On 10 Feb 2004, David McAnally wrote: >firstname.lastname@example.org (Rob Johnson) writes: > >>In article <email@example.com>, >>"G. A. Edgar" <firstname.lastname@example.org> wrote: >>>In article <email@example.com>, earl >>><firstname.lastname@example.org> wrote: >>> >>>> hey do you guys have a copy of the proof of e is transcendental? ..id really >>>> appreciate if somebody could help me. ive been searching and found nothing. >>> >>>Spivack, CALCULUS, 2nd edition, Chapter 21. > >>Are you sure that the proof there is for e being transcendental and not >>just irrational? I know that the proof of the irrationality of e is >>simple enough for a Calculus book, but that the transcendence is quite a >>bit more complicated. > >In my version of Spivak, the transcendence of e is proven in Chapter 20, >and the irrationality of pi (or actually, the stronger result, the >irrationality of pi^2) is proven in Chapter 16. Both chapters are marked >with an asterisk, presumably denoting that the chapters are intended for >the more advanced student. I note that Spivak describes these chapters as >optional. As others have noted, another good book is Irrational Numbers >by Ivan Niven, in which a proof of the transcendence of e, pi, and many >others, is provided in the first chapter. Specifically, Niven gives the >proof of the result that if a_1, ..., a_n are distinct algebraic numbers, >then exp(a_1), ..., exp(a_n) are linearly independent over the algebraic >numbers. The transcendence of e is then proven by taking a_1 = 0, a_2 = 1, >and the transcendence of pi is proven by taking a_1 = 0, a_2 = i pi.
The Greek book that I, hold by M.A. MPRIKAS (The famous Insoluble Problems of Antiquity -1970 ),refers to Lambert (1766) stating that his proofs (for the insoluble part of circle's quadrature simply)were not complete,and that Legendre (1794) completed the proofs for proving that Pi and Pi^2 are not rational.But ,still, his proofs did not show the imposibility of circle's quadrature simply. Liouville confronted ,again the same problem in 1840 , and his proposition was simplified in 1873 by G. Cantor. However ,since the transcendental numbers formulated by Liouville and G. Cantor,were somehow "TECHNICAL", Charles Hermite proved that number e ,the basis of the natural logarithms ,is not algebraic That was important since it was being proved that it was transcendental ,and related to pi via Euler's relationship
[ Eyler gave the general formula : e^[ix]=cosx+isinx in 1748 , gave e and its value 2.718 ,and new since 1728 the relationship: e^[iPi]+1 =0 ] The proof of the transcendence of e by Ch.Hermite was given in 1873(published 1874 Comptes Rendus ) Hermites proof was very complicated and lengthy , covering 31 pages in hi APANTA (all) VOL III,page 150-181. A first simplification was given by Weierstrass(Berliner Berichte ,1885) and then followed others, in 1893 by Hilbert, Hurwitz and Gordan(Mathematical Annalen,1893). Lindemann gave in 1882 his proof that number Pi is transcendental [Math.Annalen,20 (1882),p 213] a generalization of Hermites proof , which is also lengthy ang perplexed. Simplified proof was given by Hilbert [Math.Annalen(1893)P.216-219].
In his proof makes use of the relationship e^[ipi]+1=0 , or e^[ipi]=-1
My comments --------------- Since e^[iPi]=cosPi+isinPi or , e^[iPi]=-1+i then there are two solutions here, to the given equatio:
A) e^[ipi]=-1 the real part solution and
B) e^[ipi]=i , or e^[ipi]=0 the imaginary part solutio.
My question : What is the implication of this second value of e^[ipi]=0 ?