What age is a "youngster", and how "smart"? Do they already know what sine and cosine are? When I was ten my older brother asked me: "If i is the square root of -1, what is the square root of i?". I knew about the complex plane, so just started plotting various values of a+bi and their squares. An hour later I told him the answer, as near as I could tell, was about 0.7 + 0.7*i. So I think this is easy if you break it up into two pieces: Start with showing where the powers of i are. Then let A=(i+1)/sqrt(2) and plot the locations of A, A^2, A^3, A^4. If necessary you could do the same thing with A=(sqrt(3)+i)/2. The pattern is easy to notice and generalize. That gets you halfway there (establishing the sine and cosine part). I understand the other half (generalizing the exponential function to imaginary, and therefore complex, arguments) through vector calculus and the geometric interpretation of complex derivative. So, I'm no help there (-: On Sat, Jun 12, 2010 at 16:04, Dan Asimov <dasimov@earthlink.net> wrote:
I'd love to know such a way to understand their Taylor series.
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.
-- Robert Munafo -- mrob.com