Joshua wrote: << On Sat, Jun 12, 2010 at 1:04 PM, Dan Asimov <dasimov@earthlink.net> wrote:

I recently gave a talk to a bunch of smart youngsters about complex numbers, and was unable to find a truly graceful way to explain why

exp(ix) = cos(x) + i sin(x),

(without deriving their Taylor series).

So if anyone knows a way to see this, I'd love to know it.

But of course that would require giving exp a meaning on the imaginaries.

You need some physics, so that may lose some youngsters, but I find the first chapter of Tristan Needham's Visual Complex Analysis to give a very nice way of thinking about complex numbers addition, multiplication, and exponentiation, and it naturally yields this identity.

...

If you can sketch the derivation, that would be great. If on the other hand it requires reading the first chapter of that book, then I'd just as soon derive the Taylor series. Also: A physics derivation would surprise the heck out of me, though it would certainly be interesting. Still, what I'm really looking for is a math derivation. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx