Mathematica and Transcendental Numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg62880] Mathematica and Transcendental Numbers*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 6 Dec 2005 23:12:26 -0500 (EST)*References*: <dmp9na$hi2$1@smc.vnet.net> <roadnYOk3NcFDw7eRVn-jg@speakeasy.net> <200512050837.DAA08425@smc.vnet.net> <200512051840.NAA21063@smc.vnet.net> <200512060503.AAA02736@smc.vnet.net> <200512060858.DAA09071@smc.vnet.net> <E5FAFF10-6753-4DF3-B949-124CC8D8B0B6@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

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

**References**:**Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@gmail.com>

**Re: Re: Types in Mathematica thread***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@Gmail.com>

**Re: Re: Re: Re: Types in Mathematica thread***From:*ggroup@sarj.ca