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ChampernowneNumber

ChampernowneNumber

ChampernowneNumber[b]

gives the base-b Champernowne number .

ChampernowneNumber[]

gives the base-10 Champernowne number.

Details

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 $sum_(n=1)^inftyn/(b^(n+sum_(k=1)^nTemplateBox[{{{log, _, b}, (, k, )}}, Floor]))$ .
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.

Examples

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Basic Examples (2)

Evaluate to high precision:

Plot values of the first few Champernowne numbers:

Scope (3)

Evaluate for different base:

Compute continued fraction expansion:

TraditionalForm formatting:

Possible Issues (1)

The base must be an integer greater than 1:

Neat Examples (1)

Sizes of integers occurring in the first 1000 terms of continued fraction expansion of C₁₀:

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