Relevant bibliographies by topics / Rational and irrational numbers

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  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Rational and irrational numbers'

Author: Grafiati
Published: 4 June 2021
Last updated: 9 February 2022

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Journal articles on the topic "Rational and irrational numbers"

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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Rational and irrational numbers"

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Torres, Mário Régis Rebouças. "Números algébricos e transcendentes." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25736.

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TORRES, Máro Règis Rebouças. Números algébricos e transcendentes. 66 f. Dissertação (Mestrado Profissional em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
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The present work deals with algebraic and transcendent numbers characterizing them under different aspects. In particular we bring some demonstrations of the irrationality of the number π and the number of Euler, base of the natural logarithm. We will also present a demonstration of the transcendence of the number and based on the script of exercises proposed by D.G. de Figueiredo, in addition to a small historical survey on π, and, algebraic and transcendent numbers.
O presente trabalho trata sobre números algébricos e transcendentes caracterizando-os sob diferentes aspectos. Em particular trazemos algumas demonstrações da irracionalidade do número π e do número de Euler, base do logaritmo natural. Também apresentaremos uma demonstração da transcendência do número e baseada no roteiro de exercícios propostos por D.G. de Figueiredo em [4], além de um pequeno apanhado histórico sobre π, e, números algébricos e transcendentes.
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Bezerra, Rafael Tavares Silva. "Frações contínuas - um estudo sobre "boas" aproximações." Universidade Federal da Paraíba, 2016. http://tede.biblioteca.ufpb.br:8080/handle/tede/9341.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
The study of ontinued fra tions will start with some histori al fa ts, aiming at a better understanding of the subje t. We will bring the de nition of ontinued fra tions for a number α real, with the de nition for α rational and α irrational. The dis ussion will fo us on meaning results for the al ulation of redu ed and good approximations of irrational numbers, also aimed at determining the error between the redu ed and the irrational number. We will bring a study of the periodi ontinued fra tions, with emphasis on Lagrange theorem, whi h relates a periodi ontinued fra tion and a quadrati equation. Finishing with a fo us on problem solving, as the al ulation of ontinued fra tions of irrational numbers of the form √a2 + b, as well as proof of the irrationality of e by al ulating its ontinued.
O estudo das frações ontínuas terá ini io om alguns fatos históri os, visando uma melhor ompreensão do tema. Traremos a de nição de frações ontínuas para um erto número α real, apresentando a de nição para α ra ional e para α irra ional. A dis ussão será entrada em resultados importantes para o ál ulo de reduzidas e boas aproximações de números irra ionais, visando também a determinação do erro entre a reduzida e o número irra ional. Traremos um estudo sobre as frações ontínuas periódi as, om enfase ao teorema de Langrange, que rela iona uma fração ontínua periódi a e uma equação do segundo grau. Finalizando om enfoque na resolução de problemas, omo o ál ulo de frações ontínuas de números irra ionais da forma √a2 + b, assim omo a prova da irra ionalidade de e através do ál ulo de sua fração ontínua.
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Menezes, Fernanda Martinez. "Propriedades da expansão decimal." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-05102016-085553/.

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Este trabalho tem como objetivo principal o estudo da expansão decimal dos números reais. Primeiramente provamos que todo número real possui ao menos uma expansão decimal. Na sequência, um método para encontrar a expansão decimal de um número entre 0 e 1 é apresentado, bem como um estudo sobre a expansão decimal de números racionais e irracionais. Em seguida, o estudo apresenta métodos que permitem encontrar aproximações racionais de números irracionais, além dos erros cometidos por essas aproximações. Na parte final, por seu turno, o foco do trabalho recai sobre a análise da regularidade (frequência) dos dígitos das expansões decimais.
This work has as main objective the study of the decimal expansion of the real numbers. First we prove that every real number has at least one decimal expansion. Further, a method to find the decimal expansion of real numbers between 0 and 1 is provided as well as a the study of the decimal expansion of rational and irrational numbers. Next, the study presents methods that provide rational approximations to irrational numbers, in addition to the errors committed by these approximations. At the end, by its turn, the focus of the work is put on the analysis of the regularity (frequency) of the digits of the decimal expansion.
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Silva, Guimarães Vieira da. "Irracionalidade e transcendência: aspectos elementares." Universidade Federal do Tocantins, 2018. http://hdl.handle.net/11612/978.

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O presente trabalho tem como perspectiva a caracterização dos números Racionais e Irracionais, e a sua devida aplicabilidade e variações no que tange o aspecto algébrico e transcendental. Sabe-se que o Número e (de Euler), pode ser classificado como um número transcendental, isto é, aqueles que não são raízes de nenhum polinômio que possua coeficientes inteiros. Nesse pressuposto, o Número deve ser considerado existente e irracional. O objetivo desta pesquisa consiste em caracterizar os fatores que abrangem os Números Racionais e Irracionais, oferecendo a compreensão necessária referente ao Número e e a sua ação nos Números Algébricos e Transcendentes. Como recurso metodológico, utilizou-se uma revisão de literatura, com um crivo pautado nos fatores qualitativos e quantitativos, a fim de se refletir sobre a temática proposta. Assim, nesta presente pesquisa, buscouse apresentar informações dentro das melhores formas e possibilidades de favorecer a compreensão, considerando a dificuldade em torno deste respectivo tema, devido a sua característica abstrata, o que dificulta o entendimento por parte de muitos. Portanto, destacam-se as iniciativas e argumentos em torno deste princípio temático, como forma de, possivelmente, fomentar o interesse de muitos pelo mesmo, além de que, tal trabalho possa ser relevante às necessidades de investigação de outros desejosos por este universo de pesquisa.
The present work has as its perspective the characterization of Rational and Irrational numbers, and their due applicability and variations regarding the algebraic and transcendental aspects. It is known that the number e (of Euler) can be classified as a transcendental number, that is, those that are not roots of any polynomial that has integer coefficients. In this assumption, the Number should be considered existent and irrational. The objective of this research is to characterize the factors that comprise the Rational and Irrational Numbers, offering the necessary understanding regarding Number e and its action in Algebraic and Transcendent Numbers. As a methodological resource, a literature review was used, based on qualitative and quantitative factors, in order to reflect on the proposed theme. Thus, in this present research, we sought to present information within the best ways and possibilities to favor understanding, considering the difficulty around this respective theme, due to its abstract feature, which makes it difficult for many to understand. Therefore, we highlight the initiatives and arguments around this thematic principle as a way of possibly fostering the interest of many by the same, and that such work may be relevant to the research needs of others desirous by this universe of research.
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SANTOS, Ana Cláudia Guedes dos. "Uma contribuição ao ensino de números irracionais e de incomensurabilidade para o ensino médio." Universidade Federal de Campina Grande, 2013. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/2161.

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Capes
Este trabalho tem como proposta pedagógica apresentar aos alunos o conceito de segmentos comensuráveis e de segmentos incomensuráveis, mostrando a importância desses conceitos para o estudo dos números racionais e irracionais. Veremos um processo de verificação da comensurabilidade de dois segmentos, doravante P.V.C.D.S, que é um processo geométrico de verificação de comensurabilidade de dois segmentos. A partir do P.V.C.D.S, apresentamos a demonstração clássica de que p2 é irracional, com uma abordagem geométrica, mostrando que o segmento do lado de um quadrado de medida 1 e o segmento de sua diagonal são incomensuráveis. Ainda apresentamos um estudo sobre expressões decimais, no qual será apresentado um teorema que nos permite verificar se uma fração irredutível possui representação decimal finita ou infinita e periódica. Também apresentamos outro teorema que nos permite transformar expressões decimais finitas e infinitas e periódicas na sua forma de fração. Por fim, apresentaremos algumas sugestões de atividades, que englobam todo conteúdo do presente TCC. Essas atividades foram aplicadas a uma turma de 1 ano do Ensino Médio de uma escola pública, e as respostas dos alunos estão anexadas ao trabalho.
This work have pedagogical proposed to introduce the concept of commensurable segments and incommensurable segments, showing the importance of these concepts for the study of rational and irrational numbers. We will stabelish a verification process to detect the mensurability of two segments, which is a geometric process. We present the classic demonstration that root of 2 is irrational with a geometric approach, showing that the segment of the side of a square measuring its diagonal are immeasurable. We still will present a study on decimal expressions, and prove a theorem that allows to check that an irreducible fraction has decimal representation finite or infinite and periodic. We also present another theorem that allows us to turn decimal expressions finite or infinite and periodic on its fraction form. Finally we present some suggestions for activities that include all content of the TCC. These activities have been applied to a class of 1st year of high school at a public school, and the students’ answers are attached to the work.
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Santos, Simone de Carvalho. "Uma construção geométrica dos números reais." Universidade Federal de Sergipe, 2015. https://ri.ufs.br/handle/riufs/6478.

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This study aims to present a geometric construction of real numbers characterizing them as numbers that express a measure. In this construction, each point in an oriented line represents the measure of a segment (a real number). Based on ve axioms of Euclidean geometry it was de ned an order relation, a method to add and multiply points so that it was possible to demonstrate that the line has a full ordered body of algebraic structure that we call the set of real numbers. To do so, it were presented historical elements that allow us to understand the emergence of irrational numbers as a solution to the insu ciency of rational numbers with respect to the measuring problem, the evolution of the concept of number, as well as the importance that the strict construction of real numbers had to the Foundations of Mathematics. We display a construction of rational numbers from the integernumbers as motivation for construction of numerical sets. Using the notion of measure,we show a geometric interpretation of rational numbers linking them to the points of an oriented line to demonstrate that they leave holes in the line and conclude on the need to build a set that contains the rational numbers and that ll all the points of a line. The theme is of utmost importance to the teaching of mathematics because one of the major goal of basic education is to promote understanding of numbers and operations, to develop number sense and to develop uency in the calculation. To achieve this, it is necessary to assimilate the r
O presente trabalho tem por objetivo apresentar uma construção geométrica dos números reais caracterizando-os como números que expressam uma medida. Nesta construção cada ponto de uma reta orientada representa a medida de um segmento (um número real), com base nos cinco axiomas da geometria euclidiana de niu-se uma relação de ordem, um método para somar e multiplicar pontos de tal forma que fosse possível demonstrar que a reta possui uma estrutura algébrica de corpo ordenado completo a qual chamamos de conjunto dos números reais. Para tanto, foram apresentados elementos históricos que permitem compreender o surgimento dos números irracionais como solução para a insu - ciência dos números racionais no que diz respeito ao problema de medida, a evolução do próprio conceito de número, bem como a importância que a construção rigorosa dos nú- meros reais tiveram para os Fundamentos da Matemática. Exibimos uma construção dos números racionais a partir dos números inteiros como motivação para construções de conjuntos numéricos. Usando a noção de medida mostramos uma interpretação geométrica dos números racionais associando-os aos pontos de uma reta orientada para demonstrar que eles deixam buracos na reta e concluir sobre a necessidade de construir um conjunto que contenha os números racionais e que preencham todos os pontos de uma reta. O tema é de extrema importância para o ensino da matemática, visto que um dos principais objetivos do ensino básico é promover a compreensão dos números e das operações, desenvolver o sentido de número e desenvolver a uência no cálculo, sendo necessário para tal assimilar os números reais, em especial os irracionais, os quais são tratados a partir do ensino fundamental.
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Penteado, Cristina Berndt. "Concepções do professor do ensino médio relativas à densidade do conjunto dos números reais e suas reações frente a procedimentos para a abordagem desta propriedade." Pontifícia Universidade Católica de São Paulo, 2004. https://tede2.pucsp.br/handle/handle/11180.

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The work approaches the subject of the density of the real numbers, here taking in the direction of the existence of infinite rational numbers and infinite irrationals between two distinct real numbers. Some research evidences difficulties of the students in the classification of rational numbers and irrationals, as well as the unfamiliarity of the property of the density of the set of the real numbers. The objective of the study is to investigate the conception and the reaction of the teachers of high-school front to the different registers of representations of the numbers, when analyzed the property of the density, as much the density of the set of the rational numbers in the set of the real numbers how much of the irrationals in reals. Is considered to investigate it the viability of two types of distinct procedures for the attainment of real numbers between two supplied: the procedure of the arithmetic mean and other inspired in the process of diagonal line of Cantor, using the representation decimal of the real numbers. For in such a way it was carried through an intervention by means of the elaboration, application and analysis of an education sequence, composed of ten activities, based in the Theory of the Registers of Representation Semiotics of Raymond Duval. The education sequence was based on principles of the Didactic Engineering of Michèle Artigue. Although to evidence envolvement of the participants, some difficulties identified in the research persist as for example, the association of the infinite representation with irrationality and the identification of a rational number as being only that one that has finite representation. Some teachers had demonstrated the intention to apply similar questions to the ones of the sequence, to its students of high-school
O trabalho aborda o tema da densidade dos números reais, aqui tomada no sentido da existência de infinitos números racionais e infinitos irracionais entre dois números reais distintos. Várias pesquisas evidenciam dificuldades dos alunos na classificação de números racionais e irracionais, bem como o desconhecimento da propriedade da densidade do conjunto dos números reais. O objetivo do estudo é investigar a concepção e a reação dos professores do Ensino Médio frente aos diferentes registros de representações dos números, quando analisada a propriedade da densidade, tanto a densidade do conjunto dos números racionais no conjunto dos números reais quanto a dos irracionais nos reais. Propõe-se a investigar a viabilidade de dois tipos de procedimentos distintos para a obtenção de números reais entre dois dados: o procedimento da média aritmética e outro inspirado no processo de diagonal de Cantor, utilizando a representação decimal dos números reais. Para tanto foi realizada uma intervenção por meio da elaboração, aplicação e análise de uma seqüência de ensino, composta de dez atividades, embasada na Teoria dos Registros de Representação Semiótica de Raymond Duval. A seqüência de ensino foi fundamentada em princípios da Engenharia Didática de Michèle Artigue. Apesar de constatar envolvimento dos participantes, algumas dificuldades identificadas nas pesquisas persistem como por exemplo, a associação da representação infinita com irracionalidade e a identificação de um número racional como sendo somente aquele que tem representação finita. Alguns professores demonstraram a intenção de aplicar questões similares às da seqüência, aos seus alunos do Ensino Médio
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Ciano, Susan A. "Architecture, dwelling, and process: between rational and irrational." Thesis, Virginia Polytechnic Institute and State University, 1986. http://hdl.handle.net/10919/53096.

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This inquiry attempts to examine different aspects of architecture and an understanding of space by exploring architectural expectations. The process described herein depicts the struggle to find a translation between two languages, one verbal and one visual. The key to my search was a constant set of ideas. The challenge was first to discern, and then to learn to use, the tools that would become my guide to the language of architecture.
Master of Architecture
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Chu, Yim Tonia 1974. "Bidding, playing, or competing? : rational and irrational determinants in Internet auctioning." Thesis, Massachusetts Institute of Technology, 2000. http://hdl.handle.net/1721.1/8986.

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Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, Center for Transportation Studies, 2000.
Includes bibliographical references (leaves 49-50).
Auctions, especially Internet auctions, are claimed to be efficient pricing mechanisms, assuming rational behavior and recognition of individual fundamental values. This assumption and its implication are at the heart of the thesis work. The research includes both survey studies which utilize one of the most popular sites for Internet auctions, and experiments developed in the laboratory. Both field studies and laboratory experiments paid specific attention to the psychology of bidders with main focuses on starting prices, price comparison, competition, and auction formats. Two surveys and two field studies were conducted to collect statistics in real auctions and subjective opinions from real auction participants. One field experiment was performed on a real web site to test price sensitivity. A simulation bidding system was built in the laboratory to examine auction formats and the effect of competition. Four primary results are shown. First, game-playing attitude towards auctions generally exists among bidders. Second, bidders hold strong winning aspects and suffer either "winner's curse" or regret losing. Third, bidders are price sensitive when price comparison is available and their price preferences are affected by the original starting prices. Fourth, auction formats convey different information to bidders and influence the way bidders behave. (Keywords: Auctions, Internet auctioning, Individual fundamental values, Bidders' behavior)
by Yim Tonia Chu.
S.M.
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Coward, Daniel R. "Sums of two rational cubes." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320587.

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Books on the topic "Rational and irrational numbers"

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Hollander, Paul. Anti-Americanism: Irrational & rational. New Brunswick, NJ, USA: Transaction Publishers, 1995.

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Rational numbers: Poems. [Kirksville, Mo.]: Truman State University Press, 2000.

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Stein, Jeremy C. Rational capital budgeting in an irrational world. Cambridge, MA: National Bureau of Economic Research, 1996.

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Więsław, Witold. Liczby niewymierne. Warszawa: CODN, 1992.

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Flannery, David. The square root of 2: A dialogue concerning a number and a sequence. New York, NY: COPERNICUS, 2005.

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Antithetik neuzeitlicher Vernunft: "Autonomie-Heteronomie" und "rational-irrational". Göttingen: Vandenhoeck & Ruprecht, 1987.

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Bernard, Michael Edwin. Staying rational in an irrational world: Albert Ellis and rational emotive therapy. South Melbourne, Vic: McCulloch Publishing in association with Macmillan Australia, 1986.

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May, Leo. Empirical relations between some irrational transcendental mathematical universal constants. Regensburg: Roderer, 1997.

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S, Bezuk Nadine, ed. Understanding rational numbers and proportions. Reston, Va: National Council of Teachers of Mathematics, 1994.

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Pi. Greensboro, North Carolina: Morgan Reynolds Publishing, 2014.

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Book chapters on the topic "Rational and irrational numbers"

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Rosenthal, Daniel, David Rosenthal, and Peter Rosenthal. "Rational Numbers and Irrational Numbers." In Undergraduate Texts in Mathematics, 63–72. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-00632-7_8.

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Rosenthal, Daniel, David Rosenthal, and Peter Rosenthal. "Rational Numbers and Irrational Numbers." In Undergraduate Texts in Mathematics, 61–70. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-05654-8_8.

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Beck, Matthias, and Ross Geoghegan. "Rational and Irrational Numbers." In The Art of Proof, 107–12. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-7023-7_11.

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Nagar, Sandeep. "Rational and Irrational Numbers." In Beginning Julia Programming, 79–88. Berkeley, CA: Apress, 2017. http://dx.doi.org/10.1007/978-1-4842-3171-5_5.

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Erdős, Paul, and János Surányi. "Rational and Irrational Numbers. Approximation of Numbers by Rational Numbers (Diophantine Approximation)." In Topics in the Theory of Numbers, 85–107. New York, NY: Springer New York, 2003. http://dx.doi.org/10.1007/978-1-4613-0015-1_3.

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Motzer, Renate. "Irrational Numbers." In essentials, 35–39. Wiesbaden: Springer Fachmedien Wiesbaden, 2021. http://dx.doi.org/10.1007/978-3-658-32574-9_8.

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Aigner, Martin, and Günter M. Ziegler. "Some irrational numbers." In Proofs from THE BOOK, 27–34. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-22343-7_6.

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Aigner, Martin, and Günter M. Ziegler. "Some irrational numbers." In Proofs from THE BOOK, 35–41. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-00856-6_7.

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Aigner, Martin, and Günter M. Ziegler. "Some irrational numbers." In Proofs from THE BOOK, 45–51. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-662-44205-0_8.

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Aigner, Martin, and Günter M. Ziegler. "Some irrational numbers." In Proofs from THE BOOK, 47–53. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. http://dx.doi.org/10.1007/978-3-662-57265-8_8.

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Conference papers on the topic "Rational and irrational numbers"

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Chen, Franklin F. K., and B. Ronald Moncrief. "Canyon Building Ventilation System Dynamic Model Optimization Study." In ASME 1993 International Computers in Engineering Conference and Exposition. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/cie1993-0052.

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Abstract A canyon building houses special nuclear material processing facilities in two canyon like structures, each with approximately a million cubic feet of air space and a hundred thousand hydraulic equivalent feet of ductwork of various cross sections. The canyon ventilation system is a “once through” design with separate supply and exhaust fans, utilizes two large sand filters to remove radionuclide particulate matter, and exhausts through a tall stack. The ventilation equipment is similar to most industrial ventilation systems. However, in a canyon building, nuclear contamination prohibits access to a large portion of the system and therefore limits the kind of plant data possible. The facility investigated is 40 years old and is operating with original or replacement equipment of comparable antiquity. These factors, access and aged equipment, present a challenge in gauging the performance of canyon ventilation, particularly under uncommon operating conditions. The ability to assess canyon ventilation system performance became critical with time, as the system took on additional exhaust loads and aging equipment approached design maximum. Many “What if?” questions, needed to address modernization/safety issues, are difficult to answer without a dynamic model. This paper describes the development, the validation and the utilization of a dynamic model to analyze the capacity of this ventilation system, under many unusual but likely conditions. The development of a ventilation model with volume and hydraulics of this scale is unique. The resultant model resolutions of better than 0.05″wg under normal plant conditions and approximately 0.2″wg under all plant conditions achievable with a desktop computer is a benchmark of the power of micro-computers. The detail planning and the persistent execution of large scale plant experiments under very restrictive conditions not only produced data to validate the model but lent credence to subsequent applications of the model to mission oriented analysis. Modelling methodology adopted a two parameter space approach, rational parameters and irrational parameters. Rational parameters, such as fan age-factors, idle parameters, infiltration areas and tunnel hydraulic parameters are deduced from plant data based on certain hydraulic models. Due to limited accessibility and therefore partial data availability, the identification of irrational model parameters, such as register positions and unidentifiable infiltrations, required unique treatment of the parameter space. These unique parameters were identified by a numerical search strategy to minimize a set of performance indices. With the large number of parameters, this further attests to our strategy in utilizing the computing power of modern micros. Nine irrational parameters at five levels and 12 sets of plant data, counting up to 540 runs, were completely searched over the time span of a long weekend. Some key results, in assessing emergency operation, in evaluating modernization options, are presented to illustrate the functions of the dynamic model.
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Chen, Zhi-Zhong, and Ming-Yang Kao. "Reducing randomness via irrational numbers." In the twenty-ninth annual ACM symposium. New York, New York, USA: ACM Press, 1997. http://dx.doi.org/10.1145/258533.258583.

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Gielis, Johan, Bert Beirinckx, and Edwin Bastiaens. "Superquadrics with rational and irrational symmetry." In the eighth ACM symposium. New York, New York, USA: ACM Press, 2003. http://dx.doi.org/10.1145/781606.781647.

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Jing Wang, Guo-ping Jiang, and Hua Yang. "Improved DES algorithm based on irrational numbers." In 2008 International Conference on Neural Networks and Signal Processing (ICNNSP). IEEE, 2008. http://dx.doi.org/10.1109/icnnsp.2008.4590426.

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Jing Wang, Guo-ping Jiang, and Hua Yang. "Improved DES algorithm based on irrational numbers." In 2008 International Conference on Neural Networks and Signal Processing (ICNNSP). IEEE, 2008. http://dx.doi.org/10.1109/icnnsp.2008.4590427.

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Kushchenko, S. V. "Correlation of Rational and Irrational in Social Communications." In Proceedings of the Internation Conference on "Humanities and Social Sciences: Novations, Problems, Prospects" (HSSNPP 2019). Paris, France: Atlantis Press, 2019. http://dx.doi.org/10.2991/hssnpp-19.2019.17.

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Milinkovic, Luka, Marija Antic, and Zoran Cica. "Pseudo-random number generator based on irrational numbers." In TELSIKS 2011 - 2011 10th International Conference on Telecommunication in Modern Satellite, Cable and Broadcasting Services. IEEE, 2011. http://dx.doi.org/10.1109/telsks.2011.6143212.

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QUEFFÉLEC, MARTINE. "IRRATIONAL NUMBERS WITH AUTOMATON-GENERATED CONTINUED FRACTION EXPANSION." In Proceedings of the Conference in Honor of Gerard Rauzy on His 60th Birthday. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812793829_0019.

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Nagyová Lehocká, Zuzana. "LEARNING THE CHARACTERISTICS OF IRRATIONAL NUMBERS THROUGH EXPERIENCE." In 13th International Conference on Education and New Learning Technologies. IATED, 2021. http://dx.doi.org/10.21125/edulearn.2021.0321.

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Pratiwi, Yunia, and Rizky Rosjanuardi. "Error Analysis in Solving the Rational and Irrational Inequalities." In Proceedings of the 7th Mathematics, Science, and Computer Science Education International Seminar, MSCEIS 2019, 12 October 2019, Bandung, West Java, Indonesia. EAI, 2020. http://dx.doi.org/10.4108/eai.12-10-2019.2296414.

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Reports on the topic "Rational and irrational numbers"

1

Stein, Jeremy. Rational Capital Budgeting in an Irrational World. Cambridge, MA: National Bureau of Economic Research, March 1996. http://dx.doi.org/10.3386/w5496.

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Lu, Chao. A Computational Library Using P-adic Arithmetic for Exact Computation With Rational Numbers in Quantum Computing. Fort Belvoir, VA: Defense Technical Information Center, November 2005. http://dx.doi.org/10.21236/ada456488.

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