In article <pChMbSKOZRuQFwemail@example.com>, Dr J R Stockton <firstname.lastname@example.org> wrote:
> In sci.math message <email@example.com>, Wed, 28 Nov 2012 > 21:38:34, Existential Angst <firstname.lastname@example.org> posted: > > > > >So are the digits of pi random or not? > > > > The digits of pi are not random, because, unless the base is changed, > they are the same every time and can be defined relatively briefly, even > without assuming a definition of pi (which pi may formally lack). > > But any arbitrarily chosen sub-sequence of the digits of pi will, I > believe, pass an appropriate proportion of the usual tests for > randomness. Note that the full expansion of the digits of pi contains > as sub-sequences of a given length all possible digit strings of that > length, some of which will not look random to the untutored eye, such as > yours. > > A real mathematician could put that more precisely.
If the digits of a number are uniformly distributed, so that in the (infinitely) long run 0's, 1's, 2's, etc. occur equally often, then the number is said to be "simply normal" (in base 10). If, for every k = 1,2, ..., all possible groups of k digits occur equally often in the long run, then the number is said to be "normal." (Reference: Hardy and Wright, /The/ /Theory/ /of/ /Numbers/ , Fourth ed., pp. 124-5.)
Most of the irrational numbers that are interesting (such as sqrt(2) or pi) appear to be normal. Numbers that are not normal include all rational numbers, and some particular values of theta functions (for instance 0.1001000010000001 ... ).
I think most mathematicians are of the *opinion* that pi is normal. But as far as I know, nobody has proved that it is, nor that it isn't. A frustrating situation. Part of the difficulty seems to be that the property of normality does not seem to be connected to other interesting properties of an irrational number.