why are the product-rules as they are?greenspun.com : LUSENET :
quaternions : One Thread A short historical intro: When Hamilton
was thinking about the 3D-complex number x+iy+jz he (1)assumed
jj=-1 and (2)added a fourth complex number k so ij=k (et voila,
quaternions). I was wondering why he made these choices (1) and
(2). Because when you write down the productmatrix for x+iy+jz
it looks very logical to choose the following products: ii=j,
jj=-i and ij=ji=-1. Was this ever an option, does it not make
any sence, or what? PS: You can make the mandelbrot fractal in
3D with these rules (as I have done using Mathematica, notebook
available). -- Jan Harm Moes (email@example.com), January 20,
2005 Hamilton was trying to accomplish two objectives. The first
was to parallel normal complex number arithmetic where: C= a +
ib C*= a - ib (C)(C*)= a^2 + b^2 as you can see if you do the
multiplication. If you create a hyper-complex number, and want
the same result, you won't get it: H= a + ib + jc H*= a - ib -
jc (H)(H*)=