Thursday, March 14, 2013

Solution of Complex Roots of Unity Problem



Solution: We have, n = 4 with:

wn = 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:




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