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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/   
    
Associated Topics:
High School Transcendental Numbers
Middle School Number Sense/About Numbers

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