## Roots of unity.

Definition 1.5.4   Let be a natural number such that . A complex number such that is called an root of unity.

Example 1.5.5

1. is a root of unity, as .
2. is a root of unity. Please check this.

By the method of subsection 5.1, we compute all the roots of unity:

For , we have the following roots:

The images in Gauss-Argand plane of the 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 (see Figure 9(a).
(ii)
For , the roots of unity are , and (see Figure 9(b).
(iii)
For , the roots of unity are , , and (see Figure 9(c).

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

An immediate consequence is as follows:

Proposition 1.5.7   Let be a natural number such that . The sum of all roots of unity is equal to 0.

Proof. We have:

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

Proposition 1.5.8   Let be a natural number such that . Suppose that and are two complex numbers such that . Then we obtain all the roots of by separate multiplication of by all the roots of unity.

Example 1.5.9   Take the number

Squarring twice, we obtain

By Proposition 5.8, the roots of are given by:

An additional result of this example is the following: compute polar forms for the roots of . Theses roots have arguments , , and . They fit exactly the order of the roots given above.

Noah Dana-Picard 2007-12-24