So what exactly is a complex number?

[El Pitonero]

On Sep 5, 7:27 am, El Pitonero <piton... at gmail.com> wrote:
>
> I am a bit surprised that today, September 2007, in a thread about
> complex numbers, no one has mentioned about geometric algebra.

Here is a good reference for whoever is interested. It's quite
accessible to general audience.

http://www.xtec.es/~rgonzal1/treatise.pdf

If a person spends some time to look at the geometric algebra, it will
become clear that complex numbers are not that special, after all.
Hopefully the relationship between the 2-d vector plane and the
complex plane will also become more clear, as complex numbers can be
understood as rotation-dilation operators over vectors. One also
learns that complex numbers are based on a metric assumption of square
of vectors (norm) being positive (a.k.a Euclidean space). There is
nothing sacred about positively-defined metric, and in fact if one
uses mixed signature metric (pseudo-Euclidean space), one comes up
with hyperbolic numbers instead of complex numbers.