Worlds of David Darling
Encyclopedia of Science
Home > Encyclopedia of Science

normal number

A number in which digit sequences of the same length occur with the same frequency. A constant is 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, the digit 7 is expected to appear 1 million times among the first 10 million digits of its decimal expansion. It actually occurs 1,000,207 times – very 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.

Émile Borel introduced the concept of normal numbers in 1909 as a way to characterize the resemblance between the digits of a mathematical constant such as pi and a sequence of random numbers. He quickly established that there are lots of normal numbers, though finding a specific example of one proved to be a major challenge. The first to be found was Champernowne's Number, which is normal to base 10. Analogous normal numbers can be created for other bases.

To date, no specific "naturally occurring" real number has been proved to be absolutely normal, even though it is known that almost all real numbers are absolutely normal! However, in 2001, Greg Martin of the University of Toronto found some examples of the opposite extreme – real numbers that are normal to no base whatsoever.1 To start with, he noted that 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 bk eventually repeats. Martin then 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 (underlined) of the given fragment of a consists of 23,747,291,559 nines. Martin's formulation of this number and proof of its absolute abnormality involved so-called Liouville numbers.


  1. Martin, G. "Absolutely abnormal numbers." American Mathematical Monthly, 108: 746-754, 2001.

Related category