Re: e to the i pi

On Thu, 8 Feb 2007, Richard Tobin wrote:

In article <JD58Kv.J3s@xxxxxx>, *** T. Winter <***.Winter@xxxxxx> wrote:

Something like (sin x)^2 + (cos x)^2 = 1 makes sense geometically.
e^(i*x) = cos x + i * sin x, as far as I know, doesn't -- but it does
make sense algebraically.

Also geometrically. Take the complex plane with real and imaginary
axis. Draw a circle with radius 1 around the origin. exp(i * x)
circles around this circle when x increases.

It certainly *corresponds* to something geometrical - cos x and sin x
are the coordinates of exp(ix), but to *make sense* geometrically
you'd need an intuition as to why exp(ix) should be a circle for real x.

-- Richard

Do matrices look geometric enough for you?

[ 1 0] [ 0 1]
Define I = [ ] and J = [ ]. It can be proven that
[ 0 1] [-1 0]

aI + bJ is isomorphic to the complex number a + bi. It can
further be proven that

[ 0 x] [ cos x sin x]
exp [ ] = [ ]
[-x 0] [-sin x cos x]

which corresponds to exp (ix) = cos x + i sin x.

Tak-Shing
.

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