Complex Root of Unity

Description:

• The roots of $w^n=1$ where $w$ is complex number
• By de Moivre’s Theorem, $w^n=\cos (n\theta )+i.\sin (n\theta )$
  • $\to w^n=1$ only happen when $\theta =2k\pi$ where $k\in [0,...,n-1]$
• $\to w^n=1=\cos (0+2k\pi )+i\sin (0+2k\pi )$
• Then we have $n$ roots: $w_1=1,w_2=e^{\frac{2\pi }{n}i},w_3=e^{\frac{4\pi }{n}i},...$
  • Generally $w_n=w_{n-1}\times e^{\frac{2\pi }{n}}=?$
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