digitsappears

Digitsappears is a concept in the study of decimal digit sequences. A decimal expansion or a concatenation of digits is described as having the digitsappears property if every digit from 0 through 9 occurs infinitely often within the sequence.

Formally, for a real number x with decimal expansion x = 0.d1 d2 d3 ..., digitsappears means that for each digit d in {0,1,2,3,4,5,6,7,8,9}, the set {n ≥ 1 : dn = d} is infinite. In the context of finite digit strings, the property is often discussed in terms of long runs or coverage (for example, all digits appearing within a long prefix), but the standard emphasis is on infinite occurrence.

Relation to normal numbers: A number that is normal in base 10 has uniform digit frequencies and, in particular, each digit appears with limiting frequency 1/10, hence the digitsappears property holds. However, digitsappears can occur even without full normality if digits appear infinitely often but with unequal frequencies; conversely, a digit that never appears defeats the property.

Examples: The Champernowne constant, formed by writing all positive integers in base 10 in order, contains every finite block of digits, so each digit appears infinitely often and the digitsappears property holds. The decimal expansion of pi is conjectured to be normal, which would imply digitsappears, though this is unproven. A number whose expansion omits a digit clearly does not have digitsappears.

See also: normal number, Champernowne constant, digit distribution, Benford’s law.