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MathGroup Archive 2005

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Re: Re: Re: Re: Re: Types in Mathematica thread


On 6 Dec 2005, at 17:58, ggroup at sarj.ca wrote:

> On Tuesday, December 6, 2005 at 00:03 GMT -0500, Kristen W Carlson  
> wrote:
>
>> Andrzej, why there isn't a test for transcendentals--Transcendental.
>
> From Mathworld <http://mathworld.wolfram.com/ 
> TranscendentalNumber.html>:
>    A number  can then be tested to see if it is transcendental using
>    the Mathematica command Not[Element[x, Algebraics]].
>
>
>

In fact Mathematica knows quite a lot about transcendental numbers:


Not[Element[E^Pi, Algebraics]]


True

Not[Element[Sqrt[2]^Sqrt[2], Algebraics]]

True


Not[Element[E^Pi+Sqrt[2], Algebraics]]


True


Of course it won't tell you anything about the numbers that are not  
known to be transcendental (even though almost certainly are):


Element[Pi^E ,Algebraics]

Element[Pi^E,Algebraics]

(note that Pi is much harder than E, for example we do not know if

Element[Pi^Sqrt[2],Algebraics]

For E this is well known.  Everything in the case of Pi is much  
harder than in the case of E.


Of course all of this is subject to the usual limitations:

Element[Exp[Pi*I*(Cos[Pi/27]^2 + Sin[Pi/27]^2)], Algebraics]

Element[Exp[Pi*I*(Cos[Pi/27]^2 + Sin[Pi/27]^2)], Algebraics]

even though


Simplify[Exp[Pi*I*(Cos[Pi/27]^2+Sin[Pi/27]^2)]]

-1

If we then use this on an expression that Mathematica can't  
explicitly simplify to something it knows about you won't get any  
answer. Thus you certainly you are unlikely discover any new  
transcendental numbers in this way.

Andrzej Kozlowski









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