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/
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