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*To*: Alex Shinn <alexshinn@xxxxxxxxx>*Subject*: Re: infinities reformulated*From*: Per Bothner <per@xxxxxxxxxxx>*Date*: Mon, 30 May 2005 23:52:44 -0700*Cc*: Chongkai Zhu <mathematica@xxxxxxxxx>, srfi-70 <srfi-70@xxxxxxxxxxxxxxxxx>*Delivered-to*: srfi-70@xxxxxxxxxxxxxxxxx*In-reply-to*: <5fb7e08705053022211951ff45@xxxxxxxxxxxxxx>*References*: <20050531013438.308C813AD@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx> <5fb7e08705053018594e807371@xxxxxxxxxxxxxx> <429BE2BD.9080104@xxxxxxxxxxx> <5fb7e08705053022211951ff45@xxxxxxxxxxxxxx>*User-agent*: Mozilla Thunderbird 1.0.2-1.3.2 (X11/20050324)

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/

**Follow-Ups**:**Exact irrationals***From:*Aubrey Jaffer

**References**:**Re: infinities reformulated***From:*Chongkai Zhu

**Re: infinities reformulated***From:*Alex Shinn

**Re: infinities reformulated***From:*Per Bothner

**Re: infinities reformulated***From:*Alex Shinn

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