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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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