De Moivre's Theorem

Definition

De Moivre’s Theorem states that for any integer $n$: $[r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta)$ $[r(\cos \theta + i\sin \theta)]^n = r^n(\cos n\theta + i\sin n\theta)$

• How to read: “The value r times the sum of the cosine of theta and i times the sine of theta, all raised to the power of n, equals r to the power of n times the sum of the cosine of n theta and i times the sine of n theta.”
• Meaning: Raise modulus to $n$, multiply angle by $n$.

Why It Matters

De Moivre’s theorem provides an elegant and fast way to perform complex calculations involving rotation and oscillation. It is fundamental to electrical engineering and signal processing, where complex numbers model the behavior of waves and circuits.

Core Concepts

• Exponentiation: In exponential form, $(re^{i\theta})^n = r^ne^{in\theta}$. Powers simply multiply the phase and exponentiate the magnitude.
  • How to read: “The value r times e to the power of i theta, all raised to the power of n, equals r to the power of n times e to the power of i n theta.”
  • Meaning: Compact polar form of De Moivre’s theorem.
• n-th Roots: Every non-zero complex number $z = re^{i\theta}$ has exactly $n$ distinct roots in the complex plane.
• Root Formula: The $n$th roots are given by: $w_k = \sqrt[n]{r} \exp\left( i \frac{\theta + 2k\pi}{n} \right), \quad k = 0, 1, \dots, n-1$
  • How to read: “The root w k equals the n-th root of r, times e to the power of i multiplied by the ratio of theta plus two k pi over n, for k from zero to n minus one.”
  • Meaning / when to use: $n$ roots equally spaced on a circle of radius $r^{1/n}$.
• Geometric Symmetry: The roots are perfectly distributed on a circle of radius $r^{1/n}$, forming a regular $n$-gon centered at the origin.

Connected Concepts

• nth Roots
• Rational Exponents
• Polar Form of Complex Numbers
• The Binomial Theorem