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