Chapter 9.3, Problem 61AYU

To determine

To find the four complex fourth roots of unity and plot them.

Answer to Problem 61AYU

The four complex roots of unity are

Explanation of Solution

1 can be written as the complex number

Rewrite 1 in the form

The magnitude of 1,

  

The distinct complex roots of are, where is a complex number, and

The complex roots of are

  

Where

Substitute 0 for in to get the first complex root of 1.

  

Substitute 1 for k in to get the second complex root of 1.

  

Substitute 2 for k in to get the third complex root of 1.

  

Substitute 3 for k in to get the fourth complex root of 1.

  

Therefore, the four complex roots of unity are

The four complex roots has the same magnitude, 1. So, they are at the same distance from the origin. Also, the arguments of the cube roots are respectively and the argument between two consecutive pairs is