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Re: infinities reformulated



On 5/31/05, Per Bothner <per@xxxxxxxxxxx> wrote:
> Alex Shinn wrote:
> >
> > It cannot be "unlimited" in the sense that it is at least memory limited.
> > No matter how much memory you have, you can never exactly represent
> > the square root of 2 with a floating point representation
> 
> 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.  Floating point representations are limited.  And the SRFI
is specifically talking about inexact infinities.

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.

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.

-- 
Alex