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:

Wolfram Research (2008), ChampernowneNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/ChampernowneNumber.html.

Text

Wolfram Research (2008), ChampernowneNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/ChampernowneNumber.html.

CMS

Wolfram Language. 2008. "ChampernowneNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ChampernowneNumber.html.

APA

Wolfram Language. (2008). ChampernowneNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ChampernowneNumber.html

BibTeX

@misc{reference.wolfram_2024_champernownenumber, author="Wolfram Research", title="{ChampernowneNumber}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ChampernowneNumber.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_champernownenumber, organization={Wolfram Research}, title={ChampernowneNumber}, year={2008}, url={https://reference.wolfram.com/language/ref/ChampernowneNumber.html}, note=[Accessed: 22-November-2024 ]}