Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
real number
|
| gptkbp:defines |
A real number x is a Liouville number if for every positive integer n, there exist integers p and q > 1 such that |x - p/q| < 1/q^n.
|
| gptkbp:example |
gptkb:Liouville_constant
|
| gptkbp:firstDescribed |
gptkb:Joseph_Liouville
1844 |
| https://www.w3.org/2000/01/rdf-schema#label |
Liouville numbers
|
| gptkbp:isA |
irrational number
uncountable set transcendental number dense set in real numbers measure zero set |
| gptkbp:namedAfter |
gptkb:Joseph_Liouville
|
| gptkbp:property |
Almost all real numbers are not Liouville numbers.
Every Liouville number is transcendental. No algebraic number is a Liouville number. The set of Liouville numbers is a G-delta set. The set of Liouville numbers is not closed. The set of Liouville numbers is not open. The set of Liouville numbers is not measurable in the sense of Baire category. The set of Liouville numbers is invariant under rational translation. The set of Liouville numbers is invariant under negation. |
| gptkbp:bfsParent |
gptkb:Liouville's_theorem_(number_theory)
|
| gptkbp:bfsLayer |
7
|