Why are we doing this complex number stuff? Well, all of electrical engineering is done with complex arithmetic. All optics calculations are done with complex arithmetic. All quantum mechanics is done with complex arithmetic. So, we study it.
Well, as promised, now that we have something new to work with - complex numbers - one of the first thing we're going to do is exponentiate them. First, we have to learn a couple things about the exponential function.
X2 * X3 = X5. When you multiply terms, you add exponents. Similarly, X5 = X(2+3) = X2 * X3. You can take an exponent apart into the sum of two pieces, and turn this into two differene exponentials multiplied together. Similarly, eA*eB = e(A+B). Immediately we see that e(A+Bi) = eA * eBi. eA is just a number, and we're pretty much bored with this. However, this thing, eBi, this is new, so we'll look at this more closely. Anyway, we see that e(complex number) is e(real number) * e(imaginary number), so all the interesting part of exponentiating a complex number is in the imaginary part.
Euler figured out what eiB is. eiB = cosine( B ) + i sine( B ).