Thread: Question about Complex Number Calculation

So I'm asked to find the square-root of i which has negative imaginary part. Thinking geometrically, I can see immediately that this has modulus 1 and angle 5pi/4.

However, I started answering this by equating the square of the solution to i, hence z^2 = i = e^2(i theta + r) = e^(i pi/2). I then reason that 2i theta + 2r = i pi/2 thus r = 0 (and hence modulus is 1, as expected) and theta = pi/4. Now this is obviously a solution, but why does it seem from my calculation that there is only one solution? At what step are there really two ways to solve for theta when I thought there was only one?

Actually, I think I got this. I think the assumption was that the exponents must be equal, which is not necessarily true. All I know is that theta is an angle between 0 and 2pi which, when doubled, is the same angle as pi/2. This is true of both pi/4 and 5pi/4.