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MathGroup Archive 2005

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Mathematica and Transcendental Numbers


Mathematica's treatment of transcendental numbers is quite  
sophisticated . Consider these two contrasting cases:


Element[E^(Cos[(231/3577)*Pi]^2 + Sin[(231/3577)*Pi]^2),Algebraics]


False


Element[Pi^(Cos[(231/3577)*Pi]^2 + Sin[(231/3577)*Pi]^2),Algebraics]


Element[Pi^(Cos[(231/3577)*Pi]^2 + Sin[(231/3577)*Pi]^2),Algebraics]

Actually, both answers should be False, since Cos[(231/3577)*Pi]^2 +  
Sin[(231/3577)*Pi]^2) ==1 and both Pi and E are transcendental. It is  
of course not surprising that Mathematica can't get the second  
problem right since trigonometric simplifications are not performed  
automatically by the evaluator unless Simplify is used. But to deal  
with the first input correctly Mathematica makes use of some fairly  
advanced mathematical knowledge. First of all it knows that numbers  
like Cos[(231/3577)*Pi] and Sin[(231/3577)*Pi] are algebraic without  
evaluating them. Then of course it makes use of the fact that  
algebraic functions of algebraic numbers are themselves algebraic.  
And finally, having determined that the exponent is an algebraic  
number, Mathematica makes use of the fact that E^a, where a is an  
algebraic number is transcendental. For Pi the analogous result is  
unproved so the second input is returned unevaluated.
Usually, much less is known in this context about Pi then E; however,  
there is one exception, which Mathematica also knows:

Element[E^Pi,Algebraics]

False

while

Element[E^E, Algebraics]

Element[E^E, Algebraics]

is not known.

In addition to the numbers that are known to be algebraic and the  
ones known to be transcendental there is a large number of those  
about which we do not know if they are algebraic or not. Since the  
problem of deciding if a number is algebraic or not is, if I remember  
correctly, undecidable in general, I tend to refer to all numbers  
that are not known to be algebraic (such as Pi^Pi) as non-algebraic.  
Transcendental means something that has been proved to be non-algebraic.

Andrzej Kozlowski


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