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
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