Liouville number

A Liouville numberMathworldPlanetmath is an irrational number l such that for any integer n>0 there is a pair of integers j and k such that the inequality


holds. All Liouville numbers are transcendental numbersMathworldPlanetmath, but not all transcendental numbers are Liouville numbers. The first example given by Joseph Liouville was a number of the form


with an integer b>1, specifically b=10 (the resulting number is now called Liouville’s constant and is listed in A012245 of Sloane’s OEIS). In base b, a number of this form has a representation beginning 0.110001000000000000000001000… where the ith instance of the digit 1 is separated from the previous by i!-(i-1)!-1 instances of the digit 0.

The set of Liouville numbers is small in measure, having measure 0; in the measure-theoretic setting almost all numbers are not Liouville numbers. However, the set of Liouville numbers is topologically big, as it is residual and in the topological setting almost all numbers are Liouville numbers.

Title Liouville number
Canonical name LiouvilleNumber
Date of creation 2013-03-22 17:18:32
Last modified on 2013-03-22 17:18:32
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11J81
Defines Liouville’s constant