Newsgroups: sci.math.symbolic,comp.lang.functional,comp.lang.lisp
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!fas-news.harvard.edu!newspump.wustl.edu!news.ecn.bgu.edu!vixen.cso.uiuc.edu!howland.reston.ans.net!ix.netcom.com!netcom.com!NewsWatcher!user
From: hbaker@netcom.com (Henry Baker)
Subject: Re: arbitrary-precision real arithmetic
Message-ID: <hbaker-2608950928260001@192.0.2.1>
Sender: hbaker@netcom14.netcom.com
Organization: nil organization
References: <41cuio$n6q@cantaloupe.srv.cs.cmu.edu> <41g5eb$p41@sparky.franz.com> <AUGUSTSS.95Aug24234932@dogbert.cs.chalmers.se> <STEVEW.95Aug26174514@debretts.comp.vuw.ac.nz> <41mttlINNd9i@topdog.cs.umbc.edu>
Date: Sat, 26 Aug 1995 17:28:26 GMT
Lines: 35
Xref: glinda.oz.cs.cmu.edu sci.math.symbolic:18897 comp.lang.functional:6279 comp.lang.lisp:18955

In article <41mttlINNd9i@topdog.cs.umbc.edu>, pipkin@cs.umbc.edu (Stephen)
wrote:

> >
> >Computers can only handle rational numbers.
> >
>     Couldn't we write a program (say in Prolog) which would be able to
> compute such things as:
> 
>      (sqrt(2)) ^ 4    computes to  4
>      (pi ^ 2) / pi    computes to  pi
> 
> So a computer can compute with irrational numbers.

Yes and no.  There are 'logical theories' such as the theory of
'real closed fields' that have a decision procedure, thanks to Tarski
circa the late 1940's.  This allows most of classical geometry, for
example, to be shown to be amenable to a decision procedure.  Such a
theory can handle algebraic numbers such as sqrt(2).  This decision
procedure is extraordinarily expensive, however, so it isn't particularly
useful right now.

However, it has also been shown that putting complex functions -- especially
periodic functions -- into the theory makes it undecidable, because one
can now talk about integers which Goedel proved undecidable in the 1930's.

So your second example which talks about pi is off-base.

If you really care about this stuff, I would advise acquiring Macsyma or
Maple, which both know a lot about algebraic numbers and various kinds
of algebraic identities.

-- 
www/ftp directory:
ftp://ftp.netcom.com/pub/hb/hbaker/home.html
