From: "JACK L. URETSKY (C)1998; HEP DIVISION, ARGONNE NATIONAL LAB ARGONNE, IL 60439" <JLU@hep.anl.gov>
Date: Wed, 4 Nov 1998 20:47:42 -0600
Hi Bob-
The usual definition of "absolute value" (see, e.g., McCarty,
"Topology", ch V) is that "abs" is a map from the set of complex numbers
to the set of POSITIVE real numbers. The obvious generalization for any
set is that "abs" maps the set into the set of positive real numbers.
You have not, therefore, defined a "new entity" Q, but have
instead placed a different (and, I would say, strained) meaning on
the expression "absolute value".
One is free to make any definition one likes, subject to the constraint
of consistency. The trouble with unconventional definitions, however, is
that they impede communication.
Your example of the square is not a good analogy. The only necessary
property of "square" is that it be any quantity times itself. The need for
imaginaries arises when I insist on algebraic completeness, that is, that
every number have a square root. It was not a "property" of the operation
square that every square was positive, it was that there were no known
examples of quantities with negative squares. Lack of examples does not
define a "property".
The situation with absolute value is quite different. "Absolute
vaulue" is customarily <defined> to have the positivity property - it is
the positive number that defines the <size> of the complex (or other)
quantity to which it is applied.
Regards,
Jack
*******************************************************
Jack,
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
-Bob
*******************************
"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography