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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

• References:
  • Re: Re: algebraic numbers
    • From: DrMajorBob <btreat1@austin.rr.com>
  • Re: algebraic numbers
    • From: Richard Fateman <fateman@cs.berkeley.edu>