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Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106314] Re: [mg106295] Re: algebraic numbers
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Fri, 8 Jan 2010 04:14:59 -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>
On 7 Jan 2010, at 16:33, Richard Fateman wrote:
>
> Incidentally, it is not known if E+Pi is rational.
Brilliant. But even a high school child can prove that either E+Pi is
irrational or E*Pi is irrational (in fact you can replace irrational
with transcendental).
>
> If you capitalize the term and wish to say that 1.2 is not a Rational
> in Mathematica, that is just a convention based on the "type" of data
> that is input to Mathematica with a decimal point and is therefore
> stored in a memory format that is labeled "Real" which (in Mathematica)
> is a superclass of "Rational". That is, "1/2" is a Rational but is also
> a Real and is incidentally also a Complex. But 0.5 is not a Real.
You are confusing Real and Reals. This
Element[1.2, Reals]
True
has nothing to do with types because
Element[Pi, Reals]
True
even though
Head /@ {1.2, Pi}
{Real,Symbol}
and also
Element[0.5, Rationals]
False
Element[1/2, Rationals]
True
and just by the way:
Element[Pi, Rationals]
False
Element[E, Rationals]
False
Element[E + Pi, Rationals]
Element[E + Pi, Rationals]
so Mathematica knows as much as you about this. But it does not know as much as me since
Reduce[Element[E + Pi, Rationals] && Element[E Pi, Rationals]]
should return False.
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