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Journal articles on the topic 'Rational and irrational numbers'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 February 2022

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1

KWON, DOYONG. "A devil's staircase from rotations and irrationality measures for Liouville numbers." Mathematical Proceedings of the Cambridge Philosophical Society 145, no. 3 (November 2008): 739–56. http://dx.doi.org/10.1017/s0305004108001606.

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AbstractFrom Sturmian and Christoffel words we derive a strictly increasing function Δ:[0,∞) → . This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of Δ distinguishes some irrationality measures of real numbers.

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2

Lewis, Leslie D. "Irrational Numbers Can In-Spiral You." Mathematics Teaching in the Middle School 12, no. 8 (April 2007): 442–46. http://dx.doi.org/10.5951/mtms.12.8.0442.

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Introducing students to the pythagorean theorem presents a natural context for investigating what irrational numbers are and how they differ from rational numbers. This artistic project allows students to visualize, discuss, and create a product that displays irrational and rational numbers.

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3

Dubickas, Artūras. "On rational approximations to two irrational numbers." Journal of Number Theory 177 (August 2017): 43–59. http://dx.doi.org/10.1016/j.jnt.2017.01.026.

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4

Kasperski, Maciej, and Waldemar Kłobus. "Rational and irrational numbers from unit resistors." European Journal of Physics 35, no. 1 (November 13, 2013): 015008. http://dx.doi.org/10.1088/0143-0807/35/1/015008.

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5

Van Assche, Walter. "Hermite-Padé Rational Approximation to Irrational Numbers." Computational Methods and Function Theory 10, no. 2 (October 4, 2010): 585–602. http://dx.doi.org/10.1007/bf03321782.

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6

Lord, Nick. "92.75 Maths bite: irrational powers of irrational numbers can be rational." Mathematical Gazette 92, no. 525 (November 2008): 534. http://dx.doi.org/10.1017/s0025557200183846.

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7

Obersteiner, Andreas, and Veronika Hofreiter. "Do we have a sense for irrational numbers?" Journal of Numerical Cognition 2, no. 3 (February 10, 2017): 170–89. http://dx.doi.org/10.5964/jnc.v2i3.43.

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Number sense requires, at least, an ability to assess magnitude information represented by number symbols. Most educated adults are able to assess magnitude information of rational numbers fairly quickly, including whole numbers and fractions. It is to date unclear whether educated adults without training are able to assess magnitudes of irrational numbers, such as the cube root of 41. In a computerized experiment, we asked mathematically skilled adults to repeatedly choose the larger of two irrational numbers as quickly as possible. Participants were highly accurate on problems in which reasoning about the exact or approximate value of the irrational numbers’ whole number components (e.g., 3 and 41 in the cube root of 41) yielded the correct response. However, they performed at random chance level when these strategies were invalid and the problem required reasoning about the irrational number magnitudes as a whole. Response times suggested that participants hardly even tried to assess magnitudes of the irrational numbers as a whole, and if they did, were largely unsuccessful. We conclude that even mathematically skilled adults struggle with quickly assessing magnitudes of irrational numbers in their symbolic notation. Without practice, number sense seems to be restricted to rational numbers.

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8

Barbero, Stefano, Umberto Cerruti, and Nadir Murru. "Periodic representations for quadratic irrationals in the field of 𝑝-adic numbers." Mathematics of Computation 90, no. 331 (May 6, 2021): 2267–80. http://dx.doi.org/10.1090/mcom/3640.

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Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.

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Calogero, Francesco. "Cool irrational numbers and their rather cool rational approximations." Mathematical Intelligencer 25, no. 4 (September 2003): 72–76. http://dx.doi.org/10.1007/bf02984865.

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10

Belin, Mervenur, and Gülseren Karagöz Akar. "Exploring Real Numbers as Rational Number Sequences With Prospective Mathematics Teachers." Mathematics Teacher Educator 9, no. 1 (September 1, 2020): 63–87. http://dx.doi.org/10.5951/mte.2020.9999.

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The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers.

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11

Mazėtis, Edmundas, and Grigorijus Melničenko. "Rational cuboids and Heron triangles II." Lietuvos matematikos rinkinys 60 (December 5, 2019): 34–38. http://dx.doi.org/10.15388/lmr.b.2019.15233.

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We study the connection of Heronian triangles with the problem of the existence of rational cuboids. It is proved that the existence of a rational cuboid is equivalent to the existence of a rectangular tetrahedron, which all sides are rational and the base is a Heronian triangle. Examples of rectangular tetrahedra are given, in which all sides are integer numbers, but the area of the base is irrational. The example of the rectangular tetrahedron is also given, which has lengths of one side irrational and the other integer, but the area of the base is integer.

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12

Agarwal, Ravi P., and Hans Agarwal. "Origin of Irrational Numbers and Their Approximations." Computation 9, no. 3 (March 9, 2021): 29. http://dx.doi.org/10.3390/computation9030029.

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In this article a sincere effort has been made to address the origin of the incommensurability/irrationality of numbers. It is folklore that the starting point was several unsuccessful geometric attempts to compute the exact values of 2 and π. Ancient records substantiate that more than 5000 years back Vedic Ascetics were successful in approximating these numbers in terms of rational numbers and used these approximations for ritual sacrifices, they also indicated clearly that these numbers are incommensurable. Since then research continues for the known as well as unknown/expected irrational numbers, and their computation to trillions of decimal places. For the advancement of this broad mathematical field we shall chronologically show that each continent of the world has contributed. We genuinely hope students and teachers of mathematics will also be benefited with this article.

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13

Hančl, Jaroslav, Katarína Korčeková, and Lukáš Novotný. "Productly linearly independent sequences." Studia Scientiarum Mathematicarum Hungarica 52, no. 3 (September 2015): 350–70. http://dx.doi.org/10.1556/012.2015.52.3.1315.

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We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

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14

Blom, Jos. "Metrical properties of best approximants." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 1 (August 1992): 78–91. http://dx.doi.org/10.1017/s1446788700035412.

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AbstractA rational number is called a best approximant of the irrational number ζ if it lies closer to ζ than all rational numbers with a smaller denominator. Metrical properties of these best approximants are studied. The main tool is the two-dimensional ergodic system, underlying the continued fraction expansion.

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15

Loeb, Arthur L. "Buckminster Fuller Versus The Irrational : A Double Entendre (Dedicated to the Memory of R. Buckminster Fuller in the Centenary of his Birth)." International Journal of Space Structures 11, no. 1-2 (April 1996): 141–53. http://dx.doi.org/10.1177/026635119601-221.

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R. Buckminster Fuller strongly believed in the discreteness of structure, and distrusted irrational numbers. The concept of limited resolution is introduced to relate a discrete structural model to mathematical abstractions such as straight lines and circles. Fuller also made rational approximations to irrational expressions. It is shown that such approximations are unnecessary, and in point of fact at odds with Fuller's spatial rather than linear thinking. Some fundamental principles underlying an experimental Design Science are presented. It is shown that three-dimensional structural relations in forms which are identical in at least three non-planar directions are exactly expressible in terms of small rational numbers. There are four significant angles having rational trigonometric functions, in the range between 60° and 72°, which occur commonly in such structures. It is suggested that the system presented here can be the basis of a curriculum in Visual Mathematics.

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HANČL, JAROSLAV, and ONDŘEJ KOLOUCH. "ERDŐS’ METHOD FOR DETERMINING THE IRRATIONALITY OF PRODUCTS." Bulletin of the Australian Mathematical Society 84, no. 3 (July 13, 2011): 414–24. http://dx.doi.org/10.1017/s0004972711002309.

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AbstractThis paper deals with a sufficient condition for the infinite product of rational numbers to be an irrational number. The condition requires only some conditions for convergence and does not use other properties like divisibility. The proof is based on an idea of Erdős.

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17

Sen, S. K., Ravi P. Agarwal, and Raffaella Pavani. "Best k-digit rational bounds for irrational numbers: Pre- and super-computer era." Mathematical and Computer Modelling 49, no. 7-8 (April 2009): 1465–82. http://dx.doi.org/10.1016/j.mcm.2008.04.009.

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Sen, S. K., and Ravi P. Agarwal. "Best k-digit rational approximation of irrational numbers: Pre-computer versus computer era." Applied Mathematics and Computation 199, no. 2 (June 2008): 770–86. http://dx.doi.org/10.1016/j.amc.2007.10.039.

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19

LUCA, FLORIAN, and YOHEI TACHIYA. "IRRATIONALITY OF LAMBERT SERIES ASSOCIATED WITH A PERIODIC SEQUENCE." International Journal of Number Theory 10, no. 03 (March 18, 2014): 623–36. http://dx.doi.org/10.1142/s1793042113501121.

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Let q be an integer with |q| > 1 and {an}n≥1 be an eventually periodic sequence of rational numbers, not identically zero from some point on. Then the number [Formula: see text] is irrational. In particular, if the periodic sequences [Formula: see text] of rational numbers are linearly independent over ℚ, then so are the following m + 1 numbers: [Formula: see text] This generalizes a result of Erdős who treated the case of m = 1 and [Formula: see text]. The method of proof is based on the original approaches of Chowla and Erdős, together with some results about primes in arithmetic progressions with large moduli of Ahlford, Granville and Pomerance.

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Masáková, Z., J. Patera, and E. Pelantová. "Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals." Canadian Journal of Physics 79, no. 2-3 (February 1, 2001): 687–96. http://dx.doi.org/10.1139/p01-003.

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There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by Ö5, Ö2, Ö3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation. PACS Nos.: 61.90+d, 61.50-f

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21

Snow, Joanne E., and Mary K. Porter. "Math Roots: Ratios and Proportions: They Are Not All Greek to Me." Mathematics Teaching in the Middle School 14, no. 6 (February 2009): 370–78. http://dx.doi.org/10.5951/mtms.14.6.0370.

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Today, the concept of number includes the sets of whole numbers, integers, rational numbers, and real numbers. This was not always so. At the time of Euclid (circa 330-270 BC), the only numbers used were whole numbers. To express quantitative relationships among geometric objects, such as line segments, triangles, circles, and spheres, the Greeks used ratios and proportions but not real numbers (fractions or irrational numbers). Although today we have full use of the number system, we still find ratios and proportions useful and effective when comparing quantities. In this article, we examine the history of ratios and proportions and their value to people from the past through the present.

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22

Berch, Daniel B. "Why Learning Common Fractions Is Uncommonly Difficult: Unique Challenges Faced by Students With Mathematical Disabilities." Journal of Learning Disabilities 50, no. 6 (July 18, 2016): 651–54. http://dx.doi.org/10.1177/0022219416659446.

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In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students with math disabilities exhibit a significant delay or deficit in the numerical transcoding of decimal fractions, and I further maintain that they may face unique challenges in developing the ability to effectively translate between different types of fractions and other rational number notational formats—what I call conceptual transcoding. I also argue that characterizing this level of comprehensive understanding of rational numbers as rational number sense is irrational, as it misrepresents this flexible and adaptive collection of skills as a biologically based percept rather than a convergence of higher-order competencies that require intensive, formal instruction.

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23

DROSTE, MANFRED, and RÜDIGER GÖBEL. "ON THE HOMEOMORPHISM GROUPS OF CANTOR'S DISCONTINUUM AND THE SPACES OF RATIONAL AND IRRATIONAL NUMBERS." Bulletin of the London Mathematical Society 34, no. 04 (July 2002): 474–78. http://dx.doi.org/10.1112/s0024609302001066.

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24

Kanbekova, Rimma Valeevna, Elvira Albertovna Izhbulatova, and Liliya Khazinurovna Salimova. "Methods of Increasing Motivation and Quality of Potential Primary School Teachers’ Mathematical Education." Development of education, no. 4 (6) (December 18, 2019): 19–22. http://dx.doi.org/10.31483/r-53754.

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The article describes the implementation of project tasks in the study of rational and irrational numbers, presents the results of a questionnaire survey of students, potential primary school teachers, while studying the numerical line in a course on mathematics. The purpose of the article is to consider the effect of supplementing a material of a numerical line with original content on motivation when teaching mathematics to students of non-mathematical profiles. Methods. A questionnaire survey was conducted on the indifferent or negative attitude of students to the study of the rational and irrational numbers theory. Using theoretical and experimental methods, the hypothesis on the formation of positive motivation among students through the solution of project tasks, the compilation and implementation of which gives them a sense of independence, freedom of choice, success, is tested. Conclusions presented in the article are based on the results of the study. Due to the implementation of project tasks, students in the study of the discipline «Mathematics» felt the usefulness of the knowledge gained and showed high marks on exams. It is concluded that the teaching of the numerical line in the course on mathematics at the university using the techniques mentioned below in the article allowed to increase the level of positive motivation among students.

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25

Ganzburg, Michael I., and Michael Revers. "A note on Lagrange interpolation for |x|λ at equidistant nodes." Bulletin of the Australian Mathematical Society 70, no. 3 (December 2004): 475–80. http://dx.doi.org/10.1017/s0004972700034729.

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In this note, we discuss the exceptional set E ⊆ [−1, 1] of points x0 satisfying the inequality where λ > 0, λ ≠ 2, 4, … and Ln(fλ,.) is the Lagrange interpolation polynomial of degree at most n to fλ(x):= |x|λ on the interval [−1, 1] associated with the equidistant nodes. It is known that E has Lebesgue measure zero. Here we show that E contains infinite families of rational and irrational numbers.

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26

Tsujii, Masato. "Rotation number and one-parameter families of circle diffeomorphisms." Ergodic Theory and Dynamical Systems 12, no. 2 (June 1992): 359–63. http://dx.doi.org/10.1017/s0143385700006805.

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AbstractWe consider one-parameter families of circle diffeomorphisms, f1(x) = f(x) + t(t ∈ S1), where f: S1 is a Cr-diffeomorphism (r≥3). We show that, for Lebesgue almost every t ∈ S1 the rotation number of f1, is either a rational number or an irrational number of Roth type. In the former case, f1, has periodic orbits and, in the latter case, f1, is Cr − 2-conjugate to an irrational rigid rotation from well-known theorems of Herman and Yoccoz.

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Ge, Wenxu, Weiping Li, and Tianze Wang. "On Diophantine approximation by unlike powers of primes." Open Mathematics 17, no. 1 (May 30, 2019): 544–55. http://dx.doi.org/10.1515/math-2019-0045.

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Abstract Suppose that λ1, λ2, λ3, λ4, λ5 are nonzero real numbers, not all of the same sign, λ1/λ2 is irrational, λ2/λ4 and λ3/λ5 are rational. Let η real, and ε > 0. Then there are infinitely many solutions in primes pj to the inequality $\begin{array}{} \displaystyle |\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5+\eta| \lt (\max{p_j^j})^{-1/32+\varepsilon} \end{array}$. This improves an earlier result under extra conditions of λj.

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PRASAD, K. C., HRISHIKESH MAHATO, and SUDHIR MISHRA. "A NEW POINT IN LAGRANGE SPECTRUM." International Journal of Number Theory 09, no. 02 (December 5, 2012): 393–403. http://dx.doi.org/10.1142/s1793042112501382.

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Let I denote the set of all irrational numbers, θ ∈ I, and simple continued fraction expansion of θ be [a0, a1, …, an, …]. Then a0 is an integer and {an}n≥1 is an infinite sequence of positive integers. Let Mn(θ) = [0, an, an-1, …, a1] + [an+1, an+2, …]. Then the set of numbers { lim sup Mn(θ) ∣ θ ∈ I} is called the Lagrange Spectrum 𝔏. Notably 3 is the first cluster point of 𝔏. Essentially lim inf 𝔏 or [Formula: see text]. Perron [Über die approximation irrationaler Zahlen durch rationale, I, S.-B. Heidelberg Akad. Wiss., Abh. 4 (1921) 17 pp; Über die approximation irrationaler Zahlen durch rationale, II, S.-B. Heidelberg Akad. Wiss., Abh.8 (1921) 12 pp.] has found that lim inf { lim sup Mn(θ) ∣ θ = [a0, a1, a2, …, an, …] and [Formula: see text]. This article forwards the value of lim inf{lim sup Mn(θ) ∣ θ = [a0, a1, …, an, …] and an ≥ 4 frequently}, a long awaited cluster point of Lagrange Spectrum.

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29

Yaohua, Hu, and P. J. Stacey. "Toral automorphisms and antiautomorphisms of rotation algebras." Bulletin of the Australian Mathematical Society 59, no. 2 (April 1999): 247–55. http://dx.doi.org/10.1017/s000497270003286x.

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If U, V are the generators of a rational or irrational rotation C*-algebra then an automorphism φ of the algebra is determined by φ(U) = λUaVc and φ(V) = μUbVd where λ, μ are complex numbers of modulus 1 and a, b, c, d are integers with ad − bc = 1. If ad − bc = −1, then these formulae determine an antiautomorphsm of the algebra. The classification of such automorphisms and antiautomorphisms up to conjugacy by arbitrary automorphisms is studied and an almost complete classification is obtained.

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30

Jernajczyk, Jakub. "Jak pokazać to, czego pokazać nie można? O obrazowaniu liczb niewymiernych." Studia Philosophica Wratislaviensia 15, no. 3 (December 31, 2020): 17–30. http://dx.doi.org/10.19195/1895-8001.15.3.2.

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In this article, I would like to draw attention to the cognitive potential of an image, showing how significant the role of visual imagination in mathematics is. I will focus here mainly on the possibilities of visualizing irrational numbers.Our starting point is the intuitive case of the square root of two, observed in the diagonal of a square. We will also discuss a simple, geometrical method of constructing the square roots of all integers. Next, we move over to the golden ratio, hidden in a regular pentagon. We will use a looped, endless animation to visualize the irrational number φ. Then we will have a closer look at the famous number π and discuss two different attempts to find its visual representation. In the last two sections of the article, we consider the possibility of indicating rational and irrational real numbers and also grasp the whole set of real numbers.All the issues discussed in this article have inspired visual artists to create artworks that can help to understand relatively advanced mathematical problems.

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Losada, M. Piacquadio, and S. Grynberg. "Cantor Staircases in Physics and Diophantine Approximations." International Journal of Bifurcation and Chaos 08, no. 06 (June 1998): 1095–106. http://dx.doi.org/10.1142/s0218127498000887.

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For a wide class of dynamical systems the variables involved relate to one another through a Cantor staircase function. When they are time variables, the staircases have well-known universal properties that suggest a connection with certain classical problems in Number Theory. In this paper we extend some of those universal properties to certain Cantor staircases that appear in Quantum Mechanics, where the variables involved are not time variables. We also develop some connections between the geometry of these Cantor staircases and the problem of approximating irrational numbers of rational ones, classical in Number Theory.

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Grabowski, Adam, and Artur Korniłowicz. "Introduction to Liouville Numbers." Formalized Mathematics 25, no. 1 (March 28, 2017): 39–48. http://dx.doi.org/10.1515/forma-2017-0003.

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Summary The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https://en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.

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Korniłowicz, Artur, Adam Naumowicz, and Adam Grabowski. "All Liouville Numbers are Transcendental." Formalized Mathematics 25, no. 1 (March 28, 2017): 49–54. http://dx.doi.org/10.1515/forma-2017-0004.

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Summary In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.

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CIRILO, PATRÍCIA, YURI LIMA, and ENRIQUE PUJALS. "Law of large numbers for certain cylinder flows." Ergodic Theory and Dynamical Systems 34, no. 3 (January 23, 2013): 801–25. http://dx.doi.org/10.1017/etds.2012.165.

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AbstractWe construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit a law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a Gaussian distribution.

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Lai, Li, and Pin Yu. "A note on the number of irrational odd zeta values." Compositio Mathematica 156, no. 8 (August 2020): 1699–717. http://dx.doi.org/10.1112/s0010437x20007307.

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AbstractWe prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

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Ercire, Yusuf Emre, Serkan Narlı, and Esra Aksoy. "Learning Difficulties about the Relationship between Irrational Number Set with Rational or Real Number Sets." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 7, no. 2 (August 4, 2016): 417. http://dx.doi.org/10.16949/turcomat.47225.

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Quan, Le Phuong. "A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions." Mathematical and Computational Applications 23, no. 4 (October 17, 2018): 63. http://dx.doi.org/10.3390/mca23040063.

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A complete MAPLE procedure is designed to effectively implement an algorithm for approximating trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval–polynomial as its output that we can easily exploit in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both rational and irrational arguments as well as drawing the graph of the piecewise approximate functions are presented. Moreover, from the approximate integration on [ a , b ] with integrands of the form x m sin x , another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best L 2 -approximation of the sine function in the vector space P ℓ of polynomials of degree at most ℓ, a subspace of L 2 ( a , b ) .

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Abrarov, Sanjar M., Rajinder K. Jagpal, Rehan Siddiqui, and Brendan M. Quine. "Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π." Mathematics 9, no. 17 (September 4, 2021): 2162. http://dx.doi.org/10.3390/math9172162.

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In our earlier publication we have shown how to compute by iteration a rational number u2,k in the two-term Machin-like formula for π of the kind π4=2k−1arctan1u1,k+arctan1u2,k,k∈Z,k≥1, where u1,k can be chosen as an integer u1,k=ak/2−ak−1 with nested radicals defined as ak=2+ak−1 and a0=0. In this work, we report an alternative method for determination of the integer u1,k. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set ak in computation of the integer u1,k. Mathematica programs validating these results are presented.

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39

Jaeger, Peter. "Events of Borel Sets, Construction of Borel Sets and Random Variables for Stochastic Finance." Formalized Mathematics 22, no. 3 (September 1, 2014): 199–204. http://dx.doi.org/10.2478/forma-2014-0022.

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Summary We consider special events of Borel sets with the aim to prove, that the set of the irrational numbers is an event of the Borel sets. The set of the natural numbers, the set of the integer numbers and the set of the rational numbers are countable, so we can use the literature [10] (pp. 78-81) as a basis for the similar construction of the proof. Next we prove, that different sets can construct the Borel sets [16] (pp. 9-10). Literature [16] (pp. 9-10) and [11] (pp. 11-12) gives an overview, that there exists some other sets for this construction. Last we define special functions as random variables for stochastic finance in discrete time. The relevant functions are implemented in the article [15], see [9] (p. 4). The aim is to construct events and random variables, which can easily be used with a probability measure. See as an example theorems (10) and (14) in [20]. Then the formalization is more similar to the presentation used in the book [9]. As a background, further literatures is [3] (pp. 9-12), [13] (pp. 17-20), and [8] (pp.32-35).

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40

Abdo Rabbo, A. "Household survey of treatment of malaria in Hajjah, Yemen." Eastern Mediterranean Health Journal 9, no. 4 (September 21, 2003): 600–606. http://dx.doi.org/10.26719/2003.9.4.600.

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The practice of self-medication is widespread in the Republic of Yemen. The objectives of this study were to describe the treatment of malaria in households and to promote rational treatment. We surveyed 201 households with family members suffering from malaria or being treated with antimalarials. Numbers of prescribed and non-prescribed drugs were recorded and treatment rationality assessed. Common patterns of irrational treatment of malaria were observed. Polypharmacy was common, with an average of 3.8 total drugs and 1.3 antimalarials found per encounter. Misused and over use of injectables antimalarials was common. People practised self-medication because of belief, experience, lack of confidence in health services and cost of treatment. Most had no knowledge concerning possible risks of antimalarials

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41

HEATH–BROWN, D. R. "Pair correlation for fractional parts of αn2." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (January 15, 2010): 385–407. http://dx.doi.org/10.1017/s0305004109990466.

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It was proved by Weyl [8] in 1916 that the sequence of values of αn2 is uniformly distributed modulo 1, for any fixed real irrational α. Indeed this result covered sequences αnd for any fixed positive integer exponent d. However Weyl's work leaves open a number of questions concerning the finer distribution of these sequences. It has been conjectured by Rudnick, Sarnak and Zaharescu [6] that the fractional parts of αn2 will have a Poisson distribution provided firstly that α is “Diophantine”, and secondly that if a/q is any convergent to α then the square-free part of q is q1+o(1). Here one says that α is Diophantine if one has (1.1) for every rational number a/q and any fixed ϵ > 0. In particular every real irrational algebraic number is Diophantine. One would predict that there are Diophantine numbers α for which the sequence of convergents pn/qn contains infinitely many squares amongst the qn. If true, this would show that the second condition is independent of the first. Indeed one would expect to find such α with bounded partial quotients.

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42

Boso, Ana Cláudia Marassá Roza, Luís Roberto Almeida Gabriel Filho, Camila Pires Cremasco Gabriel, Bruno César Góes, and Fernando Ferrari Putti. "Applications of Continuous Fractions in Orthogonal Polynomials." International Journal for Innovation Education and Research 6, no. 12 (December 31, 2018): 284–95. http://dx.doi.org/10.31686/ijier.vol6.iss12.1245.

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Several applications of continuous fractions are restricted to theoretical studies, such as problems associated with the approximation of functions, determination of rational and irrational numbers, applications in physics in determining the resistance of electric circuits and integral equations and in several other areas of mathematics. This work aimed to study the results that open the way for the connection of continuous fractions with the orthogonal polynomials. As support, we will study the general case, where the applications of the Wallis formulas in a monolithic orthogonal polynomial, which generates a continuous fraction of the Jacobi type. It will be allowed applications with relations of recurrence of three terms in the polynomials of Tchebyshev and Legendre, through the results found, establishing connection between them with the continuous fractions. And finally, will be presented the "Number of gold", that is an application of this theory.

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43

Strauss, D. F. M. "Philosophical tendencies in the genesis of our understanding of physical nature." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 25, no. 2 (September 22, 2006): 93–110. http://dx.doi.org/10.4102/satnt.v25i2.150.

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The rise of a long-standing legacy of natural scientific thought is found in ancient Greece – the well-spring of Western civilization and the source of articulated rational reflection. The earliest phase of Greek culture already gave birth to theoretical thinking about the universe. The Pythagoreans are first of all famous for their emphasis on number as a mode of explanation. However, in their thesis that everything is number they solely acknowledged rational numbers (fractions) and this approach eventually stranded on the discovery of irrational numbers that led to the geometrization of Greek mathematics. This transition generated at once also a powerful space metaphysics overarching the entire medieval period. It was only during the early modern period that the predecessors and successors of Galileo contemplated an appreciation for motion as a new principle of explanation (compare the classical mechanistic world view of the universe as a mechanism of material particles in motion). But also this mechanistic reduction (through which all physical processes were reduced to the motion of charged or uncharged mass-points) eventually failed because it was unable to account for the irreversibility of physical processes. As a result it was only 20th century physics that managed to acknowledge the decisive qualifying role of energy-operation (thus of the physical aspect) in the existence of material things and processes. This article is concluded with an explanation of the significance of the preceding considerations for a theoretical approximation of the mysterious nature of matter.

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Naveen, Avula, and M. R. Sravani. "Study of drug utilization trends in respiratory tract infections in a tertiary care teaching hospital: a retrospective study." International Journal of Basic & Clinical Pharmacology 6, no. 11 (October 25, 2017): 2583. http://dx.doi.org/10.18203/2319-2003.ijbcp20174635.

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Background: Drug utilization studies play crucial role in the health sector and ultimately it provides insight into the efficiency of drug use and results of such research can be used to help to set priorities for the rational use of medicines and allocation of health care budgets. Respiratory tract infections (RTIs) contributing to significant mortality and morbidity of populations especially in developing countries like India. Polypharmacy and irrational prescription are significant negative fallouts in treatment of RTIs. Keeping in view of this, our study was undertaken to analyze the drug utilization pattern of RTIs.Methods: The study was conducted at Gandhi Hospital, after obtaining permission from the Institutional Ethics Committee. We have collected data of 600 case records of the patients diagnosed with respiratory tract infection and evaluated for prescribing patterns in consonance with WHO indicators.Results: Out of the total case records 348 (58%) were of male patients and 252 (42%) of female patients. Age wise distribution was done; 79 (13.16%) 0-15 years, 46 (7.67%) 16-30 years, 123 (20.50%) 31-45 years, 194 (32.33%) 46-60 years and 158 (26.33%) patients belongs to >60 years of age group respectively. A total of 4682 drugs were prescribed, 2468 (52.71%) antibiotics, 768 (16.4%) bronchodilators, 581 (12.4%) corticosteroids, 323 (6.89%) antacids, 542 (11.57%) in miscellaneous category respectively. With regard to formulations 2463 (52.60%) oral, 1463 (31.24%) injectable and 756 (16.14%) inhalational drugs were prescribed. Numbers of Fixed dose combinations were 712 (15.20%). 7.8 drugs were prescribed per prescription. 2493 (53.24%) drugs were prescribed from National Essential Medicine List. 4168 (89.02%) drugs were prescribed by their brand names.Conclusions: Prescription of drugs with branded names, Irrational prescribing, poly pharmacy were observed in our study. So there is an urgent need for creating awareness among the health care professionals regarding rational prescription by using data from from drug utilization studies.

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Beg, Mirza A., Shakti B. Dutta, Shalu Bawa, Amanjot Kaur, Subhash Vishal, and Upendra Kumar. "Prescribing trends in respiratory tract infections in a tertiary care teaching hospital." International Journal of Research in Medical Sciences 5, no. 6 (May 27, 2017): 2588. http://dx.doi.org/10.18203/2320-6012.ijrms20172452.

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Background: Respiratory tract infections are a major cause of morbidity and mortality in developing countries. Polypharmacy and irrational prescribing in respiratory diseases are common practice worldwide. Keeping in consideration this scenario, present study was undertaken to analyze the drug utilization pattern of respiratory tract infections.Methods: This drug utilization study was conducted by Pharmacology department at SGRRIM & HS to analyze drug utilization pattern of respiratory infections. A total of 585 prescriptions were collected from hospital and randomly evaluated for prescribing pattern using WHO drug indicators.Results: A total of 585 prescriptions were analyzed. Male:Female ratio was 1:0.77. Age wise distribution was done; 81(13.84%) 0-15 years, 54(9.23%) 16-30 years, 198(33.84%) 31-45 years, 75(12.82%) 46-60 years and 177(30.25%) patients belongs to >60 years of age group respectively. A total of 4869 drugs were prescribed, 2754(56.56%) antibiotics, 675(13.8%) bronchodilators, 630(12.93%) corticosteroids, 303(6.22%) antacids, 507(10.41%) in miscellaneous category respectively. 2562(52.61%) oral, 1491(30.62%) injectable and 816(16.75%) inhalational drugs were prescribed. Numbers of Fixed dose combinations were 645(13.24%). 8.32 drugs were prescribed per prescription. 2409(49.47%) drugs were prescribed from national essential medicine list 2015. 4320(88.72%) drugs were prescribed by their brand names.Conclusions: Irrational prescribing and polypharmacy was observed. The drug utilization studies are important tool to sensitize and increases awareness among physicians, which ultimately improves rational prescribing and patient care.

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46

Yang, Quan Chao, Zhi Liang Qian, and Yan Jun Gu. "Selection of the Measuring Teeth Number of Involute Cylindrical Gear Common Normal." Advanced Materials Research 479-481 (February 2012): 908–12. http://dx.doi.org/10.4028/www.scientific.net/amr.479-481.908.

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Focus on the inconvenient inquiring of non-standard gears common normal length and irrational selection of the measuring teeth number, the computational general formula of common normal length and measuring teeth number is educed. According to best value method, the scope of measuring teeth number is ascertained. A rational measuring teeth number is selected on the basis of the scope. According to Matlab-GUI, design a software which can calculate the common normal length and measuring teeth number of involute cylindrical gear. The result shows that the solving of measuring teeth number by best value method is more reasonable and the calculation of common normal length is exacter and swifter by the software.

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47

Kurent, Tine. "The Om mani padme hum, the Platonic Soul, the Tao, and the Greek Cross are an Architectural Tool." Acta Neophilologica 22 (December 15, 1989): 3–12. http://dx.doi.org/10.4312/an.22.0.3-12.

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The plan of Borobuclur conforms with two concentric octagrams. The lines of the scheme, their lengths, and their intersections, determine the articulation of the Borobudur composition, i. e. the sizes of every part and of the whole as well. The sizes of Borobudur are modular. Their modular multiples are Pell numbers, the ratios of which rationally approximate the irrational proportions in octagram If Borobudur numbers are located in the Peli number-pattern and connected with a line, the syllable OM, written in Sanskrit, appears. The word octagram is only the modern European name of the symbol of OM. The prayer OM MANI PADME HUM, translated as 'the JEWEL and the LOTOS', is a good description of octagram.

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48

Kurent, Tine. "The Om mani padme hum, the Platonic Soul, the Tao, and the Greek Cross are an Architectural Tool." Acta Neophilologica 22 (December 15, 1989): 3–12. http://dx.doi.org/10.4312/an.22.1.3-12.

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The plan of Borobuclur conforms with two concentric octagrams. The lines of the scheme, their lengths, and their intersections, determine the articulation of the Borobudur composition, i. e. the sizes of every part and of the whole as well. The sizes of Borobudur are modular. Their modular multiples are Pell numbers, the ratios of which rationally approximate the irrational proportions in octagram If Borobudur numbers are located in the Peli number-pattern and connected with a line, the syllable OM, written in Sanskrit, appears. The word octagram is only the modern European name of the symbol of OM. The prayer OM MANI PADME HUM, translated as 'the JEWEL and the LOTOS', is a good description of octagram.

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49

Slusser, George. "Descartes Meets Edgar Rice Burroughs: Beating the Rationalist Equations in Zamiatin's We." Canadian-American Slavic Studies 45, no. 3-4 (2011): 307–28. http://dx.doi.org/10.1163/221023911x570142.

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AbstractZamiatin's We presents a totalitarian world founded on an extreme application of the quantifying logic of Western rationalism. Everywhere in the rhetoric of D-503, spokesman for the One State, we hear echoes of Descartes, Leibniz, Spinoza, Pascal. Even so, from his very first utterances we see mind unable to control passions, to separate itself from the res extensa relegated beyond the Green Wall. In this “war of rhetorics” for the narrator's soul, we recognize the reason's antagonist to be pulp science fiction – the lurid purple prose and “pink Venusians” displayed on the covers of pulp SF magazines current in France and England at the time of World War I, unruly visions inspired by Verne and the American dime novel. From the evidence in his text, Zamiatin knew these magazines from his stay in England. Irrational numbers are not enough to break down the dreaded equations of rational control. For D-503, the square root of -1 is the unruly forces of pulp prose and images, the way in which mind and body free themselves to explore the “irrational” world of material phenomena, worlds of imagination without end. Pulp becomes Zamiatin's “Dionysian” element. In using it as such, however, Zamiatin makes an important connection. For science fiction itself is a literature born of Descartes's call for mind to master nature, of Pascal's claim that human reason in unique, and alone, in his infinite spaces. But since its own rationalist origins, science fiction itself has struggled against rationalist closure, seeking ways to free mind and body – science and the human condition – for open-ended exploration of worlds beyond logic.

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Vrahatis, M. N., H. Isliker, and T. C. Bountis. "Structure and Breakdown of Invariant Tori in a 4-D Mapping Model of Accelerator Dynamics." International Journal of Bifurcation and Chaos 07, no. 12 (December 1997): 2707–22. http://dx.doi.org/10.1142/s0218127497001825.

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We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic–elliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the non-smoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.

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