On the last question, I find it really interesting to note that the nth root of i will have n solutions evenly distributed around the unit circle. (As will any

On the last question, I find it really interesting to note that the nth root of i will have n solutions evenly distributed around the unit circle. (As will any unit vector.) So for the cube root of i, we'll also have -0.5√3 + 0.5i and -i as solutions, which are located at 5π/12 = 30° + 120° and 8π/12 = 30° + 2 * 120°. (On another note -i * -i * -i = -1 * -i = i. It's so simple and yet not obvious unless you do a lot of complex math...which I do not.) Read about the roots of unity (solutions to a^n = 1) to see the same concept in action. http://en.wikipedia.org/wiki/Root_of_unity http://en.wikipedia.org/wiki/Root_of_unity It all comes out of the properties of rotation. We can first show that there are n nth roots of unity. It comes down to a^n = 1 -> a^n * a^n = 1 -> (a * a)^n = 1, which effectively shows that any multiple of the angle represented by a will also be a solution. If you use the full fledged exponent formulas, it's easy to see that there will be only n of these. Then if b^n = i (our case), b^n * a^n = i * 1 -> (b * a)^n = i. So adding any of the nth roots of unity to b will get a new solution for the nth root of i. Since one of the roots of unity is always 1, aka an angle of 0, we will only get n-1 new solutions from this. Hence there are n solutions. :) EDIT: New here, didn't know the markup cues would interfere with my equations.