Chapter 9.3, Problem 65AYU

To determine

To show the complex nth roots of nonzero complex number w are equally spaced on the circle.

Answer to Problem 65AYU

the complex nth roots of a nonzero complex number w are equally spaced on the circle.

Explanation of Solution

Let be any non-zero complex number and n be any positive integer greater than or equal to 2.

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

Here the argument of the complex number is and varies as the value of k changes. Thus, the arguments of the complex roots obtained are equally spaced. Also, it is known that, the complex roots obtained from this formula when plotted on a graph form a circle with centre at origin and radius equal to the magnitude

Hence it is proved that, the complex nth roots of a nonzero complex number w are equally spaced on the circle.