The representation of rational numbers by terminating continued fractions

1996 ◽  
Vol 51 (4) ◽  
pp. 736-738 ◽  
Author(s):  
S V Konyagin ◽  
N G Moshchevitin
Keyword(s):  
Continued Fractions ◽  
Rational Numbers

scholarly journals A characterization of rational numbers by p-adic Ruban continued fractions

1985 ◽  
Vol 39 (3) ◽  
pp. 300-305 ◽  
Author(s):  
Vichian Laohakosol
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers

AbstractA type of p–adic continued fraction first considered by A. Ruban is described, and is used to give a characterization of rational numbers.

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals Application of Wynn's epsilon algorithm to periodic continued fractions

1987 ◽  
Vol 30 (2) ◽  
pp. 295-299 ◽  
Author(s):  
M. J. Jamieson
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Quadratic Surd ◽  
Partial Quotients ◽  
Infinite Continued Fraction ◽  
Image Position ◽  
Epsilon Algorithm ◽  
Periodic Continued Fractions

The infinite continued fractionin whichis periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

scholarly journals Limits of unbounded sequences of continued fractions

1990 ◽  
Vol 41 (3) ◽  
pp. 509-512
Author(s):  
Jingcheng Tong
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Quotients ◽  
Limit Points ◽  
Necessary And Sufficient ◽  
Positive Integers

Let X = {xk}k≥1 be a sequence of positive integers. Let Qk = [O;xk,xk−1,…,x1] be the finite continued fraction with partial quotients xi(1 ≤ i ≤ k). Denote the set of the limit points of the sequence {Qk}k≥1 by Λ(X). In this note a necessary and sufficient condition is given for Λ(X) to contain no rational numbers other than zero.

Encoding of Rational Numbers and Their Homomorphic Computations for FHE-Based Applications

2018 ◽  
Vol 29 (06) ◽  
pp. 1023-1044
Author(s):  
Heewon Chung ◽  
Myungsun Kim
Keyword(s):  
Continued Fractions ◽  
Homomorphic Encryption ◽  
Rational Numbers ◽  
Sensitive Information ◽  
Sensitive Data ◽  
Security Systems ◽  
Complex Arithmetic ◽  
Numeric Data ◽  
High Degree

This work addresses a basic problem of security systems that operate on very sensitive information. Specifically, we are interested in the problem of privately handling numeric data represented by rational numbers (e.g., medical records). Fully homomorphic encryption (FHE) is one of the natural and powerful tools for ensuring privacy of sensitive data, while allowing complicated computations on the data. However, because the native plaintext domain of known FHE schemes is restricted to a set of quite small integers, it is not easy to obtain efficient algorithms for encrypted rational numbers in terms of space and computation costs. For example, the naïve decimal representation considerably restricts the choice of parameters in employing an FHE scheme, particularly the plaintext size. Our basic strategy is to alleviate this inefficiency by using a different representation of rational numbers instead of naïve expressions. In this work we express rational numbers as continued fractions. Because continued fractions enable us to represent rational numbers as a sequence of integers, we can use a plaintext space with a small size while preserving the same quality of precision. However, this encoding technique requires performing very complex arithmetic operations, such as division and modular reduction. Theoretically, FHE allows the evaluation of any function, including modular reduction at encrypted data, but it requires a Boolean circuit of very high degree to be constructed. Hence, the primary contribution of this work is developing an approach to solve this efficiency problem using homomorphic operations with small degrees.

Morphing Lord Brouncker's continued fraction for π into the product of Wallis

2011 ◽  
Vol 95 (532) ◽  
pp. 17-22 ◽  
Author(s):  
Thomas J. Osler
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
The Third ◽  
Image Position

Three of the oldest and most celebrated formulae for π are:The first is Vieta's product of nested radicals from 1592 [1]. The second is Wallis's product of rational numbers [2] from 1656 and the third is Lord Brouncker's continued fraction [3,2], also from 1656. (In the remainder of the paper, for continued fractions we will use the more convenient notation

From Christoffel Words to Markoff Numbers

2018 ◽  
Author(s):  
Christophe Reutenauer
Keyword(s):  
Free Group ◽  
Continued Fractions ◽  
Quadratic Forms ◽  
Special Class ◽  
Rational Numbers ◽  
Real Numbers ◽  
Inner Automorphisms ◽  
Sturmian Words ◽  
Markoff Numbers ◽  
Lyndon Words

Christoffel introduced in 1875 a special class of words on a binary alphabet, linked to continued fractions. Some years laterMarkoff published his famous theory, called nowMarkoff theory. It characterizes certain quadratic forms, and certain real numbers by extremal inequalities. Both classes are constructed by using certain natural numbers, calledMarkoff numbers; they are characterized by a certain diophantine equality. More basically, they are constructed using certain words, essentially the Christoffel words. The link between Christoffelwords and the theory ofMarkoffwas noted by Frobenius.Motivated by this link, the book presents the classical theory of Markoff in its two aspects, based on the theory of Christoffel words. This is done in Part I of the book. Part II gives the more advanced and recent results of the theory of Christoffel words: palindromes (central words), periods, Lyndon words, Stern–Brocot tree, semi-convergents of rational numbers and finite continued fractions, geometric interpretations, conjugation, factors of Christoffel words, finite Sturmian words, free group on two generators, bases, inner automorphisms, Christoffel bases, Nielsen’s criterion, Sturmian morphisms, and positive automorphisms of this free group.

scholarly journals Two constructions of the real numbers via alternating series

1989 ◽  
Vol 12 (3) ◽  
pp. 603-613 ◽  
Author(s):  
Arnold Knopfmacher ◽  
John Knopfmacher
Keyword(s):  
Continued Fractions ◽  
Rational Numbers ◽  
Equivalence Classes ◽  
Real Numbers ◽  
The Real ◽  
New Methods ◽  
Arbitrary Choice ◽  
Ordered Field

Two further new methods are put forward for constructing the complete ordered field of real numbers out of the ordered field of rational numbers. The methods are motivated by some little known results on the representation of real numbers via alternating series of rational numbers. Amongst advantages of the methods are the facts that they do not require an arbitrary choice of "base" or equivalence classes or any similar constructs. The methods bear similarities to a method of construction due to Rieger, which utilises continued fractions.

Continued Fractions

2018 ◽  
pp. 35-42
Author(s):  
Christophe Reutenauer
Keyword(s):  
Continued Fractions ◽  
Error Term ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Lexicographical Order ◽  
Quadratic Number ◽  
Real Numbers ◽  
Irrational Numbers ◽  
Upper Limit ◽  
Convergents Of Continued Fractions

Basic theory of continued fractions: finite continued fractions (for rational numbers) and infinite continued fractions (for irrational numbers). This also includes computation of the quadratic number with a given periodic continued fraction, conjugate quadratic numbers, and approximation of reals and convergents of continued fractions. The chapter then takes on quadratic bounds for the error term and Legendre’s theorem, and reals having the same expansion up to rank n. Next, it discusses Lagrange number and its characterization as an upper limit, and equivalence of real numbers (equivalent numbers have the same Lagrange number). Finally, it covers ordering real numbers by alternating lexicographical order on continued fractions.

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