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Complex Roots of Unity

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

The equation

$\displaystyle z^n = 1

has exactly $ n$ solutions in $ \mathbb{C}$

$\displaystyle z_k = w_n^k,\quad w_n = \exp(2\pi\mathrm{i}/n),
\quad k=0,\ldots,n-1\,.

Those are called complex roots of unity


As illustrated in the figure the $ n$ -th roots of unity form a regular $ n$ -polygon inscribed in the unit circle.

(Authors: Höllig/Kopf/Abele)

see also:


  automatically generated 10/ 4/2007