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Re: apples and oranges



Concerning Joel Rauber's comments:
I seem to recall that there are only a finite set algebra's 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? If this statement is accurate then
it is quite likely that abs(Q) + abs(1) == 0 leads to nothing useful
(unless, it already is part of the quarternion or octonian properties.

As I recall this limitation to real, complex, quaternion, and octonion
number spaces defined on R, R^2, R^4, and R^8 respectively is a consequence
of some requirements made on the definition of the product of 2 numbers
requiring that the product have certain particular useful properties. One
of these properties is that the magnitude (or absolute value, if you will)
of the product of two numbers is the product of the magnitudes of each
factor, (i.e., it is an identity that |a*b| = |a|*|b| for all elements
a & b) where the absolute value is defined as a generalized version of the
Pythagorean formula: |a| =sqrt(sum_i(a_i^2)) where a_i is the real number
associated with the i_th coordinate dimension in the number space. IOW the
magnitude of a number is the square root of the sum of the squares of its
real-valued components (each stripped of any imaginary units). Another
useful restriction on the definition of the product formula is that the
product c of any 2 numbers a & b (c = a*b) has each real-valued component
of c as a *bilinear combination* of the real-valued components of the
factors a & b. For instance, as an example, consider complex numbers with
subscripts r and i representing the real-valued components for the real and
imaginary parts respectively. In this case we have:
c_r == a_r*b_r - a_i*b_i and c_i = a_i*b_r + a_r*b_i. Notice that both c_r
and c_i are linear combinations of the components (a_r, a_i) for fixed
(b_r, b_i), and are also linear combinations of the components (b_r, b_i)
for fixed (a_r, a_i). (Thus the name bilinear.)

*If* we require that: 0) the number space and product have the property
that every number, except zero, (the additive identity) have a unique
multiplicative inverse, i.e. for every nonzero a there is a unique b such
that a*b = 1 = b*a where 1 is the multiplicative identity, 1) the product
is defined so that the components of the product are bilinear in the
components of the factors, and 2) the magnitude of a number is the + square
root of the sum of the squares of the components of the number, and 3) the
magnitude and product must obey |a*b| = |a|*|b|, then the only possible
dimensionalities for the space of numbers that are consistent with these
requirements are 1, 2, 4, and 8.

If in addition to these properties we require that 4) the product be
*associative* then the octonian case becomes disallowed. If also we
require that the product be 5) *commutative*, then both the quaterion and
the octonion cases become disallowed. If also we require that the number
space be 6) *ordered* then the complex, quaternion, and octonion cases
become disallowed, leaving only the reals left. (The reals are the only
complete Archimedian ordered field there is.)

Of course if one were to relax (rather than add to) any or all of these
requirements then one may be able to define other number spaces over other
vector spaces of other dimensionalities. For instance, the algebra of
Grassman numbers, and composite numbers made of direct sums and products of
such Grassman numbers, i.e. so-called supernumbers, are the numerical
substrate on which superfields and supersymmetry theories are built.
Rather than having 'real' and 'imaginary' parts, supernumbers have a
'body' and a 'soul'.

David Bowman
dbowman@georgetowncollege.edu