Statements (136)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
Mathematician
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| gptkbp:are |
uncountably infinite
not algebraic a key example in the study of irrationality measures. a class of numbers that can be explicitly constructed. a class of numbers that challenge traditional number theory. used in the exploration of the properties of real numbers. a topic of interest in the study of mathematical structures. used to construct examples of non-algebraic numbers. used to demonstrate properties of irrational numbers. a key concept in the understanding of number systems. used to illustrate the difference between algebraic and transcendental numbers. used to demonstrate the limits of algebraic numbers. a significant concept in the study of mathematical logic. a class of numbers that have unique mathematical properties. a key concept in the theory of transcendental numbers. a key example in the exploration of irrational numbers. a class of numbers that are not easily classified. a class of numbers that have unique properties. a fundamental concept in the field of mathematics. a fundamental example in the theory of numbers. a key concept in the study of transcendentality. a key concept in transcendental number theory. a key example in the study of irrational numbers. a key example in the study of real analysis. a key example in the study of real numbers. a key example in the theory of approximation. a part of the broader study of number theory. a significant area of research in mathematics. a significant area of research in number theory. a significant area of study in mathematics. a significant area of study in modern mathematics. a significant concept in the theory of numbers. a significant topic in the history of mathematics. a significant topic in the study of real analysis. a special case of a more general class of numbers. a special case of transcendental numbers. a specific type of real number. a subject of research in modern mathematics. a subject of study in advanced mathematics. a subset of transcendental numbers a subset of transcendental numbers. a topic of discussion in advanced mathematics. a topic of discussion in mathematical circles. a topic of interest for mathematicians. a topic of interest in mathematical analysis. a topic of interest in mathematical logic. a topic of interest in mathematical research. a topic of interest in the field of number theory. a topic of ongoing mathematical exploration. a topic of ongoing mathematical research. a topic of research in modern mathematics. a topic of research in the field of mathematics. a type of irrational number. a type of number that has unique properties. a type of number that is not algebraic. a type of number that is not easily categorized. a type of number that is not easily defined. a type of number that is not easily understood. a type of number that is not finite. a type of number that is not rational. defined by their approximation properties. used to demonstrate the existence of transcendental numbers. a subject of interest in the field of number theory. important in number theory. important in the field of mathematical logic. not algebraic. not algebraically independent. not computable. not countable. not rational. related to the concept of measure in mathematics. related to the study of irrationality measures. uncountably infinite. used in proofs of various mathematical theorems. used in the analysis of irrationality. used in the context of mathematical proofs. used in the study of irrationality. used in the study of transcendental functions. used in various mathematical proofs. used to demonstrate the diversity of real numbers. used to explore the nature of irrationality. used to illustrate the properties of real numbers. used to illustrate the complexity of number systems. a significant concept in the exploration of number theory. a significant concept in the study of real analysis. a fundamental concept in the study of real analysis. a key concept in the understanding of transcendental numbers. used to demonstrate the richness of the real number line. used in the context of transcendental number classification. a class of numbers that have important implications in mathematics. a key concept in the study of irrationality measures. a fundamental concept in the field of transcendental number theory. a significant area of study in mathematical research. a topic of interest in the field of mathematical analysis. used to explore the boundaries of number classification. a class of numbers that challenge conventional understanding. used in the proof of the existence of transcendental numbers. related to the concept of density in number theory. used to illustrate the limitations of algebraic numbers. a key example in the field of mathematical analysis. a key concept in the study of transcendental numbers. constructed by a specific method involving sequences. named_after_a_French_mathematician. related_to_Diophantine_approximation. used_in_proofs_of_the_Lindemann–Weierstrass_theorem. |
| gptkbp:can_be |
used to explore the limits of rational approximation.
used to study the relationships between different mathematical concepts. approximated by rational numbers. constructed using continued fractions. constructed using sequences of rational numbers. expressed in the form of a series. expressed in the form of infinite series. used to analyze the properties of sequences. used to construct other transcendental numbers. used to explore the nature of irrationality. used to explore the properties of real numbers. used to demonstrate the density of irrational numbers. used to illustrate the complexity of number systems. used to explore the relationships between different types of numbers. used to demonstrate the richness of the real number line. used to illustrate properties of irrational numbers. used to illustrate the concept of density in mathematics. used to analyze the properties of irrational numbers. used to explore the limits of mathematical understanding. used to illustrate the complexity of number theory. used to study the relationships between different types of numbers. |
| gptkbp:characteristics |
for every positive integer n, there exists a constant c_n such that |x - p/q| < c_n/q^n for infinitely many rational approximations p/q.
for any positive integer n, there exists a constant c such that |x - p/q| < c/q^n for infinitely many rational p/q. |
| gptkbp:defines |
real numbers that can be approximated by rational numbers with a certain degree of accuracy.
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| gptkbp:examples |
The number L = 0.1100010000000000000000010000000000000000001... is a Liouville number.
the number L = 0.1100010000000000000000010000000000000000001... (with n zeros between each 1) |
| gptkbp:has |
applications in number theory.
|
| https://www.w3.org/2000/01/rdf-schema#label |
Liouville numbers
|
| gptkbp:previousName |
gptkb:Joseph_Liouville
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| gptkbp:related_to |
algebraic numbers
|