@Julian: Thanks for the note! I’d like to cover the Fourier Transform eventually :).
@Farid: Appreciate the note – I’m still a little confused, but as you say, you can plot out “e” with smaller approximations for n (like n = 10) and see the imaginary interest “wrapping around the circle”. The larger your n, the closer you follow the circle [with n=1, your interest is very “chunky” and doesn’t keep turning you in micro-increments]. But thanks for the note!
@mra: Great question. I actually don’t see “e” as a number by itself – it’s the result of starting with 1, and growing at 100% interest as fast as you can. The integral of 1/x can also get you to this “infinite growth” process, see this article:
http://betterexplained.com/articles/developing-your-intuition-for-math/
So instead of “why e?” think “Why is it this process of continuous growth?”. It’s important not see “e” as a magic spell :).
Sine is actually a similar process of continuous change as well, which makes it fit with e much better. See http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/
@Tim: Great quote. I can’t say I’ve really learned something unless it is intuitive to me. Otherwise, I’m just parroting facts someone else found out.