Re: Possreps and numeric types

"Bob Badour" <bbadour_at_pei.sympatico.ca> wrote in message news:HfPNh.15436$PV3.159008_at_ursa-nb00s0.nbnet.nb.ca...
> David Cressey wrote:
>
> > "Marshall" <marshall.spight_at_gmail.com> wrote in message
> > news:1174863816.647794.146930_at_d57g2000hsg.googlegroups.com...
> >
> >>On Mar 25, 1:30 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
> >>
> >>>Marshall wrote:
> >>>
> >>>
> >>>>So what if we had an internal representation for
> >>>>integer similar to java.math.BigInteger, and an
> >>>>internal representation for rational that was a pair
> >>>>of integers. We can define *exact* operators for
> >>>>these types for basic arithmetic functions.
> >>>
> >>>I disagree. Unless one has infinite precision, rational is
> >>>always an approximation.
> >>
> >>"Approximation" is perhaps not the best choice of words.
> >>We certainly have resource limits in our finite computers.
> >>There are computations that we can't do because
> >>we don't have the resources. For example, a computer
> >>might be able to add together two one billion digit
> >>integers, but not be able to add together two ten billion
> >>digit integers because it didn't have enough memory.
> >>That doesn't mean the result of adding the two
> >>one billion digit integers is approximate; on the
> >>contrary, it is precise and exact. Or consider
> >>java.util.BigInteger. Any answer you get from it
> >>will be precise, and it can handle up to four
> >>billion digit numbers. If it can't give you an answer,
> >>it'll fail in a way that can't be mistaken for an
> >>answer.
> >
> > I disagree with Bob. If pairs of integers are used to represent
rationals,
> > rational(N,M) = N/M,
> > Then any rational can be represented exactly (not an approximation)
within
> > the scheme, provided that N and M can both be represented within the
> > scheme.
> >
> > If the representation scheme for integers is indefinitely extensible,
then
> > the field of rationals representable is likewise indefinitely
representable.
> > Common decimal notation of integers is indefinitely extensible. There
are
> > other schemes.
> >
> > In any finite computer, it is only possible to actually represent a
finite
> > subset of the integers, and thus it is only possible to represent
exactly a
> > finite subset of the rationals. The problem is that the finite subset
of
> > rationals will not, in general, exhibit closure under addition. Thus
one is
> > forced into the realm of approximation as soon as one begins to store
the
> > results of arithmetic computation.
>
> How exactly are you disagreeing with me? It seems to me you agree
entirely.

I must have misread you. Sorry. Received on Mon Mar 26 2007 - 15:30:13 CEST