Here’s a short mathematical riddle for you.

Let i be the square root of -1. Then i2 = -1, i3 = i . i2 = -i and i4 = i2 . i2 = 1, and so on. Thus we know how to calculate i to any integer power you might care about.

A popular way to interpret complex numbers of the form a + i b, where a and b are real numbers, is as points in a two-dimensional plane. In this plane, such complex numbers correspond to points with coordinates (a, b). Therefore multiplying any number by i corresponds to an anticlockwise rotation by 90o.

Now, how should one calculate i to the ith power?

Hint: You can find at least one real number n such that ii = n.

Update: I’ll throw in Euler’s formula for free:

tex:e^{i \theta} = \cos \theta + i \sin \theta