Home page for accesible maths Math 101 Chapter 4: Taylor series and complex numbers 4.38 De Moivre’s Theorem 4.40 Simplifying multiple angles in terms of trig powers

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4.39 Complex roots of unity

For $n\in{\textbf{N}}$ , an $n^{th}$ root of unity is a complex number $z$ such that $z^{n}=1$ .

Proposition

For $n\in{\textbf{N}}$ , there are $n$ distinct complex $n^{th}$ roots of unity given by $z=e^{2\pi ik/n}$ for $k=0,1,\dots,n-1$ . These points are equally spaced round the unit circle in the Argand diagram. Proof. Consider the polar form $z=re^{i\theta},$ so by De Moive’s theorem $z^{n}=r^{n}e^{in\theta}$ . We have $z^{n}=1$ if and only if $r^{n}=1$ and $e^{in\theta}=1$ , so $r=1$ and

in\theta=0,2\pi i,4\pi i,\dots,2(n-1)\pi i;

this gives $n$ distinct solutions

z=1,e^{2\pi i/n},e^{4\pi i/n},\dots,e^{2(n-1)\pi i/n};

an polynomial of degree $n$ has at most $n$ roots.