φ + 2kπ)/n. Taking k = 0, 1, 2, …, n−1 produces exactly n distinct roots. For k ≥ n the angles repeat, cycling back to the same complex numbers. The n roots are equally spaced at angular intervals of 2π/n on a circle of radius R^{1/n}.

The nth roots of unity are the solutions to z^n = 1. Here R = 1 and φ = 0, so all roots lie on the unit circle (radius 1) with arguments θ_k = 2kπ/n for k = 0, 1, …, n−1. In exponential notation these are ω^k where ω = e^{2πi/n} = cos(2π/n) + i sin(2π/n) is called the primitive nth root of unity. For example, the cube roots of unity are 1, e^{2πi/3} = −1/2 + i√3/2, and e^{4πi/3} = −1/2 − i√3/2 — the vertices of an equilateral triangle inscribed in the unit circle. A key identity: 1 + ω + ω² + … + ω^{n−1} = 0, meaning the roots sum to zero.

De Moivre's theorem also generates multiple-angle formulas. Since (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), expanding the left side with the binomial theorem and equating real and imaginary parts gives identities like cos(3θ) = 4cos³θ − 3cos θ and sin(3θ) = 3sin θ − 4sin³θ. This technique replaces pages of trigonometric manipulation with straightforward algebra, illustrating why complex numbers simplify so many areas of mathematics.

• r — modulus (dimensionless): The absolute value |z| of the complex number; always a non-negative real number equal to √(a² + b²).
• \theta — argument (radians): The angle the complex number makes with the positive real axis in the complex plane, measured in radians.
• n — exponent / degree (dimensionless): A positive integer indicating the power (for De Moivre's theorem) or the number of roots being computed.
• k — root index (dimensionless): An integer from 0 to n − 1 selecting each of the n distinct nth roots; different values of k give different equally-spaced roots.
• z_k — kth nth root (dimensionless): The kth distinct nth root of z, located on a circle of radius r^{1/n} at angle (θ + 2kπ)/n.

Problem. Find all fourth roots of z = −8 + 8i√3, expressing each root in polar and rectangular form.

Answer. The four fourth roots of −8 + 8i√3 are √3 + i, −1 + i√3, −√3 − i, and 1 − i√3.

• Polar Form of Complex Numbers — De Moivre's Theorem is stated in polar form; understanding modulus and argument is an essential prerequisite.
• Euler's Formula — Euler's formula e^{iθ} = cos θ + i sin θ provides the slickest proof of De Moivre's Theorem and unifies exponential and trigonometric representations.
• Multiple-Angle Trigonometric Identities — De Moivre's Theorem can derive identities like cos(nθ) and sin(nθ) in terms of cos θ and sin θ, extending standard double-angle formulas.
• Roots of Polynomials — The nth roots of unity are exactly the roots of the polynomial z^n − 1 = 0, connecting De Moivre's theorem to the Fundamental Theorem of Algebra.