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From: rjnoe@ihlts.UUCP (Roger Noe @ N41:48.5, W88:07.2)
Newsgroups: net.math,net.philosophy
Subject: Re: i**i, foundations and philosophy
Message-ID: <363@ihlts.UUCP>
Date: Wed, 15-Feb-84 11:49:54 EST
Article-I.D.: ihlts.363
Posted: Wed Feb 15 11:49:54 1984
Date-Received: Thu, 16-Feb-84 02:44:13 EST
References: <230@houem.UUCP>
Organization: AT&T Bell Labs, Naperville, IL
Lines: 64

To begin with, one must define elementary functions for complex numbers
in such a way as to be consistent with their real counterparts.  Among
other things, this means than ln must be the inverse of exp.  Further, the
same rules of logarithms and exponents must apply.  For example,
ln(a^b) == b*ln a, e^(x+y) == e^x * e^y, etc.  Any other technique
leads to inconsistencies, forcing one to treat numbers with complex
components as entities entirely different from real numbers.  This
defeats the purpose of complex numbers.  Similar problems are encountered
by treating negative integers as fundamentally different, subject to
different laws of mathematics than positive integers.  A better idea
was to treat them all as simply integers.  The same tendency to make
mathematics general-purpose can be seen in the integration of irrational
numbers with the rationals to make the reals.  It took many people a
long time to accept the concept of zero, then rational numbers, then
negative numbers, then irrationals, and finally complex numbers.  (Let's
skip extended reals for the moment.)  But the wisest course is to for-
mulate general rules in math to apply to all these sets of numbers.

Until we get to complex numbers, we have a problem in mapping.  With any
of the other subsets, elementary functions can easily give us results
that don't map into the same set; subtraction on positive numbers,
division on integers, and both square root and logarithms on reals.
But with extended complex numbers, there is no such problem.  Any
combination of elementary functions one can imagine will map onto this
same set.  What still causes people some problems is that not all of
these map one-to-one.

Mappings that are not bijections are really rather common, even when one
is not used to dealing with anything but reals.  The square root function
is bivalued on the reals (except at zero, of course), and the square
(x^2) function maps two-to-one.  It is not surprising to find that
logarithms on complex numbers are multivalued.  To reject this is
equivalent to rejecting the Euler formula:
		e^(i*t) = cos t  +  i * sin t
Because e^[i*(t+2*PI*n)] has a single value for all integers n.  Thus the
exponential function is an infinite-to-one mapping.  To reject this is to
reject the basic concept of complex numbers.  Real numbers are on a line,
so complex numbers are on a plane.  Such a concept is very useful, very
natural for our Euclidean minds.  Certainly one can come up with a
self-consistent, but very different concept of complex numbers, but
human beings would have a difficult time using it.  I think mathematics
exists independently of any intelligence, but that doesn't mean that
all MODELS of it are the same.  The model must be suited to our minds.

Accepting the Euler formula means accepting traditional rules for a real
raised to a complex power.  But this also means accepting that
		ln i = i * (2n + .5) * PI
because
		e^[i*(2n+.5)*PI] = i
for all integers n.  It is fallacious to presume that just because certain
mappings are one-to-one in the real domain that the same must hold true
in the complex domain.  Keeping the utility of mathematics, we must say
that ln(i^i) = i * ln i.  Similar rules for multiplication of complex
numbers require that i * i = -1, a pure real.  The fact that i^i is
infinitely multivalued is inescapable from the accepted model (or
representation) of mathematics.  One can come up with a different repre-
sentation, but that does not make it useful or even consistent.

The best approach, for those who have trouble with i^i, is to use the
principal value, where n=0.  Then the modulus of all complex numbers must
be within the interval (-PI, PI].  Of course, any single half-open real
interval with a 2*PI width works just fine.
		Roger Noe		ihnp4!ihlts!rjnoe
		AT&T Bell Laboratories