Concept
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: C10 = 0.12345678910111213141516... . Champernowne constants can also be constructed in other bases, similarly, for example: C2 = 0.11011100101110111... 2 C3 = 0.12101112202122... 3. The Champernowne word or Barbier word is the sequence of digits of C10 obtained by writing it in base 10 and juxtaposing the digits: 12345678910111213141516... More generally, a Champernowne sequence (sometimes also called a Champernowne word) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order. For instance, the binary Champernowne sequence in shortlex order is 0 1 00 01 10 11 000 001 ... where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated. 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 10, 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 is normal in base 10, while Nakai and Shiokawa proved a more general theorem, a corollary of which is that is normal in base for any b. It is an open problem whether is normal in bases . Kurt Mahler showed that the constant is transcendental. The irrationality measure of is , and more generally for any base . The Champernowne word is a disjunctive sequence.