Re: less than or equal to formula for complex numbers

[Lukas Reichlin]

On 18.12.2013, at 03:31, Alasdair McAndrew <address@hidden> wrote:

> The octave definition is pretty arbitrary.  You may like to consider which of 
> the following properties it satisfies:
> 
> !x<x
> x<y & y<z -> x<z
> For any x,y then one of the following must hold: x<y, y<x, x=y
> If x<y then x+a<y+a for all a
> If x<y and a>0 then xa<ya.
> 
> In fact, it can be proved that there is no possible ordering on the complex 
> numbers which satisfies all of these (that is, makes the complex numbers an 
> ordered field).  So you can pick any ordering you like, and decide which of 
> the above ordering properties you're prepared to live without.  One standard 
> ordering is lexicographic, which can easily be adapted to quaternions:
> 
> a+bi+cj+dk<A+Bi+Cj+Dk iff a<A or, a=A and b<B, or a=A,b=B and c<C, or 
> a=A,b=B,c=C,d<D.
> 
> (This won't satisfy the last property above).
> 
> cheers,
> Alasdair

Thanks for the hint about the ordered fields! I noticed some oddities with 
complex numbers and quaternions, but I thought I made a silly mistake when 
comparing them; e.g. script [1]. I'm indecisive which ordering to choose, but 
maybe I should accept that there is no such thing as "quaternion 1 is greater 
than quaternion 2" and accordingly, should only provide the comparison 
operators == and != in my quaternion package.

Best regards,
Lukas


[1]
http://sourceforge.net/p/octave/quaternion/ci/default/tree/devel/test_eq.m