Proving a Number is Transcendental

Date: 4/12/96 at 21:55:58
From: Janet Sutorius
Subject: transcendental numbers

Hi, I'm a high school math teacher, but I am also just finishing 
up a masters in math education.  I like my students to know that 
learning is a continual process.  

My question is, how does one prove a number is transcendental?  
It's relatively easy to prove a number is not transcendental.  All 
one needs to do is find a polynomial with integral coefficients 
which has as one of its solutions the number one doesn't want to 
be transcendental.  That can often be done simply by using the 
number as one of the factors and multiplying to find a polynomial.  
One needs to be wise in choosing the factors but the method is 
pretty straightforward and works in many instances with irrational 
numbers and even i.  

But proving a given number is transcendental is much harder.  Does 
each individual number have to be proven individually or can the 
entire set of transcendentals be proven as a body?  Are there 
places where the individual proofs have been written?  I have been 
searching, but it seems no one is very interested in 
transcendental numbers because I keep running into a dead end.

Date: 4/17/96 at 14:37:11
From: Doctor Jodi
Subject: Re: transcendental numbers

Thanks for your question... From 

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Lindemann.html   

I was able to get some information on Lindemann, who was the first 
person to prove that pi is transcendental.  The proof is published 
in an 1882 paper, "Uber die Zahl Pi" (in German obviously with the 
correct spelling, etc. on the Web page).

A university librarian should be able to assist you in finding 
this paper, which might be a good place to start.

I'd really interested in hearing what you find out...

You may also be interested in searching for transcendentals and 
proofs in the MathSearch at 

http://www.ms.maths.usyd.edu.au:8000/MathSearch.html   

Good luck!

-Doctor Jodi,  The Math Forum

Date: 4/1/99 at 01:25:15
From: souppe
Subject: Re: transcendental numbers

I wanted simply to point out that the demonstration of the 
Hermite-Lindemann transcendence theorem and its inferences about Pi 
and e is available in _100 Great Problems of Elementary Mathematics_
collection Dover (just $9.95).

Francois