Alex Shinn wrote:
> On 5/31/05, Per Bothner <xxxxxx@bothner.com> wrote:
>>The message you quoted did not say "floating point".  There are finite
>>ways of representing trancendentals.  "The square root of 2" is one.
>
> Chongkai was talking about the BigFloat implementations in MacLisp
> and Perl.

I would read Chongkai's posting as making separate points about
MacLisp bigfloats (in reponse to your explicit question), followed
by a more general point about "unlimited precision numbers", which
aren't necessarily floating-point.

> And the SRFI is specifically talking about inexact infinities.

But exact infinities have also been proposed and discussed.

> Symbolic manipluation systems can implement irrationals and transcendentals
> in many ways, but these are not inexact numbers.  The very concept of
> inexact implies you're using a limited representation which loses information.

Yes and yes.

> But for the sake of argument, even exact numbers are limited on a finite
> computer architecture.  We like to pretend bignums are unbounded, but
> they aren't.  BigRationals have the further problem that even if the computation
> itself isn't getting any larger, repeated arithmatic can cause the
> representation
> to require more and more memory.  More complete symbolic representation
> systems such as algebraic roots or Taylor series can become exponentially
> larger in simple repeated calculations when even fewer terms are able to cancel
> out.  Unless you have the good fortune to be using a Turing machine everything
> is limited.

Yes.  But orthogonal to the issue of exactness and precision of
non-rational real numbers: A language implementation could have exact
"infinite-precision" real arithmetic to the same extent that it has
"infinite-precision" rational arithmetic.  The former is even more
resource-hungry, and has some serious limitations in that comparing
two exact real numbers isn't always possible.  But it still makes sense
to allow for exact real arithmetic.
--
	--Per Bothner
xxxxxx@bothner.com   http://per.bothner.com/