Liouville numbers
E637310
Liouville numbers are real numbers that can be approximated extremely closely by rationals, making them a classic example of transcendental numbers in number theory.
Observed surface forms (3)
| Surface form | Occurrences |
|---|---|
| Liouville number | 1 |
| Liouville constant | 0 |
| Liouville | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Liouville number
ⓘ
real number ⓘ transcendental number ⓘ |
| approximationProperty | approximable by rationals faster than any power of denominator ⓘ |
| belongsTo | uncountable dense subset of R with measure zero ⓘ |
| cardinality | uncountable ⓘ |
| characterizedBy | can be approximated extremely closely by rational numbers ⓘ |
| closureProperty |
product of a nonzero rational number and a Liouville number is a Liouville number
ⓘ
set of Liouville numbers is closed under addition with rationals ⓘ set of Liouville numbers is closed under multiplication by nonzero rationals ⓘ sum of a Liouville number and a rational number is a Liouville number ⓘ |
| complexExtension | can be considered as complex numbers with zero imaginary part ⓘ |
| contrastWith |
algebraic irrational numbers
ⓘ
badly approximable numbers ⓘ normal numbers ⓘ |
| definedAs | sum from k=1 to infinity of 10^{-k!} ⓘ |
| definition | a real number x is a Liouville number if for every positive integer n there exist integers p and q>1 such that 0 < |x - p/q| < 1/q^n ⓘ |
| densityProperty | between any two real numbers there exists a Liouville number ⓘ |
| disjointFrom | algebraic numbers ⓘ |
| example | Liouville constant 0.1100010000000000000000010000... NERFINISHED ⓘ |
| field | number theory ⓘ |
| firstUse | to prove existence of transcendental numbers ⓘ |
| hasIrrationalityMeasure | infinite ⓘ |
| historicalSignificance | provided first explicit examples of transcendental numbers ⓘ |
| introducedBy | Joseph Liouville NERFINISHED ⓘ |
| introducedInYear | 1844 ⓘ |
| logicalStatus | proper subset of transcendental numbers ⓘ |
| measure | Lebesgue measure zero ⓘ |
| namedAfter | Joseph Liouville NERFINISHED ⓘ |
| notClosedUnder |
addition of two Liouville numbers in general
ⓘ
multiplication of two Liouville numbers in general ⓘ |
| property |
every Liouville number is transcendental
ⓘ
every algebraic irrational has finite irrationality measure ⓘ not every transcendental number is a Liouville number ⓘ set of Liouville numbers is a dense G-delta set in R ⓘ set of Liouville numbers is of first Baire category ⓘ set of Liouville numbers is uncountable but meagre ⓘ |
| relatedTo |
Diophantine approximation
NERFINISHED
ⓘ
irrationality measure ⓘ |
| researchArea |
metric Diophantine approximation
ⓘ
transcendental number theory ⓘ |
| subsetOf |
real numbers
ⓘ
transcendental numbers ⓘ |
| symbolicNotation | often denoted by L in examples ⓘ |
| topologicalCategory | F-sigma-delta set in R ⓘ |
| topologicalProperty | dense in the real numbers ⓘ |
| usedIn | construction of explicit transcendental numbers ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Liouville
this entity surface form:
Liouville number