Champernowne constant

In mathematics, the Champernowne constant C₁₀ is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.

For base 10, the number is defined by concatenating representations of successive integers:

Champernowne constants can also be constructed in other bases, similarly, for example:

The Champernowne constant can be expressed exactly as an infinite series:

and this series generalizes to arbitrary bases b by replacing 10 and 9 with b and b − 1 respectively.

The Champernowne word or Barbier word is the sequence of digits of C_k.

A real number x is said to be normal if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. x is said to be normal in base b if its digits in base b follow a uniform distribution.

If we denote a digit string as [a₀,a₁,...], then, in base ten, we would expect strings [0],[1],[2],...,[9] to occur 1/10 of the time, strings [0,0],[0,1],...,[9,8],[9,9] to occur 1/100 of the time, and so on, in a normal number.

Champernowne proved that ${\displaystyle C_{10}}$ is normal in base ten, while Nakai and Shiokawa proved a more general theorem, a corollary of which is that ${\displaystyle C_{b}}$ is normal for any base ${\displaystyle b}$. It is an open problem whether ${\displaystyle C_{k}}$ is normal in bases ${\displaystyle b\neq k}$.

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