De Moivre's Theorem

Complex Analysis · Axiom Academy

To raise a complex number to a power, raise the length and multiply the angle — a spiral, not a slog. 1. Powering Up Is Spiraling Out Multiplying by a complex number does two things at once: it scales length by r and rotates by the angle . So ( n times) repeats that move n times — the arrow walks outward in a spiral, scaling to r^n and turning through . Polar / trig form of the theorem The same statement in exponential form 2. Two Dials: Length and Angle The power n hits the modulus and the argument separately . The length is squeezed or stretched geometrically — r , then r^2 , then r^3 — while the angle is simply added to itself — , then , then . Watch each dial move on its own. Raised to the power: |z^n| = r^n . With r > 1 the point races outward; with r < 1 it spirals inward toward 0 . Multiplied by the power: . Each power turns the arrow one more notch of around the origin. The length never changes, so z^n stays on the unit circle and only the angle winds — pure rotation. The rule holds for every integer n — negative powers turn the angle the other way and shrink the length. In exponential form , so — the law of exponents (a^m)^n = a^ mn multiplies the exponent, sending the angle to . That is the whole proof. Take and cube it. De Moivre says: cube the modulus, triple the angle. The arrow stretches from length 2 to length 8 and swings from to in a single move, landing on the answer. The modulus step: 2^3 = 8 — the arrow's length cubes.

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