12 Jun
2010
12 Jun
'10
3:06 p.m.
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. --Joshua