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

  • Kurt Mahler


Let (n)q(n)_q denote the base-qq expansion of the integer nn. The Champernowne number to the base qq is the concatenation of the base-qq expansions of the positive integers after a radix point; that is, the number 0.(1)q(2)q(3)q(n)q.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)].