gives the base-b Champernowne number .
gives the base-10 Champernowne number.
- Mathematical constants treated as numeric by NumericQ and as constants by D.
- ChampernowneNumber[b] is a normal transcendental real number whose base-b representation is obtained by concatenating base-b digits of consecutive integers.
- ChampernowneNumber can be evaluated to arbitrary numerical precision.
- ChampernowneNumber automatically threads over lists.
Background & Context
- ChampernowneNumber[b] represents the base-b Champernowne constant, defined as the concatenation of the base-b digits of consecutive positive integers placed to the right of a decimal point. The base-10 Champernowne constant may be computed using ChampernowneNumber and has value 0.1234567891011…. A concise nested sum for ChampernowneNumber[b] is given by .
- ChampernowneNumber[b] is both irrational and transcendental, meaning it can be expressed neither as a ratio of integers nor as the root of any integer polynomial. In addition, as a result of its definition, ChampernowneNumber[b] is normal (meaning the digits in its base-b expansion are equally distributed) in base b.
- For specific base b, ChampernowneNumber[b] is treated as numeric by NumericQ and as a constant by D. ChampernowneNumber automatically threads over lists and can be evaluated to arbitrary numerical precision using N. RealDigits can be used to return a list of digits of ChampernowneNumber and ContinuedFraction to obtain terms of its continued fraction expansion. The continued fractions for ChampernowneNumber[b] contain very large sporadic terms, resulting in excellent rational approximations but making them potentially challenging to calculate.
Examplesopen allclose all
Introduced in 2008