De Moivre's Theorem and nth Roots

De Moivre's Theorem and nth Roots of Complex Numbers

1. De Moivre's Theorem

Statement:
For any complex number in polar form $z = r (\cos \theta + i \sin \theta)$ and any integer $n$,
$z^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right).$

Proof:

• For positive integers, use mathematical induction:
  • Base case ($n = 1$): Trivially true.
  • Inductive step: Assume true for $n = k$, then show for $n = k+1$ using angle addition formulas.
• For negative integers, use $z^{-n} = \frac{1}{z^n}$ and trigonometric identities.

Applications:

• Efficiently compute powers of complex numbers.
• Derive multiple-angle trigonometric identities (e.g., $\cos(3\theta)$).

2. Finding nth Roots of a Complex Number

Given $z = r (\cos \theta + i \sin \theta)$, its $n$-th roots are:
$\sqrt[n]{z} = r^{1/n} \left[ \cos\left( \frac{\theta + 2k\pi}{n} \right) + i \sin\left( \frac{\theta + 2k\pi}{n} \right) \right],$
where $k = 0, 1, 2, \dots, n-1$.

Key Properties:

• Number of roots: Exactly $n$ distinct roots (due to periodicity of $2\pi$).
• Geometric interpretation: Roots lie on a circle of radius $r^{1/n}$ in the complex plane, spaced at angles of $\frac{2\pi}{n}$.

Example (Cube Roots of Unity):
Solve $z^3 = 1$:

• Polar form: $1 = 1 (\cos 0 + i \sin 0)$.
• Roots:
  $\omega_k = \cos\left( \frac{2k\pi}{3} \right) + i \sin\left( \frac{2k\pi}{3} \right), \quad k = 0, 1, 2.$
  Explicitly:
  • $\omega_0 = 1$,
  • $\omega_1 = -\frac{1}{2} + i \frac{\sqrt{3}}{2}$,
  • $\omega_2 = -\frac{1}{2} - i \frac{\sqrt{3}}{2}$.

3. Connection to Polynomial Equations

• The equation $z^n = w$ has $n$ solutions (roots) in the complex plane.
• Fundamental Theorem of Algebra: Every non-constant polynomial of degree $n$ has exactly $n$ roots in $\mathbb{C}$.

4. Practical Computations

1. Step 1: Convert the complex number to polar form $z = r (\cos \theta + i \sin \theta)$.
2. Step 2: Apply De Moivre’s formula for roots.
3. Step 3: Evaluate for $k = 0$ to $k = n-1$ to list all roots.

Example: Find all 4th roots of $z = 16 (\cos \pi + i \sin \pi)$.

• Roots:
  $2 \left[ \cos\left( \frac{\pi + 2k\pi}{4} \right) + i \sin\left( \frac{\pi + 2k\pi}{4} \right) \right], \quad k = 0, 1, 2, 3.$
• Explicit solutions:
  • $k=0$: $2 \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) = \sqrt{2} + i \sqrt{2}$,
  • $k=1$: $2 \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) = -\sqrt{2} + i \sqrt{2}$,
  • $k=2$: $2 \left( \cos \frac{5\pi}{4} + i \sin \frac{5\pi}{4} \right) = -\sqrt{2} - i \sqrt{2}$,
  • $k=3$: $2 \left( \cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right) = \sqrt{2} - i \sqrt{2}$.

5. Applications

• Engineering: Solving AC circuit problems with phasors.
• Computer Graphics: Rotations and scaling using complex multiplication.
• Quantum Physics: Wavefunction symmetries and eigenvalue problems.

De Moivre’s Theorem and nth roots unify algebraic and geometric perspectives of complex numbers, enabling elegant solutions to polynomial equations and periodic systems.