On 12/01/2012 07:21 PM, Christopher J. Henrich wrote: > 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. >
Another question, this time on approximability by rational numbers, can be asked for pi, for example.
Thm.: There are infinitely many pairs (m, n) of positive integers such that
| pi - m/n | < 1/(n^2).
if 1/(n^2) is replaced by 1/(n^d), for a given real number d >=2, one has the question:
Is it true that there are infinitely many pairs (m, n) of positive integers such that | pi - m/n | < 1/(n^d) ?