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 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.

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

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.

scholarly journals Non-trivial matrix actions preserve normality for continued fractions

2017 ◽  
Vol 153 (2) ◽  
pp. 274-293 ◽  
Author(s):  
Joseph Vandehey
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Rational Numbers ◽  
Continued Fraction Expansion

A seminal result due to Wall states that if $x$ is normal to a given base $b$, then so is $rx+s$ for any rational numbers $r,s$ with $r\neq 0$. We show that a stronger result is true for normality with respect to the continued fraction expansion. In particular, suppose $a,b,c,d\in \mathbb{Z}$ with $ad-bc\neq 0$. Then if $x$ is continued fraction normal, so is $(ax+b)/(cx+d)$.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Calculating astronomical refraction by means of continued fractions

1979 ◽  
Vol 89 ◽  
pp. 95-101
Author(s):  
S. Mikkola
Keyword(s):  
Asymptotic Expansion ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Astronomical Refraction

A continued fraction was derived for the summation of the asymptotic expansion of astronomical refraction. Using simple approximations for the last denominator of the fraction, accurate formulae, useful down to the horizon, were obtained. The method is not restricted to any model of the atmosphere and can thus be used in calculations based on actual aerological measurements.

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