[FOM] Wittgenstein's '770' in pi

Stephen G Simpson simpson at math.psu.edu
Mon Jul 16 09:52:35 EDT 2007

Harold Teichman writes:
 > No such statements (about decimal expansions) figure prominently in
 > the texts on number theory that I have seen.  Could someone refer
 > me to some actual literature in number theory (or any other part of
 > mathematics) that has a bearing on this question?

The literature on so-called "normal numbers" is surely relevant.  Some
references are in this on-line article:


Quoting from the article:

  A normal number is an irrational number for which any finite pattern
  of numbers occurs with the expected limiting frequency in the
  expansion in a given base (or all bases). For example, for a normal
  decimal number, each digit 0-9 would be expected to occur 1/10 of
  the time, each pair of digits 00-99 would be expected to occur 1/100
  of the time, etc.

The experts in this kind of number theory have a strong intuition that
numbers such as pi, the square root of 2, ... are normal, but this has
not been proved.  Obviously, if pi is normal then it would follow that
770 occurs infinitely often in the decimal expansion of pi.

Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University
research specialties: mathematical logic, foundations of mathematics

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