Reprint: On the decimal expansion of certain irrational numbers (1937)

Abstract

Let $(n{)}_q$ denote the base-$q$ expansion of the integer $n$. The Champernowne number to the base $q$ is the concatenation of the base-$q$ expansions of the positive integers after a radix point; that is, the number $0.(1{)}_q(2{)}_q(3{)}_q\cdots (n{)}_q\cdots .$ In this paper, Mahler shows that each of these numbers is transcendental, but is not a Liouville number.

Reprint of the author's paper [Mathematica B, Zutphen 6, 22--36 (1937; Zbl 0018.11102; JFM 63.0155.03)].