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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Liouville | 1 |
| Liouville number | 1 |
| Liouville numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7030807 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Liouville numbers Context triple: [Diophantine approximation, hasKeyConcept, Liouville numbers]
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A.
Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
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B.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
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C.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
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D.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
-
E.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Liouville numbers Target entity description: Liouville numbers are real numbers that can be approximated extremely closely by rationals, making them a classic example of transcendental numbers in number theory.
-
A.
Khinchin–Lévy constant
The Khinchin–Lévy constant is a mathematical constant arising in metric number theory and continued fractions, describing the typical exponential growth rate of the denominators of convergents for almost all real numbers.
-
B.
Lindemann–Weierstrass theorem precursor
The Lindemann–Weierstrass theorem precursor is an early foundational result in transcendental number theory developed by Ferdinand von Lindemann that paved the way for the full Lindemann–Weierstrass theorem on the algebraic independence of exponentials of algebraic numbers.
-
C.
Khinchin's constant
Khinchin's constant is a mathematical constant that arises in metric number theory, describing the almost-sure geometric mean of the partial quotients in the continued fraction expansions of real numbers.
-
D.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
-
E.
Continued Fractions
Continued Fractions is a classic mathematical monograph by Aleksandr Khinchin that systematically develops the theory and applications of continued fraction expansions in number theory and analysis.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Liouville numbers Description of subject: Liouville numbers are real numbers that can be approximated extremely closely by rationals, making them a classic example of transcendental numbers in number theory.
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