Problem Solution: Basics of Complex Numbers (Part 4) - Complex Roots

 Solution:  

The diagram:

We have, n = 4 with:

w_n = cos (2 p)k/ n + isin(2 pk)/n

For k = 0, 1, 2 and 3 then,

The first root:: w₀ = cos(0) + isin(0) = 1

The second root:  w₁ = cos (p/2) + isin(p/2) = 1i = i

The third root:  w₂ = cos (p) + isin(p) = -1

And the fourth root:

w₃ = cos(3 p/2) + isin(3p/2) = -1i = -i

All of which can be checked using the accompanying diagram which is part of the problem solution.

Note the roots correspond to successive increases of the angle by p/2 = 90 degrees: