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From: hbaker@netcom.com (Henry Baker)
Subject: Re: arbitrary-precision real arithmetic
Message-ID: <hbaker-2708951704280001@192.0.2.1>
Sender: hbaker@netcom11.netcom.com
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References: <41cuio$n6q@cantaloupe.srv.cs.cmu.edu> <STEVEW.95Aug26174514@debretts.comp.vuw.ac.nz> <41mttlINNd9i@topdog.cs.umbc.edu> <hbaker-2608950928260001@192.0.2.1> <DDzExB.KGv@bath.ac.uk>
Date: Mon, 28 Aug 1995 01:04:28 GMT
Lines: 42
Xref: glinda.oz.cs.cmu.edu sci.math.symbolic:18905 comp.lang.functional:6284 comp.lang.lisp:18968

In article <DDzExB.KGv@bath.ac.uk>, masjhd@bath.ac.uk (James Davenport) wrote:

> In the referenced article, hbaker@netcom.com (Henry Baker) writes:
> >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.

I forgot to say that for algebraic numbers (roughly solutions of polynomials
with integer coefficients -- e.g., sqrt(2)), there is a very straightforward
way to calculate with them -- by representing such an algebraic number by a
square matrix whose characteristic polynomial is the minimal polynomial which
has the given algebraic number as a root.

The Cayley-Hamilton theorem tells you that one of these matrices satisfies
its own characteristic polynomial, so you can now calculate with the set
of square matrices generated by one of these particular matrices.

For example, the complex numbers are isomorphic to the set of numbers
generated by

[0 -1]
[1  0]

whose characteristic polynomial is 1+x^2 = 0.

I.e., the isomorphism looks like:

a + bi => 

[a -b]
[b  a]

This method is not terribly efficient, but it is understandable to a high
school student who goes to school outside of the LA Unified School District.

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