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Frequency of Digits in Pi


Date: 04/05/2001 at 04:37:54
From: kate
Subject: Pi

What is the least occurring digit in pi?


Date: 04/05/2001 at 12:05:01
From: Doctor Rob
Subject: Re: Pi

Thanks for writing to Ask Dr. Math, Kate.

Since Pi has infinitely many digits, of which we only know a finite
number, this question cannot be answered.

We can talk about the asymptotic density of the digits of a decimal
number.  That is found by taking the number of a certain digit in
the first n digits of the decimal, and then letting n approach
infinity.  A decimal is called normal if the asymptotic densities
of the ten digits are all 1/10.

As an example, 1/7 = .142857142857142857...  The asymptotic
densities of the ten digits 0 through 9 are 0, 1/6, 1/6, 0, 1/6, 1/6, 
0, 1/6, 1/6, and 0, respectively. Thus 1/7 is not a normal decimal.

It is suspected, but not proven, that Pi is a normal decimal, so
that all digits of Pi have asymptotic density 1/10.  If that is so,
then you could say that no digit occurs in Pi with lower frequency
than any other.

You can answer the question asked for any known initial segment of
the decimal expansion of Pi, by counting. For the first 200
billion digits of Pi - 3, see this page from the Information 
Technology Center or the Computer Centre Division, University of 
Tokyo, for the counts:

  ftp://pi.super-computing.org/README.our_latest_record   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
Middle School Pi

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