Re: Possreps and numeric types

From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Mon, 26 Mar 2007 14:42:29 GMT
Message-ID: <pZQNh.15473$PV3.159172_at_ursa-nb00s0.nbnet.nb.ca>

David Cressey wrote:

> "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.

I must not have been clear enough. Sorry. Received on Mon Mar 26 2007 - 16:42:29 CEST