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### Proving a Number is Transcendental

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

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
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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

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Associated Topics:
High School Transcendental Numbers

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