Editing Champernowne constant (section)

{{Short description|Transcendental number(s) with all positive integers  in order}}
In [[mathematics]], the '''Champernowne constant''' {{math|''C''<sub>10</sub>}} is a [[transcendental number|transcendental]] [[real number|real]] [[mathematical constant|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.<ref name=Cha33>{{harvnb|Champernowne|1933}}</ref>  The number is defined by [[concatenation|concatenating]] the [[base-10]] representations of the positive integers:
:{{math|1=''C''<sub>10</sub> = 0.1234567891011121314151617181920... }} {{OEIS|A033307}}.

Champernowne constants can also be constructed in other bases similarly; for example,
:{{math|1=''C''<sub>2</sub> = 0.11011100101110111... <sub>2</sub>}}
and
:{{math|1=''C''<sub>3</sub> = 0.12101112202122... <sub>3</sub>}}.

The '''Champernowne word''' or '''Barbier word''' is the sequence of digits of ''C''<sub>10</sub> obtained by writing it in base 10 and juxtaposing the digits:<ref>Cassaigne & Nicolas (2010) p.165</ref><ref>{{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 | page=299 }}</ref>

:{{math| 12345678910111213141516... }} {{OEIS|A007376}}

More generally, a ''Champernowne sequence'' (sometimes also called a ''Champernowne word'') is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order.<ref>
{{citation
 | last1 = Calude | first1 = C. | author1-link = Cristian S. Calude
 | last2 = Priese | first2 = L. | author2-link = Lutz Priese
 | last3 = Staiger | first3 = L. | author3-link = Ludwig Staiger
 | publisher = University of Auckland, New Zealand
 | pages = 1–35
 | title = Disjunctive sequences: An overview
 | year = 1997 | citeseerx = 10.1.1.34.1370 }}</ref>
For instance, the binary Champernowne sequence in [[shortlex order]] is

:{{math|1= 0 1 00 01 10 11 000 001 ...}} {{OEIS|A076478}}
where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.