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

*To*: mathgroup at smc.vnet.net*Subject*: [mg62852] Re: [mg62844] Re: [mg62824] Re: [mg62787] Re: [mg62781] Re: [mg62708] Types in Mathematica thread*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 6 Dec 2005 23:10:25 -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>*Sender*: owner-wri-mathgroup at wolfram.com

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

**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