[ExI] imaginary numbers: RE: new entry from symphony of science

[Bill Hibbard]

On Sat, 27 Nov 2010, Bill Hibbard wrote:
> Spike wrote:
>> Here's a mathematical comment that will blow your mind
>> if you think about it hard enough, and one which also
>> has real world applications:
>> 
>> e^(i*pi) = -1
>> 
>> . . .
>> But this is a very exciting equation!  It is a result
>> of the fact that
>> 
>> e^(i*theta) = cosine(theta) + i*sin(theta)
>> 
>> !!
>> 
>> Is this cool or what?
>
> Way cool, Spike. When I learned this stuff as a tadpole
> I was fascinated by the fact that:
>
>  i^i = e^(-pi/2) = 0.207879576...
>
> is real.

I should point out that i^i has multiple values because
its computation requires taking a complex logarithm.

> Not only do complex variables have lots of practical uses,
> they are also a source of great mathematical beauty.

The first cool thing is that adding the square
root of -1 to the real numbers suffices to give
the roots of any polynomial.

The second cool thing comes when you extend
calculus to complex variables. For real variables
the limit of:

   ( f(x + delta.x) - f(x) ) / delta.x

has to be consistent for delta.x approaching
zero from the negative and positive directions.
For complex variables this limit must be
consistent approaching zero from any direction
in the complex plane. This constrains
differentiable functions to have a great deal
of structure.

If you already know calculus then complex
variables aren't that big a leap, and the
beauty is worth the effort.

Bill