Statements (49)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:application |
Analysis of periodicity in quadratic irrationals
Best rational approximations Computing square roots Solving Diophantine equations |
| gptkbp:defines |
Expression obtained through an iterative process of representing numbers as the sum of their integer part and the reciprocal of another number
|
| gptkbp:field |
gptkb:Mathematics
|
| https://www.w3.org/2000/01/rdf-schema#label |
Continued Fractions
|
| gptkbp:notation |
[a0; a1, a2, a3, ...]
|
| gptkbp:property |
Can be finite or infinite
Can be used to represent transcendental numbers Can be used to solve Pell's equation Expansion is unique for positive real numbers Related to Euclidean algorithm Related to Farey sequences Related to continued fraction algorithm Used in computer algebra systems Used in cryptography Used in the analysis of dynamical systems Used in the study of Markov numbers Used in the study of ergodic theory Used in the study of fractals Used in the study of irrationality measure Used in the study of measure theory Used in the study of modular forms Used in the study of periodic orbits Used in the study of random processes Quadratic irrationals have periodic continued fraction expansions Convergents provide best possible rational approximations Used in the proof of irrationality of certain numbers Every irrational number has an infinite continued fraction expansion Every rational number has a finite continued fraction expansion Used in the study of continued fraction digits distribution |
| gptkbp:relatedTo |
gptkb:Generalized_continued_fraction
gptkb:Regular_continued_fraction Convergent Partial quotient Simple continued fraction |
| gptkbp:represents |
Rational numbers
Irrational numbers |
| gptkbp:studiedBy |
gptkb:Joseph-Louis_Lagrange
gptkb:Leonhard_Euler gptkb:Roger_Cotes gptkb:John_Wallis |
| gptkbp:usedIn |
gptkb:Number_theory
Analysis Approximation theory |
| gptkbp:bfsParent |
gptkb:A._Ya._Khinchin
|
| gptkbp:bfsLayer |
6
|