Midnight Math

I made the mistake of drinking coffee tonight and couldn't sleep. Trying to understand math usually works, so I started thinking about the famous formula:

exp(i*x) = cos(x) + i*sin(x)

It seemed to me that that equation combined simplicity and great beauty. So how to understand it? If you realize that exp(i*x) lies on the unit circle in the complex plain at angle x wrt the real axis, it's obvious, but how do you get to that? Comparing the power series works, but somehow appealing to the full machinery of differential calculus seemed like a cheat too. I wanted to understand it in more fundamental terms.

Suppose you multiply a real number a by the imaginary unit i. If you think of a as a point a distance a to to right of zero on the real axis, i*a is a similar distance up the imaginary axis. Similarly, the point a + i*b can be reached from zero by going 'a" to the right and then up 'b'. It thus lies at an angle arctan(b/a) from the real axis at distance sqrt(a^2+ b^2) from zero. If we multiply a complex number z by another complex number a + i*b, the answer is another complex number lengthened by a factor of sqrt(a^2 + b^2) and rotated an angle arctan(b/a) in the counterclockwise direction.

Consider the complex number 1 + i*x/n. Multiply it by anything and you get that rotation by arctan(x/n) and magnification by sqrt(1 + (x/n)^2). If I recall correctly, the fundamental definition of exp(x) is the limit as n->infinity of
[1 + x/n]^n. In the limit as n -> infinity, arctan(x/n) -> (x/n).

[1 + x/n]^n is thus a vector in the complex plain rotated by n angles of arctan(x/n) and length of the product of n factors of sqrt(1 + (x/n)^2), which in the limit as n -> inf becomes rotated by angle x and of length 1 - that is, a point on the unit circle in the complex plain at angle x from the real axis.

I probably should check my work but maybe I can sleep now instead.