Editing Champernowne constant (section)

== Properties ==
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. A number ''x'' is said to be normal in [[radix|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 10, 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 10,<ref name=Cha33>{{harvnb|Champernowne|1933}}</ref> while Nakai and Shiokawa proved a more general theorem, a corollary of which is that <math>C_{b}</math> is normal in base <math>b</math> for any ''b''.<ref name=Nak92>{{harvnb|Nakai|Shiokawa|1992}}</ref> It is an open problem whether <math>C_{k}</math> is normal in bases <math>b \neq k</math>. For example, it is not known if <math>C_{10}</math> is normal in base 9. For example, 54 digits of <math>C_{10}</math> is 0.123456789101112131415161718192021222324252627282930313. When we express this in base 9 we get <math>{0.10888888853823026326512111305027757201400001517660835887}_9</math>.

[[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>
The [[irrationality measure]] of <math>C_{10}</math> is <math>\mu(C_{10})=10</math>, and more generally <math>\mu(C_b)=b</math> for any base <math>b\ge 2</math>.<ref>[[Masaaki Amou]], ''[http://www.sciencedirect.com/science/article/pii/S0022314X05800393 Approximation to certain transcendental decimal fractions by algebraic numbers]'', [[Journal of Number Theory]], Volume 37, Issue 2, February 1991, Pages 231–241</ref>

The Champernowne word is a [[disjunctive sequence]]. A '''disjunctive sequence''' is an infinite [[Sequence#Infinite sequences in theoretical computer science|sequence]] (over a finite [[Alphabet (computer science)|alphabet]] of [[Character (computing)|characters]]) in which every [[String (computer science)#Formal theory|finite string]] appears as a [[substring]].