Doesn't Pi Ever Repeat?Date: 7/8/96 at 10:58:22 From: Anonymous Subject: Pi Doesn't Repeat? How do people know that pi goes on forever without repeating? Date: 7/11/96 at 14:28:15 From: Doctor Ceeks Subject: Re: Pi Doesn't Repeat? Hi, Numbers which have terminating decimal expansions or which have repeating decimal expansions are rational numbers, i.e. ratios of integers. Lindemann proved that in fact, Pi is transcendental, which means that it is not a root of any polynomial equation with rational coefficients. (Note that every rational number P/Q, with P and Q integers, is a root of QX-P = 0) He did this by an elaborate proof by contradiction; that is, he assumed Pi was rational, and deduced a contradiction. His proof uses integrals. You can read the proof in the book, _Irrational Numbers_, by Ivan Niven. Or, you can try Serge Lang's, _Algebra_, which contains a generalization of the theorem in an Appendix. Unfortunately, the proof is far too long to replicate here (and uses too many mathematical symbols). -Doctor Ceeks, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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