Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Prepcourse Mathematics - Basics - Complex Numbers

Complex Roots of Unity

[previous page] [next page] [table of contents][page 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

\includegraphics[width=0.5\linewidth]{Bild_Einheitswurzel.eps}

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)
Calculating the third roots of unity, the general formula

$\displaystyle z_k = \exp{\frac{2 \pi \mathrm{i} k}{3}} \qquad k = 0,1,2 \,, $

yields
$\displaystyle z_0$ $\displaystyle =$ $\displaystyle \exp{0} = 1$  
$\displaystyle z_1$ $\displaystyle =$ $\displaystyle \exp{\frac{2 \pi \mathrm{i}}{3}} = \cos{\frac{2 \pi}{3}} + \mathrm{i} \sin{\frac{2 \pi}{3}} = -\frac{1}{2} + \mathrm{i} \frac{\sqrt{3}}{2}$  
$\displaystyle z_2$ $\displaystyle =$ $\displaystyle \exp{\frac{4 \pi \mathrm{i}}{3}} = \cos{\frac{4 \pi}{3}} + \mathrm{i} \sin{\frac{4 \pi}{3}} = -\frac{1}{2} - \mathrm{i} \frac{\sqrt{3}}{2}\, .$  


So the root $ z^{\frac{1}{3}}$ is ambiguous, there exist $ 3$ different values.

Analogously the fourth roots of unity are $ 1, \mathrm{i}, -1, -\mathrm{i}$.

(Authors: Höllig/Kopf/Abele)
[previous page] [next page] [table of contents][page overview]
  automatically generated 10/23/2009