polar coordinates

Question

Write in polar coordinates all 8-th roots of unity. Find all pairs of roots r1 and r2
such that r1 = r2

Answer 1

To multiply complex numbers in polar coordinates we multiply the magnitude and add the angle. In the case of magnitude 1, this is de Moivre's theorem that Mark M. referes to. So for example (-1)²=1 because -1 in polar is (1,\pi) and (1,\pi) squared =(1*1,\pi + \pi). This isn't great notation, so written as complex numbers in polar form, we have

 

-1 = cos \pi + i sin \pi   ( = -1 + 0i)

 

 

That's a second, or square, root of  unity. The other one (there are two) is 1 itself.

 

An eighth root of unity w must satisfy w⁸=1. Just as there were 2 second roots of unity, there are 8 eighth roots of unity. So, find an eighth of 2\pi (an eighth of a circle), and then twice that, three times that, etc. up to the full circle (1) itself. This gives the eight angles; the magnitudes must be real eighth roots of 1, so they must be 1 (or -1); we use 1, and get the -1 from when we reach the opposite angle.

 

To prove this approach is correct, we go back to de Moivre's theorem that

(cos x + i sin x)⁸ = (cos 8x + i sin 8x) = 1

and solve for all values of x that satisfy sin 8x = 0, cos 8x = 1. Since cos u = 1 implies sin u = 0, we solve for cos 8x = 1, or 8x = some multiple of 2\pi. We use 2\pi itself, then 2*2\pi, then 3*2\pi, etc. until we have completed a full circle with the values of x we obtain.

I hope that helps.