The thing about Euler's identity is it's impossible to derive the number e without some use of infinitesimals, be it the limit definition, integration, or infinite series. That makes Euler's Identity REALLY HARD TO TALK ABOUT IN LAYMAN'S TERMS. That all said, I'm really happy how this turned out.

Enjoy.

Euler’s Identity: 'The Most Beautiful Equation'

http://www.livescience.com/51399-eulers-identity.html