Re: Re: Types in Mathematica thread

*To*: mathgroup at smc.vnet.net*Subject*: [mg62929] Re: [mg62891] Re: Types in Mathematica thread*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 8 Dec 2005 06:25:33 -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> <dn3jsl$8s0$1@smc.vnet.net> <5_ydnSmM8KqB-gjenZ2dnUVZ_v6dnZ2d@speakeasy.net> <dn5npi$nef$1@smc.vnet.net> <200512080504.AAA11638@smc.vnet.net> <7A0DFBF7-1606-40E2-995E-7AB809E68662@mimuw.edu.pl>*Sender*: owner-wri-mathgroup at wolfram.com

On 8 Dec 2005, at 17:24, Andrzej Kozlowski wrote: >>> >>> Exp[I Pi]==-1. >> >> Yes. >> >>> In that case would you say that you can also "derive" I form Pi and >>> E? What do you mean by deriving a number from another number? >> >> I meant to say that Pi can be defined in terms of E. I am >> assuming the >> definition of complex numbers as a prerequisite. It's something >> that's >> been in the back of my mind for quite some time. > > > But there is no reason at all to think that this would help in > deducing that Pi is transcendental form the fact that E is. The > fact that neither Lindemann's nor Hilbert could do this using this > ancient formula of Euler, which they certainly new, would make most > people hesitate in claiming that it should be done once you have > got complex numbers "as a prerequisite". > > But if you really have an idea how to deduce that Pi is > transcendental from the fact that E is then perhaps you might wish > to prove that E+Pi is transcendental because somehow nobody has so > far been able to do it. > > Andrzej Kozlowski Although, I have to admit, that if you make use of the Lindemann- Weierstrass theorem (which was proved by Weierstrass much later than Lindemann's proof of the transcendentality of Pi) than indeed transcendentality of Pi does follow from the transcendentality of E and the relationship Exp[I Pi]= -1. But the Liendemann-Weierstrass theorem is far from easy to prove; and certainly requires more prerequisites than "complex numbers", see for example Alan Baker, "Transcendental Number Theory" (which I just have been looking at). 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: Types in Mathematica thread***From:*"Steven T. Hatton" <hattons@globalsymmetry.com>