Re: Representing Infinity

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Date: Thu, 08 Jul 2004 22:41:44 GMT

Barry Margolin <barmar@alum.mit.edu> writes:

> In article <bY2Ec.3108$nc.522@fed1read03>,
> Ari Johnson <ari_j@hotmail.com> wrote:
>
> > Mark McConnell wrote:
> > > Ari Johnson <ari_j@hotmail.com> wrote in message
> > > news:<2XrDc.1389$nc.986@fed1read03>...
> > >
> > >>Infinite numerical values. IEEE floating point +inf and -inf, for example.
> > >
> > >
> > > I also wish there were an integer infinity.
> >
> > I'd just be happy with a printed representation of the IEEE values. I
> > also don't see a good reason why you can't have a "rational infinity" of
> > 1/0 for +Inf and -1/0 for -Inf.
>
> Because infinities aren't rational numbers. They're acceptible in
> floating point computations because these are already approximations;
> infinities are an extension of that, as an approximation for something
> that in algebra would be represented as a limit expression.
>
> Rational arithmetic, on the other hand, is required to be exact. As
> soon as an infinity gets into the equation, all exactness is lost
> (unless you manipulate them symbolically, like Macsyma can).

I don't think it has anything to do with being exact or not. It is
just that what people think of as a "number" (this is a vague idea)
doesn't include "infinity". This topic came up in sci.math a few
weeks ago.

Consider rational numbers with an infinity, call it "inf". Let the
usual rules of aritmetic work for addition, subtraction,
multiplication and division of pairs of rationals. For combination of
a rational r and inf, set

(+ r inf) => inf
(- r inf) => inf
(* r inf) => inf
(/ r inf) => 0

(< r inf) => T

and so forth

then let (- inf inf), (/ inf inf), (< inf inf) be undefined and throw
some sort of math exception. After all, we already handle division by
zero. Nothing strikes me as inexact. However, the rationals with inf
(or +inf and -inf) would no longer be a "field".

Also, do you want one infinity or two (or more)? Rationals with inf
is distinct from rationals with +inf and -inf.

In complex analysis, the Reimann sphere is a useful concept. This is
where you adjoin a point at infinity to the usual set of complex
numbers and extend the topology such that neighborhoods of infinity
contain everything of sufficiently large magnitude. This shows that
having an infinity isn't completely useless. But the traditional
definition of complex numbers without the infinite point hints that it
is perhaps not as useful.

I think that the answer is somewhat part of
*) it is traditional to exclude "infinity" from the concept of
   "number"
*) no integer machine representation for "infinity" (but IEEE floats
   do have +inf, -inf and, i think, a combined inf)
*) did you want one infinity or two?
*) being a field seems more useful than having an infinite quantity

-- 
Johan KULLSTAM

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