Liouville number

As a Liouville number, named after Joseph Liouville, is known in the theory of numbers is a real number satisfying the condition that integers and exist for all positive integers such that

Irrationality and transcendence

All Liouville numbers are irrational: For each rational number with an integer numerator and denominator positive integer there exists a positive integer Now if and are integers and are then

1844 showed Liouville that numbers with this property are not only irrational but also transcendental. This was the first proof of the transcendence of a number, the Liouville constant:

All Liouville numbers are transcendental, but not all transcendental numbers are liouvillesch. For example, the Euler number is transcendent, but not Liouvillesch.