in mathematics, the champernowne constant  is a transcendental real constant whose decimal expansion has important properties. it is named after mathematician d. g. champernowne, who published it as an undergraduate in 1933. for base&nbsp;10, the number is defined by concatenating representations of successive integers: : . champernowne constants can also be constructed in other bases, similarly, for example: : :. the champernowne constant can be expressed exactly as an infinite series: :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}} and this series generalizes to arbitrary bases&nbsp; by replacing 10 and 9 with  and  respectively. the champernowne word or barbier word is the sequence of digits of c'k.cassaigne & nicolas (2010) p.165* == normality == a real number x is said to be 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 b if its digits in base b follow a uniform distribution. if we denote a digit string as [a0,a1,...], 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 c_{10} is normal in base ten, although it is possible that it is not normal in other bases.d. g. champernowne, the construction of decimals normal in the scale of ten, journal of the london mathematical society, vol. 8 (1933), pp. 254â260 == continued fraction expansion == 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.  the simple continued fraction expansion of champernowne's constant has been studied as well. kurt mahler showed that the constant is transcendental;k.&nbsp;mahler
Mathematical_constants Number_theory