Roots of unity.

Definition 1.5.4   Let $n$ be a natural number such that $n \geq 2$ . A complex number $u$ such that $u^n=1$ is called an $n^{th}-$ root of unity.

Example 1.5.5       

1. $i$ is a $4^{th}$ root of unity, as $i^4=1$ .
2. $-1/2 +i\sqrt{3}/2$ 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:

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

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

$$z_0 = \cos 0 + i \; sin 0 =1,$$

$$z_1 =\cos \frac {2\pi}{n}+ i \sin \frac {2 \pi}{n},$$

$$z_2 =\cos \frac {4 \pi}{n}+ i \sin \frac {4 \pi}{n},$$

$$\dots \dots$$

$$z_{n-1} = \cos \frac {2(n-1) \pi}{n} + i \; \sin \frac {2(n-1) \pi}{n}.$$

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 ($n=2$ ) are $1$ and $-1$ (see Figure 9(a).

(ii)

For $n=3$ , the roots of unity are $1$ , $-1/2+i \; \sqrt{3}/2$ and $-1/2-i \; \sqrt{3}/2$ (see Figure 9(b).

(iii)

For $n=4$ , the roots of unity are $1$ , $i$ , $-1$ and $-i$ (see Figure 9(c).

Figure 9: The images of the roots of unity.

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

$$z_2 = z_0^2,$$

$$z_3 = z_0^3,$$

$$\dots \dots$$

$$z_{n-1} = z_0^{n-1},$$

$$z_0 = z_0^n.$$

An immediate consequence is as follows:

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

Proof. We have:

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

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

$$\quad =0.$$

$\qedsymbol$

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

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

Example 1.5.9   Take the number

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

Squarring twice, we obtain

$$v^4=16i.$$

By Proposition 5.8, the $4^{th}$ roots of $16i$ are given by:

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

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

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

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

An additional result of this example is the following: compute polar forms for the $4^{th}$ roots of $16i$ . 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.