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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 [[mathematician]] [[D. G. Champernowne]], who published it as an undergraduate in 1933.
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For [[base&nbsp;10]], the number is defined by concatenating representations of successive integers:
:{{math|1=''C''<sub>10</sub> = 0.12345678910111213141516… }} {{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>}}
:{{math|1=''C''<sub>3</sub> = 0.12101112202122… <sub>3</sub>}}.
 
The Champernowne constant can be expressed exactly as an [[infinite series]]:
 
:<math>C_{10}=\sum_{n=1}^\infty\sum_{k=10^{n-1}}^{10^n-1}\frac{k}{10^{n(k-10^{n-1}+1)+9\sum_{l=1}^{n-1}10^{l-1}l}}</math>
 
and this series generalizes to arbitrary bases&nbsp;{{mvar|b}} by replacing 10 and 9 with {{mvar|b}} and {{math|''b'' − 1}} respectively.
 
The '''Champernowne word''' or '''Barbier word''' is the sequence of digits of ''C''<sub>''k''</sub>.<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>
 
== Normality ==
A [[real number]] ''x'' is said to be [[normal number|normal]] if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. ''x'' is said to be normal in [[base (exponentiation)|base]] ''b'' if its digits in base b follow a uniform distribution.
 
If we denote a digit string as [''a''<sub>0</sub>,''a''<sub>1</sub>,...], then, in base ten, we would expect strings [0],[1],[2],...,[9] to occur 1/10 of the time, strings [0,0],[0,1],...,[9,8],[9,9] to occur 1/100 of the time, and so on, in a normal number.
 
Champernowne proved that <math>C_{10}</math> is normal in base ten, although it is possible that it is not normal in other bases.<ref name="champernowne33">D. G. Champernowne, ''The construction of decimals normal in the scale of ten'', Journal of the London Mathematical Society, vol. 8 (1933), pp. 254–260</ref>
 
== Continued fraction expansion ==
[[Image:Champernowne constant.svg|right|300px|thumb|The first 161 quotients of the continued fraction of the Champernowne constant. The 4th, 18th, 40th, and 101st are (much) bigger than 270, so they don't appear in the graph.]]
[[Image:Champernowne constant logscale.svg|right|300px|thumb|The first 161 quotients of the continued fraction of the Champernowne constant represented using the [[logarithmic scale]].]]
The [[continued fraction|simple continued fraction]] expansion of Champernowne's constant has been studied as well. [[Kurt Mahler]] showed that the constant is [[transcendental number|transcendental]];<ref name="mahler">K.&nbsp;Mahler, ''Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen'', Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p.&nbsp;421–428.</ref> therefore its continued fraction does not [[continued fraction#Finite continued fractions|terminate]] (because it is not [[rational number|rational]]) and is [[continued fraction#Periodic continued fractions|aperiodic]] (because it is not an irreducible quadratic).
 
The terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10,
: ''C''<sub>10</sub> = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
:::4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,
:::6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...].
 
The large number at position&nbsp;19 has 166 digits, and the next very large term at position&nbsp;41 of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good [[diophantine approximation|approximation]] of the Champernowne constant. For example, truncating before the 4th partial quotient, we obtain the partial sum <math>10/81 = 0.\overline{123456790}</math>, which approximates Champernowne's constant with an error of about {{math|1 × 10<sup>−9</sup>}}, while truncating just before the 18th partial quotient, we get
:<math>\frac{60499999499}{490050000000} =
0.123456789\overline{101112\ldots9900010203040506070809}
,</math>
which approximates Champernowne's constant with error approximately {{math|9 × 10<sup>−190</sup>}}.
 
==See also==
*[[Copeland–Erdős constant]], a similar normal number, defined using the [[prime number]]s
*[[Liouville number#Liouville constant|Liouville's constant]], another constant defined by its decimal representation
 
== References ==
{{reflist}}
* {{cite book | last1=Cassaigne | first1=J. | last2=Nicolas | first2=F. | chapter=Factor complexity | pages=163–247 | editor1-last=Berthé | editor1-first=Valérie | editor2-last=Rigo | editor2-first=Michel | title=Combinatorics, automata, and number theory | series=Encyclopedia of Mathematics and its Applications | volume=135 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-51597-9 | zbl=1216.68204  }}
 
==External links==
*{{MathWorld|urlname=ChampernowneConstant|title=Champernowne constant}}
*[http://www.matifutbol.com/en/pencil.html The fantastic pencil and the Champernowne constant]
[[Category:Mathematical constants]]
[[Category:Number theory]]

Latest revision as of 14:26, 7 October 2014

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