Champernowne constant

Champernowne constant is a real number constructed by concatenating the base-10 representations of the positive integers in order. It was first studied in detail by the English economist and mathematician David Gawen Champernowne, who published his analysis in the Journal of the London Mathematical Society in 1933. This constant provides a fundamental example of a normal number in base 10, a concept central to the fields of number theory and analytic number theory.

Definition and construction

The constant is formally defined as the decimal expansion obtained by writing the sequence of natural numbers in their standard decimal representation and concatenating them. The construction begins with the units 1 through 9, followed by the two-digit numbers 10 through 99, then the three-digit numbers 100 through 999, and so on indefinitely. This process yields the infinite decimal 0.12345678910111213141516..., where each block of digits corresponds to a successive integer. The definition can be generalized to other bases, such as base 2 or hexadecimal, by concatenating the base-b representations of the integers.

Properties

As a transcendental number, it is not a root of any polynomial with integer coefficients, a fact later established building upon foundational work by Kurt Mahler in Diophantine approximation. It is an irrational number, which follows directly from its non-repeating, non-terminating decimal pattern. The constant is also a computable number, meaning there exists a finite algorithm to calculate its digits to any desired precision. Its value has been calculated to high precision by projects like the Inverse Symbolic Calculator and is listed in the On-Line Encyclopedia of Integer Sequences under sequence A033307.

Normality

A major result concerning this constant is that it is a normal number in base 10. This means that in its decimal expansion, every finite string of digits (or block) occurs with the frequency expected if the digits were produced by a random source with equal probability for each digit from 0 to 9. Champernowne proved this normality in his 1933 paper, making it the first explicitly constructed provable example of a normal number. This property connects deeply to questions in probability theory, ergodic theory, and the work of Émile Borel, who first defined normal numbers.

Continued fraction expansion

The continued fraction representation of the constant, denoted C₁₀, begins with the sequence [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, ...]. This expansion is notable for containing exceptionally large partial quotients early in the sequence, such as 149083, which is a hallmark of numbers that are exceptionally well-approximated by rational numbers at certain points. The structure of this continued fraction has been analyzed in relation to the general theory of Lagrange's theorem on Diophantine approximation and the geometry of numbers.

Generalizations

The concept has been extended in several significant directions. The Copeland–Erdős constant is formed by concatenating the prime numbers instead of all integers. Mathematicians have also studied constants created by concatenating sequences like the Fibonacci numbers or values of specific polynomials. For an arbitrary base b, the base-b Champernowne constant is defined analogously and is known to be normal in that base. These generalizations are studied within the broader context of normality and automata theory, with connections to the work of Gregory Chaitin on algorithmic information theory and the halting problem.

Category:Mathematical constants Category:Real numbers Category:Number theory