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Roots of Unity
Re: Roots of Unity
If we write a complex number in polar form, that is, $z = r (\cos(\theta) + i \sin(\theta))$, de Moivre's Theorem says that $z^n = r^n (\cos(n\theta) + i \sin(n\theta))$. Thus, the solutions to the equation $z^n = m$ ($z = m (\cos(2\pi) + i \sin(2\pi))$) are precisely $$z = m^{\frac{1}{n}} \left(\cos\left(\frac{2k\pi}{n}\right) + i \sin\left(\frac{2k\pi}{n}\right)\right),$$ for $k = 0, \dots, n-1$.
This is exactly why the roots of unity are evenly spaced around the unit circle, and the roots of the equation $z^n = m$ are evenly spaced around a circle with radius $m^{\frac{1}{n}}$.
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