# ChampernowneNumber

gives the base-b Champernowne number .

gives the base-10 Champernowne number.

# Details • Mathematical constants treated as numeric by NumericQ and as constants by D.
• 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

• 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 and has value 0.1234567891011. A concise nested sum for is given by .
• 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, is normal (meaning the digits in its base-b expansion are equally distributed) in base b.
• For specific base 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 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:

## 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:

Introduced in 2008
(7.0)