Thanks! No, I worked it out for myself while I was thinking about what the exponential of a linear operator is. The explanation at my school was the one with Taylor Series.

By the time I got to university it was just taken as a given, so nobody explained it at all.

My explanation seems to have attracted a lot of positive votes in a few minutes. I've written a more detailed version here if anyone doesn't get it:

http://johnlawrenceaspden.blogspot.com/2009/10/ei-pi-1.html

If it gets much attention I might draw the diagrams that go with it.

I'm always bewildered how many scientific types (and even some mathematicians) find complex numbers mystical. They were originally discovered by mystic methods, but Argand showed us what was really going on, and they're no more weird than 2-dimensional vectors.

Actually a lot less weird than the Reals, which really are magical and mysterious, but which everyone seems to be quite happy with!