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How Can Pi Not Repeat?

Date: 06/01/99 at 04:29:41
From: Aaron Chan
Subject: How can pi not repeat?

Dear Dr. Math,

How can pi not repeat? Won't it repeat when it runs out of number 


Date: 06/01/99 at 10:47:10
From: Doctor Rob
Subject: Re: How can pi not repeat?

We must be careful to distinguish between a repeat of any finite 
pattern of digits and periodicity of digits.

Let the nth decimal digit of a number be d(n).

A decimal is called periodic if there are whole numbers N >= 0 and 
k > 0 such that for every n > N, d(n+k) = d(n). This is the situation 
with numbers like 47/66, whose decimal expansion is 0.712121212121... 
In this case, N = 1 and k = 2, so for all n > 1, d(n+2) = d(n). There 
is a string of k digits in a row such that all the digits from the Nth 
on are gotten by writing down infinitely many copies of that single 

A decimal may have repeated strings without being periodic, as long as 
the strings are followed by different digits. Thus 
0.347329384732046... has the repeated digit string 4732, but the first 
is followed by a 9, and the second by a 0.

Only rational numbers (fractions) have periodic decimal expansions. Pi 
is not rational, so its decimal expansion is not periodic. Its decimal 
expansion does have fairly long repeated digit strings, but they are 
always followed by different digits. Thus Pi repeats (for a little 
while) but is not periodic.

Here is another example of a non-periodic but repeating decimal:


Here the strings of zeroes have lengths 1, 2, 3, 4, 5, ..., and so on. 
The digit string 00000, for example, repeats infinitely often in this 
number, but it is not periodic.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Transcendental Numbers

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