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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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