MathWorld Headline News

Bailey and Crandall Discover a New Class of Normal Numbers

By Eric W. Weisstein

October 4, 2001 (Revised March 17, 2003)--A normal number is defined as a real number whose digits are truly random--in the sense that each base-b digit, pair of digits, and so on, occur with a limiting uniform distribution. For example, in order for a number to be normal in base 10, the digits 0-9 would each appear 1/10 of the time, the digits 00-99 would each appear 1/100 of the time, and so on. Proving the normality of numbers is extremely difficult and, prior to new research announced today, all known numbers having this property had been artificially constructed. For example, determining if the famous constants pi and e are normal in any base remains an open problem to this day.

In work to appear in the journal Experimental Mathematics, D. H. Bailey and R. E. Crandall built on their previous chaotic-dynamical modeling of random digits to prove that certain general classes of numerical constants have uniformly distributed digits. In particular, while the b-normality of the natural logarithm of 2

Log[2] == Sum[1/(n 2^n), {n, Infinity}]

remains unknown for any b, Bailey and Crandall have generalized previous results of Stoneham (1973) to show that, for any relatively prime positive integers b and c

alpha[b, c] == Sum[1/(c^k b^(c^k)), {k, Infinity}]

is b-normal. The new results also establish the b-normality for constants of the form

Sum[1/(b^m[[i]] c^n[[i]]), {i, Infinity}]

for certain sequences (m_i) and (n_i) of integers.

Bailey and Crandall have also proved a number of related results, including the determination of a spigot algorithm for alpha[2, 3], thus establishing that the googol-th binary bit is 0.