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Re: algebraic numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg106294] Re: algebraic numbers
• From: Richard Fateman <fateman at cs.berkeley.edu>
• Date: Thu, 7 Jan 2010 02:33:20 -0500 (EST)
• References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <hi1qer$ent$1@smc.vnet.net>

Noqsi wrote:
...

> 
> Machine reals are not the reals or the rationals: they are themselves,
> with their own special properties. Those who reason as if machine
> reals are either real or rational often suffer adverse consequences.

If you treat each "machine real"  (that is, hardware or software FLOAT 
format object) as a representation of an exact rational number, then I 
think that you are in much better shape in terms of numerical analysis 
than if you treat each object as some kind of fuzz-ball.

> Much of the art of numerical analysis depends upon understanding these
> special properties and their consequences.
> 
I agree.  But I disagree with the assertion that you do better with 
fuzz-balls.  You especially can do careful analysis using a computer 
algebra system where you can carry around symbolic "machine epsilon" 
data, and do arithmetic on such expressions.

RJF