In mathematics, the Champernowne constant is a transcendentalreal 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 constants can be expressed exactly as infinite series: where ceiling, in base 10, and is the base of the constant. A slightly different expression is given by Eric W. Weisstein : where floor.
The Champernowne word or Barbier word is the sequence of digits of C10, obtained writing n in base 10 and juxtaposing the digits: More generally, a Champernowne sequence is any sequence of digits obtained by concatenating all finite digit-strings in some recursive order. For instance, the binary Champernowne sequence in shortlex order is where spaces have been inserted just to show the strings being concatenated.
Normality
A real numberx 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 , then, in base 10, we would expect strings ,,,..., to occur 1/10 of the time, strings ,,...,, 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 for any base. It is an open problem whether is normal in bases. It is also disjunctive sequence.
The simple continued fraction expansion of Champernowne's constant has been studied as well. Kurt Mahler showed that the constant is transcendental; therefore its continued fraction does not terminate and is aperiodic. The terms in the continued fraction expansion exhibit very erratic behaviour, with extremely large terms appearing between many small ones. For example, in base 10, The large number at position 18 has 166 digits, and the next very large term at position 40 of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good approximation of the Champernowne constant. It can be understood from infinite series expression of : for a specified we can always approximate the sum over by setting the upper limit to instead of. Then we ignore the terms for higher. For example, if we keep lowest order of n, it is equivalent to truncating before the 4th partial quotient, we obtain the partial sum which approximates Champernowne's constant with an error of about. While truncating just before the 18th partial quotient, we get the approximation to second order: which approximates Champernowne's constant with error approximately. The first and second incrementally largest terms after the initial zero are 8 and 9, respectively, and occur at positions 1 and 2. Sikora noticed that the number of digits in the high-water marks starting with the fourth display an apparent pattern. Indeed, the high-water marks themselves grow doubly-exponentially, and the number of digits in the nth mark for are: whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by: However, it is still unknown as to whether or not there is a way to determine where the large terms occur, or their values. The high-water marks themselves, however, are located at positions:
