holds. All Liouville numbers are transcendental numbers, but not all transcendental numbers are Liouville numbers. The first example given by Joseph Liouville was a number of the form
with an integer , specifically (the resulting number is now called Liouville’s constant and is listed in A012245 of Sloane’s OEIS). In base , a number of this form has a representation beginning 0.110001000000000000000001000… where the th instance of the digit 1 is separated from the previous by 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.
|Date of creation||2013-03-22 17:18:32|
|Last modified on||2013-03-22 17:18:32|
|Last modified by||PrimeFan (13766)|