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Re: infinities reformulated



Alex Shinn wrote:
On 5/31/05, Per Bothner <per@xxxxxxxxxxx> 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
per@xxxxxxxxxxx   http://per.bothner.com/