On Jun 12, 2010, at 4:04 PM, Dan Asimov 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.
I've sometimes explained e^(i x) by a (admittedly strange) economic metaphor for imaginary interest. Imagine a cooperative bank where you deposit or borrow money, with interest paid in gold = imaginary money. There is a property tax on gold at the same rate, e.g. $1 earn .01 gold pieces/year, and each gold piece is taxed at $.01 / year. You can take a short position in gold, with the same terms. This is just to dress up the differential equation f'(t) = i f(t) in a more concrete form. You can then plot what happens over time in the (money, gold) plane, and it's easy to see the solutions of the differential equation go around in a circle, hence sine and cosine. Alternatively: f'(t)=i f(t) => f''(t) = -f(t). This latter equation works independently on the real and imaginary parts, and is one of the reasonable ways to define the cosine and sine functions. Bill