# [FOM] 770 in Pi

David Ross ross at math.hawaii.edu
Mon Jul 16 05:36:20 EDT 2007

> 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?

This is probably a reference to Borel's Normal Number Theorem, which
asserts that almost all real numbers are normal.  The number x is normal
(base 10)provided for any sequence of digits (like "770"), the sequence
appears in the decimal expansion of x as often as one would expect if
the digits were random, ie,
as n\to\infty, the number of times 770 appears in the first n decimal
digits of x divided by n tends to 1/1000.  You can do this with respect
to any base, and in fact almost all real numbers are normal with respect
to all bases at the same time.  As far as I know, it is not yet known
whether \pi (or any other well-known transcendental constant) is normal.

David Ross