Absolutely Abnormal

Identifying the normal (or even the abnormal) in mathematics can pose serious difficulties.

In 1909, mathematician Émile Borel (1871–1956) introduced the concept of normality as one way to characterize the resemblance between the digits of a mathematical constant such as pi (the ratio of a circle’s circumference to its diameter) and a sequence of random numbers.

If a number is normal, digit sequences of the same length occur with the same frequency. A constant would be considered normal to base 10 if any single digit in its decimal expansion appears one-tenth of the time, any two-digit combination one-hundredth of the time, any three-digit combination one-thousandth of the time, and so on.

In the case of pi, you would expect the digit 7 to appear 1 million times among the first 10 million decimal digits of pi. It actually occurs 1,000,207 times–close to the expected value. Each of the other digits also turns up with approximately the same frequency, showing no significant departure from predictions.

A number is said to be “absolutely normal” if its digits are normal not only to base 10 but also to every integer base greater than or equal to 2. In base 2, for example, the digits 1 and 0 would appear equally often.

Borel established that there are lots of normal numbers. Finding a specific example of a normal number, however, proved much more difficult. In 1933, D.G. Champernowne showed that the carefully constructed number 0.12345678910111213. . ., created by writing all the positive integers in a row as a single decimal, is normal to base 10. You can construct analogous normal numbers for other bases.

At the same time, although it is known that almost all real numbers are absolutely normal, no one has yet proved even a single, “naturally occurring” real number to be absolutely normal.

Mathematician Greg Martin of the University of Toronto recently turned his attention to the opposite extreme–real numbers that are normal to no base whatsoever. He described his successful search for such an “absolutely abnormal” number in the October American Mathematical Monthly.

To start with, every rational number is absolutely abnormal. For example, the fraction 1/7 can be written in decimal form as 0.1428571428571. . . . The digits 142857 just repeat themselves. Indeed, an expansion of a rational number to any base b or b^k eventually repeats. Hence, a rational number “is about as far from being simply normal to the base b^k as it can be,” Martin remarked.

Martin focused on constructing a specific irrational absolutely abnormal number. He nominated the following candidate, expressed in decimal form, for the honor:

a = 0.6562499999956991999999. . .9999998528404201690728. . .

The middle portion (shown in bold) of the given fragment of a consists of 23,747,291,559 9s.

Martin’s formulation of this number and proof of its absolute abnormality involved so-called Liouville numbers, named for Joseph Liouville (1809–1882). One example of a Liouville number is 0.1100010000000000000000010000. . ., where there is a 1 in place 1, 2, 6, 24,. . . n! and 0 elsewhere. Liouville had introduced such numbers as examples of transcendental numbers–real numbers that are not roots of polynomial equations with integer coefficients.

The given construction, Martin concluded, “easily generalizes to a construction giving uncountably many absolutely abnormal numbers.”