De Moivre's theorem says that raising a point on the unit circle to the nth power simply multiplies its angle by n. If you start at angle θ and apply the operation n times, you end at angle nθ. This is the geometric heart of complex number arithmetic.
Starting at angle θ=40° on the unit circle. Squaring doubles the angle to 80° (green). Cubing triples it to 120° (red). The point just rotates: its distance from the origin stays 1.
The theorem follows instantly from Euler's formula e^(iθ) = cosθ + i sinθ. Raising both sides to the power n: (e^(iθ))ⁿ = e^(inθ) = cos(nθ) + i sin(nθ). De Moivre stated his result in 1707, 41 years before Euler published the formula, making the proof feel like magic rather than mechanics.
The 6th roots of unity form a regular hexagon on the unit circle. The nth roots of z^n = 1 always form a regular n-gon, equally spaced at angles 2πk/n = τk/n.
De Moivre's theorem is the key tool for computing powers and roots of complex numbers, deriving multiple-angle formulas (cos 3θ = 4cos³θ - 3cosθ), and finding the n equally-spaced nth roots of any complex number. It connects the algebra of complex numbers to the geometry of rotation.
When you multiply two complex numbers, their angles (arguments) add and their magnitudes multiply. If both numbers sit on the unit circle (magnitude 1), only the angles change. Multiplying n times adds the angle n times: that is De Moivre's theorem.
De Moivre's theorem shows that cos(n*theta) can always be written as a polynomial in cos(theta). These are the Chebyshev polynomials T_n: T_n(cos theta) = cos(n*theta). For example, cos(2*theta) = 2*cos^2(theta) - 1, so T_2(x) = 2x^2 - 1. They appear in numerical analysis, filter design, and approximation theory.