Complex Numbers

De Moivre's Theorem and Roots of Unity

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De Moivre's Theorem and Roots of Unity

De Moivre's Theorem and Roots of Unity

This lesson covers the following key concepts:

  • De Moivre's theorem for integer powers
  • nth roots of a complex number
  • nth roots of unity: solutions of z^n = 1
  • Properties of roots of unity (sum, product)
  • Geometric interpretation: regular n-gon
  • Cube roots of unity: 1, ω, ω²
  • Applications to trigonometric identities

Important Formulas

  • (cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)
  • z1/n=r1/n[cos⁡θ+2kπn+isin⁡θ+2kπn],k=0,1,...,n−1z^{1/n} = r^{1/n}\left[\cos\frac{\theta + 2k\pi}{n} + i\sin\frac{\theta + 2k\pi}{n}\right], k = 0,1,...,n-1
  • nth roots of unity: e2πik/n,k=0,1,...,n−1\text{nth roots of unity: } e^{2\pi ik/n}, k = 0,1,...,n-1
  • 1+ω+ω2+...+ωn−1=0 (for nth root of unity)1 + \omega + \omega^2 + ... + \omega^{n-1} = 0 \text{ (for nth root of unity)}
  • ω3=1,1+ω+ω2=0 (cube roots)\omega^3 = 1, 1 + \omega + \omega^2 = 0 \text{ (cube roots)}
  • ω2=ω‾ (for cube roots of unity)\omega^2 = \overline{\omega} \text{ (for cube roots of unity)}