Re: Possreps and numeric types
From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Mon, 26 Mar 2007 12:45:27 GMT
Message-ID: <HfPNh.15436$PV3.159008_at_ursa-nb00s0.nbnet.nb.ca>
>
> I disagree with Bob. If pairs of integers are used to represent rationals,
> rational(N,M) = N/M,
> Then any rational can be represented exactly (not an approximation) within
> the scheme, provided that N and M can both be represented within the
> scheme.
>
> If the representation scheme for integers is indefinitely extensible, then
> the field of rationals representable is likewise indefinitely representable.
> Common decimal notation of integers is indefinitely extensible. There are
> other schemes.
>
> In any finite computer, it is only possible to actually represent a finite
> subset of the integers, and thus it is only possible to represent exactly a
> finite subset of the rationals. The problem is that the finite subset of
> rationals will not, in general, exhibit closure under addition. Thus one is
> forced into the realm of approximation as soon as one begins to store the
> results of arithmetic computation.
Date: Mon, 26 Mar 2007 12:45:27 GMT
Message-ID: <HfPNh.15436$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. Received on Mon Mar 26 2007 - 14:45:27 CEST