My goal is to get all these concepts out of awe (which I had too) into a real sense of "Ah, I get it!". It doesn't help anyone to memorize incantations.

Two other insights I love:

* e as continuous growth: http://betterexplained.com/articles/an-intuitive-guide-to-ex... * radians as the perspective of the mover: http://betterexplained.com/articles/intuitive-guide-to-angle...

I realized that I didn't really get what e and radians were about -- I memorized them, but I wasn't comfortable. Once you have the right analogies, Euler's formula starts making sense (without resorting to "Oh, take the Taylor series expansion of each and see how they match up", which is essentially an incantation).