# Complex roots of unity

• Xeract
In summary, the person is trying to find the real and imaginary roots of z^4 = -1 and has attempted to use polar coordinates to solve it. However, the method does not seem to give the correct roots. The conversation then discusses the concept of multiplying complex numbers and finding fourth roots of (-1) by using angles to the positive real axis, which is necessary and sufficient to find all the roots.

## Homework Statement

I need to find the real and imaginary roots of z^4 = -1.

## The Attempt at a Solution

The polar coordinates of -1 are at (-1, pi), (-1, 3pi) etc so if I assume the solutions take the form z = exp[i n theta] then

n theta = pi + 2npi

This dosen't seem to give the correct roots though, what am I doing wrong? I don't want the solution, just the method so I can work it through for myself if anyone can help.

Thanks

Are you familiar with the fact that:

Multiplying two complex numbers yields a complex number whose angle to the positive real axis is the sum of the factors' angles to the same axis (while the modulus/length of the complex number gained is the product of the facturs' moduli)?

Thus, the fourth roots of a complex number must have one fourth the angle that complex number may have, as measured to the positive real axis.

Remember that the number (-1) can be said to have the angles $\pi,3\pi,5\pi,7\pi$ to the positive x-axis.
Ask yourself why these four angular representations are both necessary and sufficient to find ALL fourth roots of (-1)!

## What are complex roots of unity?

Complex roots of unity are solutions to the equation x^n = 1, where n is a positive integer. These solutions are complex numbers that, when raised to the nth power, result in 1.

## What is the significance of complex roots of unity?

Complex roots of unity are important in mathematics because they have special properties that are used in many areas, such as number theory, algebra, and geometry. They also have applications in physics and engineering.

## How many complex roots of unity are there?

There are n distinct complex roots of unity, where n is the exponent in the equation x^n = 1. These roots are equally spaced around the unit circle in the complex plane.

## What is the relationship between complex roots of unity and the unit circle?

Complex roots of unity are closely related to the unit circle in the complex plane. They lie on the unit circle and their angles correspond to the nth roots of unity. This relationship is used to visualize and understand the properties of complex roots of unity.

## How are complex roots of unity used in mathematics?

Complex roots of unity have many applications in mathematics, including in number theory, algebra, and geometry. They are also used in signal processing, Fourier analysis, and other areas of mathematics and engineering.

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