# Complex roots of unity

1. Feb 22, 2007

### Xeract

1. The problem statement, all variables and given/known data

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

3. 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

2. Feb 22, 2007

### arildno

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)!