So what exactly is a complex number?

[El Pitonero]

On Sep 1, 3:54 am, Grzegorz S odkowicz <jerg... at wp.pl> wrote:
>
> You're mixing definition with application. You didn't say a word about
> what complex numbers are, not a word about the imaginary unit, where
> does it come from, why is it 'imaginary' etc.  
> ...
> I'd also like to see a three-dimensional vector
> represented by a complex number.

Well, maybe you'd like to learn something about Geometric Algebra. :)

I am a bit surprised that today, September 2007, in a thread about
complex numbers, no one has mentioned about geometric algebra. There
is an older way of looking at complex numbers: the imaginary unit as
square root of -1. And then there is a new way of looking at complex
numbers: as the multi-vector space associated to the two-dimensional
vector space. So, yes, complex numbers are a bit like vectors, but
more precisely, they are "multi-vectors", where the first component
(the real part) is a "scalar", and the second part (the imaginary
part) is an "area".

This may all be just paraphrasing. But it gets more interesting when
you go to higher dimensions. You'd like to know whether there are
extension of complex numbers when you go to three dimensional space,
and the answer is definitely YES! But the new multivectors live in 8
dimensional space. Geometric product not only make this possible, but
this product is invertible. Moreover, complicated equations in
electromagnatism in physics (Maxwell's equations) can be written in a
single line when you use geometric algebra. When you see some of the
features of geometric algebra, you will realize that complex number
are but a small part of it. (There is a paper with the title
"Imaginary numbers are not real...", I guess the title says it all.)

Anyway, there are always many ways of looking at the same thing.
Geometric algebra is one. Who knows what tomorrow brings? But as of
today, I'd say that it's better to teach school children about
geometric algebra, instead of the present way of introducing imaginary
unit. Just my opinion.