On Sat, Jun 12, 2010 at 1:04 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Mike wrote:
<< Is there a geometric way to understand the Taylor series for sin and cos? The closest I've been able to find is a combinatorical explanation (below), but it doesn't seem to help much.
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.
I think the Greeks would have had a hard time with Taylor series, because they insisted that all numbers have units. You can't have X cm + (X cm)^2. Can one do unitless calculations geometrically? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com