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Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106313] Re: [mg106295] Re: algebraic numbers
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Fri, 8 Jan 2010 04:14:48 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net> <E44EA2F2-1274-43E8-93DE-DC5BD31884A5@mimuw.edu.pl> <op.u52ai6jwtgfoz2@bobbys-imac.local> <504E0A05-61DB-4A43-9637-68216076623C@mimuw.edu.pl> <hi1qit$etn$1@smc.vnet.net> <201001070733.CAA24037@smc.vnet.net>
Richard Fateman wrote:
> Andrzej Kozlowski wrote:
>> [...]
> Can Mathematica represent Reals that are NOT RATIONAL? Sure. Here are
> examples: Sqrt[2], 3*Pi, 4*E. 3*E +4*E^E + 5*E^E^E.
> Incidentally, it is not known if E+Pi is rational.
It is known whether this thread is rational. Empirical evidence seems to
argue against it.
> [...]
> Maybe you think that Mathematica has a human mind?
Of course she does.
> (A better example would be 1.25, since 1.2 is not representable exactly
> in binary. This example of 1.2 actually reveals a "misfeature of
> mathematica.
>
> 1.2==5404319552844595/4503599627370496
> True.
>
> So 1.2 is actually Mathematica-equal to another rational number. Many,
> in fact.
> )
That (a misfeature), or maybe it's a missing feature in some other
programs. I rather like this behavior of Equal, though I agree there is
good sense behind some recent criticisms to the effect that maybe it
should be configurable (regarding bits of slop, or relative or absolute
error specifications).
> [...]
> the explanation is that Mathematica takes numbers written with a decimal
> point and labels them "Real". This has nothing to do with their values,
> which are, most assuredly, equal to rational numbers. And in
> particular, 1.2==12/10 in Mathematica should trouble you if you believe
> Mathematica speaks meaningfully on these issues.
I would be far more troubled if 1.2===12/10 gave True (that is, they
were deemed SameQ rather than just Equal).
Much of the town is shut down, including schools (though, alas, not the
HS drama club trip). I had to shovel out this morning before work. I'll
have to shovel again when I get home. So here I am, and it feels like I
am still shovelling. Such sound and fury...
Daniel Lichtblau
Wolfram Research
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