Complex Numbers

Cube Roots and nth Roots of Unity

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Cube Roots and nth Roots of Unity

Cube Roots and nth Roots of Unity

This lesson covers the following key concepts:

  • Cube roots of unity: 1, ω, ω²
  • Properties of cube roots: ω³ = 1, 1 + ω + ω² = 0
  • ω² = conjugate of ω for cube roots
  • nth roots of unity and their properties
  • Sum and product of nth roots of unity
  • Geometric representation: vertices of regular polygon
  • Applications to solving higher degree equations

Important Formulas

  • ω=e2πi/3=−12+32i\omega = e^{2\pi i/3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i
  • ω2=e4πi/3=−12−32i\omega^2 = e^{4\pi i/3} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i
  • 1+ω+ω2=01 + \omega + \omega^2 = 0
  • ω3=1, hence ω3k=1,ω3k+1=ω,ω3k+2=ω2\omega^3 = 1, \text{ hence } \omega^{3k} = 1, \omega^{3k+1} = \omega, \omega^{3k+2} = \omega^2
  • nth roots: zk=e2πik/n,k=0,1,...,n−1\text{nth roots: } z_k = e^{2\pi ik/n}, k = 0, 1, ..., n-1
  • ∑k=0n−1zk=0,∏k=0n−1zk=(−1)n−1\sum_{k=0}^{n-1} z_k = 0, \prod_{k=0}^{n-1} z_k = (-1)^{n-1}
  • xn−1=(x−1)(x−ω)(x−ω2)...(x−ωn−1)x^n - 1 = (x-1)(x-\omega)(x-\omega^2)...(x-\omega^{n-1})