Re: apples and oranges

[Phil Parker <PPARKER@TWSUVM.UC.TWSU.EDU>]

> Date: Tue, 03 Nov 1998  9:05 -0600
> From: "Rauber, Joel              Phys" <RAUBERJ@mg.sdstate.edu>
>
>
> > Before the introduction of imaginary numbers it was a property of the
> > operation "square" that the square of any number was positive.  The
> > extension to imaginary numbers abandons this property and proposes
> > defining NEW entities whose square is negative.
> > So too, here it is proposed to define NEW entities whose absolute value
> > is negative.
>
> > abs(Q) + abs(1) = 0 is no more perverse than i^2 +1^2 = 0
>
> > Whether this leads to anything useful is another question, but it is
> > conceptually feasible.
>
> I seem to recall that there are only a finite set algebras for numbers,
> which is why we have real numbers, (obeying one algebra); complex numbers,
> obeying the property listed above; quarternions and Octonians, which have
> properties I've forgotten.  And no more?
Correct.  Loosely, complex numbers lose the ordering, quaternions lose
the commutativity, and octonians lose the associativity.  After that,
there are no more (so-called) division algebras. (There are actually
several different theorems to the same overall effect.)
   But the Clifford algebras in many ways are the infinite tower of
algebras that one should have been considering instead.
   Having abs(Q) < 0 is probably best realized with indefinite metric
tensors (e.g., the Lorentzian ones, with Minkowski space as primary
examples).  But you have to go to at least two real dimensions, and you
lose most of the algebraic properties of abs.  Again, the Clifford
algebras are the best one can do.

---------------------------------------------
Phil Parker        pparker@twsuvm.uc.twsu.edu
Random quote for this second:
  Half the work that is done in the world
  is to make things appear what they are not.
  ---E.R. Beadle