Notes

Section 5: Mathematical Constants

Digit sequence properties

Empirical evidence for the randomness of the digit sequences of Sqrt[n], π, etc. has been accumulating since early computer experiments in the 1940s. The evidence is based on applying various standard statistical tests of randomness, and remains somewhat haphazard. (Already in 1888 John Venn had noted for example that the first 707 digits of π lead to an apparently typical 2D random walk.) (See page 1089.)

The fact that Sqrt is not a rational number was discovered by the Pythagoreans. Numbers that arise as solutions of polynomial equations are called algebraic; those that do not are called transcendental. E and π were proved to be transcendental in 1873 and 1882 respectively. It is known that Exp[n] and Log[n] for whole numbers n (except 0 and 1 respectively) are transcendental. It is also known for example that Gamma[1/3] and BesselJ[0, n] are transcendental. It is not known for example whether EulerGamma is even irrational.

A number is said to be "normal" in a particular base if every digit and every block of digits of any length occur with equal frequency. Note that the fact that a number is normal in one base does not imply anything about its normality in another base (unless the bases are related for example by both being powers of 2). Despite empirical evidence, no number expressed just in terms of standard mathematical functions has ever been rigorously proved to be normal. It has nevertheless been known since the work of Emile Borel in 1909 that numbers picked randomly on the basis of their value are almost always normal. And indeed with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal. An example of this is the number 0.1234567891011121314... obtained by concatenating the digits of successive integers in base 10 (see below). This number was discussed by David Champernowne in 1933, and is known to be transcendental. A few other results are also known. One based on gradual extension of work by Richard Stoneham from 1971 is that numbers of the form Sum[1/(pn bpn), {n, }] for prime p > 2 are normal in base b (for GCD[b, p] 1), and are transcendental.

From Stephen Wolfram: A New Kind of Science [citation]