# Champernowne word

From Encyclopedia of Mathematics

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The infinite word over the alphabet $B = \{0,1,\ldots,b-1\}$ for given $b \ge 2$, obtained by concatenating the representations of the natural numbers with respect to base $b$. Thus for $b=2$ the word begins $$ 0\,1\,10\,11\,100\,101 \ldots \ . $$

This is a disjunctive word, that is, contains each finite sequence over $B$ as a factor infinitely often.

The **Champernowne number** is obtained by interpreting the word as the sequence of digits of a number $c_b$ in the interval $(0,1)$. The Champernowne number is a weakly normal number base $b$.

#### References

- Bugeaud, Yann. "Distribution modulo one and Diophantine approximation", Cambridge Tracts in Mathematics
**193'**Cambridge University Press (2012)**ISBN**978-0-521-11169-0 Zbl 1260.11001 - Berthé, Valérie; Rigo, Michel. "Combinatorics, automata, and number theory". Encyclopedia of Mathematics and its Applications
**135**Cambridge University Press**ISBN**978-0-521-51597-9 Zbl 1216.68204

**How to Cite This Entry:**

Champernowne number.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Champernowne_number&oldid=39295