Roots of unity

Roots of unity

Roots of unity

Definition 1.5.4 Let

be a natural number such that $n \geq 2$ . A complex number

such that

is called an $n^{th}-$ root of unity.

Example 1.5.5

is a $4^{th}$ root of unity, as .
$-1/2 +i\sqrt{3}/2$ is a $3^{rd}$ root of unity. Please check this.

By the method of subsection 5.1, we compute all the $n^{th}-$ roots of unity:

$\displaystyle z^n=1 \Leftrightarrow \begin{cases}\vert z^n\vert=1 \ \operatorn... ...ert z\vert=1 \ \operatorname{arg}\nolimits (z)= \frac {2k \pi}{n} \end{cases}.$

For $k=0, \dots n-1$ , we have the following roots:

$\displaystyle z_0$	$\displaystyle = \cos 0 + i \; sin 0 =1,$
$\displaystyle z_1$	$\displaystyle =\cos \frac {2\pi}{n}+ i \sin \frac {2 \pi}{n},$
$\displaystyle z_2$	$\displaystyle =\cos \frac {4 \pi}{n}+ i \sin \frac {4 \pi}{n},$
$\displaystyle \dots$	$\displaystyle \dots$
$\displaystyle z_{n-1}$	$\displaystyle = \cos \frac {2(n-1) \pi}{n} + i \; \sin \frac {2(n-1) \pi}{n}.$

The images in Gauss-Argand plane of the $n^{th}$ roots of unity are the vertices of a regular polygone inscribed in the unit circle.

Example 1.5.6

(i) The square roots of unity (

) are

and

.
(ii) For

, the roots of unity are

, $-1/2+i \; \sqrt{3}/2$ and $-1/2-i \; \sqrt{3}/2$
(iii) For

, the roots of unity are

**Figure 9:**
$\begin{figure}\mbox{ \subfigure[$n=2$]{\epsfig{file=2RootsUnity.eps,height=4cm}... ...ad \subfigure[$n=4$]{\epsfig{file=4RootsUnity.eps,height=4cm}} }\end{figure}$

By De Moivre's formula De Moivre's formula, the following holds:

$\displaystyle z_2$	$\displaystyle = z_0^2,$
$\displaystyle z_3$	$\displaystyle = z_0^3,$
$\displaystyle \dots$	$\displaystyle \dots$
$\displaystyle z_{n-1}$	$\displaystyle = z_0^{n-1},$
$\displaystyle z_0$	$\displaystyle = z_0^n.$

An immediate consequence is as follows:

Proposition 1.5.7 Let be a natural number such that $n \geq 2$ . The sum of all $n^{th}$ roots of unity is equal to 0.

Proof

$\displaystyle z_0+z_1+z_2+ \dots +z_{n-1}$ $\displaystyle = 1+ z_1 + z_1^2 + z_1^3 + \dots + z_1^{n-1}$

$\displaystyle \quad$ $\displaystyle = \frac {1 -z_1^n}{1-z_1}$

$\displaystyle \quad$ $\displaystyle =0.$

$\qedsymbol$

Now, let us see a nice application of these roots of unity.

Proposition 1.5.8 Let be a natural number such that $n \geq 2$ . Suppose that are two complex numbers such that . Then we obtain all the $n^{th}$ roots of by separate multiplication of by all the $n^{th}$ roots of unity.

Example 1.5.9

$\displaystyle v = \sqrt{2+\sqrt{2}} + i \; \sqrt{2-\sqrt{2}}.$

Squarring twice, we obtain

$\displaystyle v^4=16i.$

By Proposition 5.8, the $4^{th}$ are given by:

$\displaystyle 1 \cdot v$ $\displaystyle = \sqrt{2+\sqrt{2}} + i \; \sqrt{2-\sqrt{2}}$

$\displaystyle i \cdot v$ $\displaystyle = -\sqrt{2-\sqrt{2}} + i \; \sqrt{2+\sqrt{2}}$

$\displaystyle (-1) \cdot v$ $\displaystyle = -\sqrt{2+\sqrt{2}} - i \; \sqrt{2-\sqrt{2}}$

$\displaystyle (-i) \cdot v$ $\displaystyle = \sqrt{2-\sqrt{2}} - i \; \sqrt{2+\sqrt{2}}$

An additional result of this example is the following: compute polar forms for the $4^{th}$ roots of . Theses roots have arguments $\pi /8$ , $5\pi /8$ , $9\pi /8$ and $13\pi /8$ . They fit exactly the order of the roots given above.