Associated Topics || Dr. Math Home || Search Dr. Math

### 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