Applications and Extensions

Question

Find the four complex fourth roots of unity (1) and plot them.

Answer 1

Step 1 :

Theorem :

If n is a positive integer, the complex number has exactly n distinct copmplex roots.

The complex roots are  , where .

Step 2 :

The complex number is .

First convert the complex number into polar form.

Compare the complex number with .

.

The angle is .

The polar form of is .

The complex fourth roots of  are :

, here n = 4 and k = 0, 1, 2 and 3.

Step 3 :

The four complex roots are :

For k = 0,

For k = 1,

For k = 2,

For k = 3,

The complex fourth roots of are

Step 4 :

Graph the complex number  is,

Solution :

The complex fourth roots of unity are

Graph the complex number  is,