COmplex numbers?

Question

FInd the 8th roots of 1 expressing your answer in x+iy form

Answer 1

These are the eight solutions to z^8 - 1 = 0. These are

z = exp(i k pi/4), k = 0, 1, ..., 7.

Now convert to rectangular form using exp ia = cos a + i sin a.

Answer 2

1
-1
i
-i
(1/√2)(1 + i)
(1/√2)(1 - i)
(1/√2)(-1 + i)
(1/√2)(-1 - i)

They're at 8 points of the unit circle in the complex plane. In complex analysis, multiplication of unit vectors is simple as adding angles, so it's easy to see that for any integer n, multiplying n/8 by 8 will get you an integer, which puts the final vector as a postive real, or 1 in this case.