Relevant bibliographies by topics / Operations on rational numbers

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Academic literature on the topic 'Operations on rational numbers'

Author: Grafiati
Published: 2 June 2025
Last updated: 5 July 2025

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Journal articles on the topic "Operations on rational numbers"

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Christou, Konstantinos P., Courtney Pollack, Jo Van Hoof, and Wim Van Dooren. "Natural number bias in arithmetic operations with missing numbers – A reaction time study." Journal of Numerical Cognition 6, no. 1 (2020): 22–49. http://dx.doi.org/10.5964/jnc.v6i1.228.

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When reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning.
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Moss, Joan. "Research, Reflection, Practice: Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations." Teaching Children Mathematics 9, no. 6 (2003): 335–39. http://dx.doi.org/10.5951/tcm.9.6.0335.

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How do we foster computational fluency with rational numbers when this topic is known to pose so many conceptual challenges for young students? How can we help students understand the operations of rational numbers when their grasp of the quantities involved in the rational-number system is often very limited? Traditional instruction in rational numbers focuses on rules and memorization. Teachers often give students instructions such as, “To add fractions, first find a common denominator, then add only the numerators” or “To add and subtract decimal numbers, line up the decimals, then do your calculations.”
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Schwarzweller, Christoph, and Artur Korniłowicz. "Characteristic of Rings. Prime Fields." Formalized Mathematics 23, no. 4 (2015): 333–49. http://dx.doi.org/10.1515/forma-2015-0027.

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Summary The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.
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Putra, Zetra Hainul. "Didactic Transposition of Rational Numbers: a Case From a Textbook Analysis and Prospective Elementary Teachers’ Mathematical and Didactic Knowledge." Revija za elementarno izobraževanje 13, no. 4 (2020): 365–94. http://dx.doi.org/10.18690/rei.13.4.365-394.2020.

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This study aims to present a detailed analysis of didactic transposition of rational numbers from knowledge to be taught into taught knowledge occurring in a teacher education institution. The knowledge to be taught of rational numbers is analysed from the mathematics textbook used by prospective elementary teachers in a mathematics education course. The analysis focuses on mathematical praxeology, especially the type of task and technique. Then, the taught knowledge is investigated from 32 prospective elementary teachers’ collaborative work on two hypothetical teacher tasks (HTT) related to operations with rational numbers.
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Nowlin, Douald. "Division with Fractions." Mathematics Teaching in the Middle School 2, no. 2 (1996): 116–19. http://dx.doi.org/10.5951/mtms.2.2.0116.

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Although operations with fractions have been in the elementary and middle school curriculum for many years, finding examples of practical problems that illustrate the usefulness of division with fractions and mixed numbers is not easy. Most real-world applications of rational numbers involve decimal numerals (Usiskin and Bell 1984), but examples of division with fractions and mixed numbers are often obviously contrived. In addition, many teachers and prospective teachers have difficulty constructing examples and concrete models for the operation of division with fractions (Borko 1992; Ball 1988).
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Gorbunov, Konstantin, and Vassily Lyubetsky. "Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions." Mathematics 8, no. 11 (2020): 2001. http://dx.doi.org/10.3390/math8112001.

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We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.
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Chung, Heewon, and Myungsun Kim. "Encoding of Rational Numbers and Their Homomorphic Computations for FHE-Based Applications." International Journal of Foundations of Computer Science 29, no. 06 (2018): 1023–44. http://dx.doi.org/10.1142/s0129054118500193.

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This work addresses a basic problem of security systems that operate on very sensitive information. Specifically, we are interested in the problem of privately handling numeric data represented by rational numbers (e.g., medical records). Fully homomorphic encryption (FHE) is one of the natural and powerful tools for ensuring privacy of sensitive data, while allowing complicated computations on the data. However, because the native plaintext domain of known FHE schemes is restricted to a set of quite small integers, it is not easy to obtain efficient algorithms for encrypted rational numbers in terms of space and computation costs. For example, the naïve decimal representation considerably restricts the choice of parameters in employing an FHE scheme, particularly the plaintext size. Our basic strategy is to alleviate this inefficiency by using a different representation of rational numbers instead of naïve expressions. In this work we express rational numbers as continued fractions. Because continued fractions enable us to represent rational numbers as a sequence of integers, we can use a plaintext space with a small size while preserving the same quality of precision. However, this encoding technique requires performing very complex arithmetic operations, such as division and modular reduction. Theoretically, FHE allows the evaluation of any function, including modular reduction at encrypted data, but it requires a Boolean circuit of very high degree to be constructed. Hence, the primary contribution of this work is developing an approach to solve this efficiency problem using homomorphic operations with small degrees.
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Yakes, Christopher. "Rational Number Operations on the Number Line." Mathematics Enthusiast 14, no. 1-3 (2017): 309–24. http://dx.doi.org/10.54870/1551-3440.1400.

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Jay, Ar I. Laranang Edd, En Jay B. Santiago Prince, Louise Oca Danielle, and Vincent T. Bongon Patrick. "Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner's Excellence in Mathematics)." International Journal of Social Science and Education Research Studies 05, no. 05 (2025): 435–44. https://doi.org/10.5281/zenodo.15469834.

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Abstract : This action research explores the impact of an innovative game-based intervention entitled “Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics),” on the academic performance of grade 7 students at Mount Carmel College of Casiguran, Inc., academic year 2024-2025. The research employs a quasi-experimental design with purposive sampling consisting of an experimental and control group, wherein the experimental group undergoes the intervention. A validated 30-item pre-test and post-test was the main instruments for this study and administered to 52 participants (26 experimental group and 26 control group) to measure the academic performance of both group. This game-based intervention was designed to foster critical thinking and strengthen mathematical skills over 20 days. Results revealed significant differences among learners exposed to Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics). The experimental group’s post-test mean score increased to outstanding, compared to the control group’s Did Not Meet Expectations. Statistical analysis confirmed a significant difference between the two groups’ performances post-intervention, with a very large effect size indicated. Additionally, a dependent t-test for the experimental group showed a notable improvement from their pre-test mean score. The study’s findings highlighted the effectiveness of Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics) as an innovative educational tool addressing mathematical challenges and enhancing foundational skills. It also offers valuable insights for educators, emphasizing the importance of engaging and interactive pedagogies in improving learner outcomes in mathematics.
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Carvalho, Alexia Bezerra de, Rita de Cássia da Costa Guimarães, William Vieira, Emanoel Fabiano Menezes Pereira, and Roberto Seidi Imafuku. "Sobre dificuldades de ingressantes no ensino médio na compreensão de números racionais." ForScience 9, no. 1 (2021): e00878. http://dx.doi.org/10.29069/forscience.2021v9n1.e878.

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Apresenta-se, neste artigo, uma análise da resolução de uma questão sobre números racionais, exposta por estudantes ingressantes no Ensino Médio. O objetivo foi identificar as principais dificuldades e erros apresentados pelos participantes sobre o conceito e operações de números racionais. Para isso, realizou-se uma análise de erros das resoluções apresentadas pelos participantes. A interação de aspectos algorítmicos, intuitivos e formais é o referencial teórico adotado nas análises. Verificou-se que os estudantes não possuem clareza sobre o conceito de número racional e apresentam, em decorrência, dificuldades no uso dos procedimentos algorítmicos necessários para a resolução da questão proposta. Palavras-chave: Números racionais. Análise de erros. Dificuldades. Interpretação. Difficulties of high school entrants in understanding rational numbers Abstract This paper presents an analysis of the resolution of a question about rational numbers, posed by students entering high school. The goal was to identify the main difficulties and errors presented by the participants about the concept and operations of rational numbers. To do so, an error analysis of the resolutions presented by the participants was performed. The interaction of algorithmic, intuitive and formal aspects is the theoretical framework adopted in the analysis. It was found that students do not have clarity about the concept of rational numbers and present, as a result, difficulties in the use of algorithmic procedures necessary to solve the proposed question. Keywords: Rational numbers. Error analysis. Difficulties. Interpretation.
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Dissertations / Theses on the topic "Operations on rational numbers"

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Dopico, Evelyn. "The Impact of Small Group Intervention Focusing on Operations with Rational Numbers on Students' Performance in the Florida Algebra I End-of-Course Examination." Thesis, Nova Southeastern University, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10845405.

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<p> In Florida, passing the Algebra I end-of-course examination (EOCE) is a graduation requirement. The test measures knowledge of basic algebra. In spring 2015, the Department of Education introduced a different version of the test. For the first two administrations of the new test, the failure rate for 9th-grade students in the state was almost 50%. In contrast, the failure rate for students in the school where this study was implemented exceeded 70%. The purpose of this study was to determine the outcome of small group intervention focusing on operations with rational numbers of high school students&rsquo; performance on the Algebra I EOCE. </p><p> After analyzing several potential methods of instruction, small group instruction with the incorporation of the use of manipulatives, visuals, and guided inquiry was selected. In addition, the focus of the study was chosen to be operations with rational numbers, an area many researchers have identified as critical for student understanding of algebraic concepts. Twenty students from the target population of 600 10th and 11th grade students volunteered to participate in the study. These participants received three to six small group instruction sessions before retaking the test. In Sept 2016, all the students in the target population were administered the Algebra I EOCE again. A t-test yielded no significant difference in the learning gains of those who participated in the study and the other students in the target population. The implications of the results were that the interventions had no significant impact on student achievement. A possible reason for the lack of success could have been that six intervention sessions were not enough to produce significant results. It is recommended that future research includes a substantially larger number of interventions.</p><p>
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Lopes, Ana Paula. "Desenvolvimento do sentido de número no ensino básico: um estudo no sétimo ano de escolaridade." Master's thesis, Universidade de Évora, 2010. http://hdl.handle.net/10174/20773.

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A sociedade contemporânea exige do cidadão raciocínio quantitativo e os novos paradigmas económico-sociais colocam a Matemática escolar perante um novo desafio: desenvolver a literacia matemática dos alunos. A literacia matemática contempla um vasto conjunto de conhecimentos e capacidades entre eles o sentido de número. O desenvolvimento do sentido de número dos alunos tem suscitado alguns trabalhos de pesquisa, em particular ao nível do primeiro e segundo ciclos do Ensino Básico. O objectivo desde estudo é recolher evidências sobre o sentido de número de alunos do terceiro ciclo, mais concretamente sobre o sentido de número racional dos alunos do sétimo ano de escolaridade. Para tal foi necessário considerar uma vasta e complexa rede de competências que caracterizam o sentido de número, entre elas o sentido de operação, mas também os subconstructos que definem os números racionais. Desta forma emergem deste objectivo duas questões: (i) que compreensão têm os alunos das formas equivalentes (inteiros, fracções, decimais, percentagens) dos números? (ii) como entendem os alunos o efeito das operações nos números, as propriedades e as relações entre as operações? Este estudo é uma investigação de natureza qualitativa com recurso ao design de caso. A recolha de dados empíricos foi realizada ao longo do ano lectivo 2007/2008 numa turma de sétimo ano de escolaridade onde a investigadora era a docente da disciplina de Matemática. A investigadora assumiu um papel de observadora participante e foram considerados três estudos de caso: dois alunos e a turma. A informação analisada resultou de vários métodos de recolha: a) inquéritos­ por questionário e entrevista com tarefas; b) observação da resolução das tarefas em ambiente de sala de aula; c) análise documental. A análise deste conjunto de contributos foi feita tendo em conta as questões de investigação e de forma individual para cada um dos casos. O estudo mostrou que os alunos atribuem pouco significado aos números, às operações e aos contextos. Revelam alguma compreensão dos números racionais escritos na forma de fracção e na forma decimal, mas têm dificuldade em compreender e manipular racionais escritos com recurso a outras formas de representação e a maioria dos alunos revela poucas competências de comparação e ordenação de números racionais. Nesta investigação ressalta também o fraco sentido de operação dos alunos. Mostram pouca compreensão do efeito das operações nos números e têm dificuldade em reconhecer, no contexto do problema, a operação que mais se adequa à situação, manifestando dificuldade quer na interpretação das situações quer na definição de estratégias apropriadas. ABSTRACT: Nowadays society requires of the citizen quantitative reasoning and the new economic-social paradigms place the school Mathematics before a new challenge: to develop the mathematical literacy of the pupils. The mathematical literacy contemplates a vast set of knowledge and capacities such as number sense. The number sense development has excited some works of research, in particular referring to grades 1-6. The aim of this study is to collect evidences about the number sense of pupils of grades 7-9, more concretely on the 7th grade pupils' sense of rational numbers. For such it was necessary to consider vast and complex net of abilities that characterize number sense, including operation sense, but also the rational numbers' sub constructs. Therefore, of this aim two questions emerge: (i) which understanding pupils have of the equivalents forms (entire, fractions, decimals, percentages) of numbers? (ii) how pupils understand the effect that operations have on numbers, the properties and relations between the operations? This study is a qualitative nature investigation that resorts to case studies. Empirical data collection was carried along the school year of 2007/2008 in a 7th grade class where the investigator was the Mathematics teacher. The investigator assumed the role of participant observer and had been considered three studies of case: two pupils and the class. The analyzed data was collected by means of: a) survey - questionnaire and interview with tasks; b) observation of the resolution of the tasks in classroom environment; c) documentary analysis. The data analysis was made according to the questions of the study and individually for each one of the cases. The study showed that pupils attribute slight meant to the numbers, to the operations and the contexts. They reveal some understanding of rational numbers represented as fractions or decimal numbers but disclose difficulty in understanding and manipulating rational numbers' other forms of representation. The majority of the pupils shows few abilities of comparison and ordinance of rational numbers. This investigation also revealed the weak sense of operation of the pupils. They show little understanding of the effect that operations have on numbers as well as difficulty in recognizing, according to the problem's context, the operation that more suits the situation. Pupils disclose difficulty in the interpretation of the situations and in the definition of appropriate strategies.
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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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Ketkar, Pallavi S. (Pallavi Subhash). "Primitive Substitutive Numbers are Closed under Rational Multiplication." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278637/.

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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.
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Brown, Bruce John Lindsay. "The initial grounding of rational numbers : an investigation." Thesis, Rhodes University, 2007. http://hdl.handle.net/10962/d1006351.

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This small scale exploratory research project investigated the grounding of rational number concepts in informal, everyday life situations. A qualitative approach was taken to allow for the identification and then in depth investigation, of issues of importance for such a grounding of rational number understanding. The methodology followed could be seen as a combination of grounded theory and developmental research. And the data was generated through in-depth and clinical interviews structured around a number of grounded tasks related to rational numbers. The research comprised three cycles of interviews that were transcribed and then analysed in detail, interspersed with periods of reading and reflection. The pilot cycle involved a single grade three teacher, the second cycle involved 2 grade three teachers and the third cycle involved 2 grade three children. The research identified a number of different perspectives that were all important for the development of a fundamental intuitive understanding that could be considered personally meaningful to the individual concerned and relevant to the development of rational number concepts. Firstly in order to motivate and engage the child on a personal level the grounding situation needed to be seen as personally significant by the child. Secondly, coordinating operations provided a means of developing a fundamental intuitive understanding, through coordination with affording structures of the situation that are relevant to rational numbers. Finally, goal directed actions that are deliberately structured to achieve explicit goals in a situation are important for the development of more explicit concepts and skills fundamental for rational number understanding. Different explicit structures give rise to different interpretations of rational numbers in grounding situations. In addition to these perspectives, it became evident that building and learning representations was important for developing a more particularly mathematical understanding, based on the fundamental understanding derived from the child's grounded experience. The conclusion drawn in this research as a result of this complexity, is that to achieve a comprehensive and meaningful grounding, children's learning of rational numbers will not follow a simple linear trajectory. Rather this process forms a web of learning, threading coordinating operations for intuitive development, interpretations for explicit grounding and representations to develop more formal mathematical conceptions.
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LORIO, MARCELO NASCIMENTO. "APPROXIMATIONS OF REAL NUMBERS BY RATIONAL NUMBERS: WHY THE CONTINUED FRACTIONS CONVERGING PROVIDE THE BEST APPROXIMATIONS?" PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2014. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=23981@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Frações Contínuas são representações de números reais que independem da base de numeração escolhida. Quando se trata de aproximar números reais por frações, a escolha da base dez oculta, frequentemente, aproximações mais eficientes do que as exibe. Integrar conceitos de aproximações de números reais por frações contínuas com aspectos geométricos traz ao assunto uma abordagem diferenciada e bastante esclarecedora. O algoritmo de Euclides, por exemplo, ao ganhar significado geométrico, se torna um poderoso argumento para a visualização dessas aproximações. Os teoremas de Dirichlet, de Hurwitz-Markov e de Lagrange comprovam, definitivamente, que as melhores aproximações de números reais veem das frações contínuas, estimando seus erros com elegância técnica matemática incontestável.<br>Continued fractions are representations of real numbers that are independent of the choice of the numerical basis. The choice of basis ten frequently hides more than shows efficient approximations of real numbers by rational ones. Integrating approximations of real numbers by continued fractions with geometrical interpretations clarify the subject. The study of geometrical aspects of Euclids algorithm, for example, is a powerful method for the visualization of continued fractions approximations. Theorems of Dirichlet, Hurwitz-Markov and Lagrange show that, definitely, the best approximations of real numbers come from continued fractions, and the errors are estimated with elegant mathematical technique.
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Trappmann, Henryk. "Arborescent numbers : higher arithmetic operations and division trees." Phd thesis, Universität Potsdam, 2007. http://opus.kobv.de/ubp/volltexte/2007/1524/.

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The overall program "arborescent numbers" is to similarly perform the constructions from the natural numbers (N) to the positive fractional numbers (Q+) to positive real numbers (R+) beginning with (specific) binary trees instead of natural numbers. N can be regarded as the associative binary trees. The binary trees B and the left-commutative binary trees P allow the hassle-free definition of arbitrary high arithmetic operations (hyper ... hyperpowers). To construct the division trees the algebraic structure "coppice" is introduced which is a group with an addition over which the multiplication is right-distributive. Q+ is the initial associative coppice. The present work accomplishes one step in the program "arborescent numbers". That is the construction of the arborescent equivalent(s) of the positive fractional numbers. These equivalents are the "division binary trees" and the "fractional trees". A representation with decidable word problem for each of them is given. The set of functions f:R1->R1 generated from identity by taking powers is isomorphic to P and can be embedded into a coppice by taking inverses.<br>Baumartige Zahlen und höhere arithmetische Operationen Von Schülern und Laienmathematikern wird oft die Frage gestellt, warum nach den Operationen Addition (1. Stufe), Multiplikation (2. Stufe), Potenzieren (3. Stufe) keine Operationen der 4. oder höheren Stufen betrachtet werden. Jede Operation der nächsthöheren Stufe ist die Wiederholung der vorhergehenden Operation, z.B. n * x = x + x + ... + x x^n = x * x * ... * x Das offensichtliche Problem mit der Wiederholung des Potenzierens besteht darin, dass das Potenzieren nicht assoziativ ist und es somit mehrere Klammerungsmöglichkeiten für die Wiederholung dieser Operation gibt. Wählt man eine spezifische Klammerungsmöglichkeit aus, z.B. x^^n = (x^(x^(x^(......)))), gibt es jedoch wieder verschiedene Möglichkeiten, diese Operation auf rationale oder reelle n fortzusetzen. In der Tat kann man im Internet verschiedene solcher Fortsetzungen beschrieben finden und keine scheint besonders ausgezeichnet zu sein. Das ganze Dilemma der verschiedenen Klammerungen kann man jedoch überwinden, in dem man den Zahlenbereich abstrakter macht. So dass statt nur der Anzahl auch eine Klammerungsstruktur in einer Zahl kodiert wird. Die ganz natürliche Verallgemeinerung der natürlichen Zahlen in dieser Hinsicht sind die Binärbäume. Und in der Tat lassen sich die 4. und höhere Operationen in einer eindeutigen Weise auf den Binärbäumen erklären. Vielmehr stellt sich sogar heraus, dass die Binärbäume zu viel Information mit sich tragen, wenn es nur darum geht, die höheren Operationen zu definieren. Es gibt eine Spezialisierung der Binärbäume, die aber allgemeiner als die natürlichen Zahlen (die die assoziative Spezialisierung der Binärbäume sind) ist, und die die passende Informationsmenge zur Definition der höheren Operationen kodiert. Dies sind die so genannten linkskommutativen Binärbäume. Es stellt sich heraus, dass die (linkskommutativen) Binärbäume viele Eigenschaften der natürlichen Zahlen teilen, so z.B. die Assoziativität der Multiplikation (die Operation der 2. Stufe) und eine eindeutige Primzahlzerlegung. Dies motiviert die Frage, ob man die Erweiterungskonstruktionen der Zahlen: „natürliche Zahlen zu gebrochenen Zahlen“ (macht die Multiplikation umkehrbar) „gebrochene Zahlen zu positiven reellen Zahlen“ (macht das Potenzieren umkehrbar und erlaubt Grenzwertbildung) auch ausgehend von (linkskommutativen) Binärbäumen vornehmen kann. In der vorliegenden Arbeit wird (neben unzähligen anderen Resultaten) gezeigt, dass die Zahlenbereichserweiterung „natürliche Zahlen zu gebrochenen Zahlen“ auch analog für (linkskommutative) Binärbäume möglich ist. Das Ergebnis dieser Konstruktion sind die Divisionsbinärbäume (bzw. die gebrochenen Bäume). Letztere lassen sich unerwartet in der Form von Brüchen darstellen, sind jedoch als Verallgemeinerung der gebrochenen Zahlen sehr viel komplexer als dieser. (Das kann man live nachprüfen mit dem dafür erstellten Online-Rechner für gebrochene Bäume (auf englisch): http://math.eretrandre.org/cgi-bin/ftc/ftc.pl ) Damit wird ein Programm „baumartige Zahlen“ gestartet, indem es darum geht, auch die Erweiterung „gebrochene Zahlen zu positiven reellen Zahlen“ für die Divisionsbinärbäume (bzw. die gebrochenen Bäume) durchzuführen, wobei die höheren Operationen auf dieser Erweiterung definiert werden könnten und umkehrbar sein müssten. Ob dies wirklich möglich ist, ist derzeit unklar (neben diversen anderen direkt aus der Dissertation sich ergebenden Fragen) und eröffnet damit ein enorm umfangreiches Feld für weitere Forschungen.
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Hellman, Noah. "Mitchell-Based Approximate Operations on Floating-Point Numbers." Thesis, Linköpings universitet, Datorteknik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-178882.

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By adapting Mitchell's algorithm for floating-point numbers, one can efficiently perform arithmetic floating-point operations in an approximate logarithmic domain in order to perform approximate computations of functions such as multiplication, division, square root and others. This work examines how this algorithm can be improved in terms of accuracy and hardware complexity by applying a set of various methods that are parametrized and offer a large design space. Optimal coefficients for a large portion of this space is determined and used to synthesize circuits for both ASIC and FPGA circuits using the bfloat16 format\@. Optimal configurations are then extracted to create an optimal curve where one can select an acceptable error range and obtain a circuit with a minimal hardware cost.
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Carbone, Rose Elaine. "Elementary Teacher Candidates’ Understanding of Rational Numbers: An International Perspective." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79565.

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This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.
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Clark, David Alan. "The Euclidean algorithm for Galois extensions of the rational numbers." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=39408.

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Let K be a totally real, quartic, Galois extension of $ doubq$ whose ring of integers R is a principal ideal domain. If there is a prime ideal p of R such that the unit group maps onto $(R/{ bf p} sp2$)*, then R is a Euclidean domain. This criterion is generalized to arbitrary Galois extensions.<br>Let E be an elliptic curve over a number field F. Suppose ($F: doubq rbrack le 4$ and $F(E lbrack q rbrack ) not subseteq F$ for all primes q such that F contains a primitive $q sp{ rm th}$ root of unity, then the reduced elliptic curve $ tilde{E}(F sb{ bf p})$ is cyclic infinitely often. In general, if $ Gamma$ a subgroup of $E(F)$ with the range of $ Gamma$ sufficiently large, there are infinitely many prime ideals p of F such that the reduced curve $ tilde{E}(F sb{ bf p}) = Gamma sb{ bf p}$, where $ Gamma sb{ bf p}$ is the reduction modulo p of $ Gamma$.
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Books on the topic "Operations on rational numbers"

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

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Mary, Stroh, and Sopris West Inc, eds. TransMath: Making sense of rational numbers. Cambium Learning/Sopris West, 2010.

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P, Carpenter Thomas, Fennema Elizabeth, and Romberg Thomas A, eds. Rational numbers: An integration of research. Lawrence Erlbaum Associates, 1992.

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Lappan, Glenda. Bits and pieces I: Understanding rational numbers. Dale Seymour Publications, 1998.

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Glenda, Lappan, and Michigan State University, eds. Bits and pieces II: Using rational numbers. Dale Seymour Publications, 1998.

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Lappan, Glenda. Bits and pieces I: Understanding rational numbers. Dale Seymour Publications, 1998.

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Glenda, Lappan, and Michigan State University, eds. Bits and pieces II: Using rational numbers. Prentice Hall, 2002.

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1948-, Silver Edward A., and Stein Mary Kay, eds. Improving instruction in rational numbers and proportionality. Teachers College Press, 2005.

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Bellos, Alex. Here's looking at Euclid: A surprising excursion through the astonishing world of math. Free Press, 2010.

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Bellos, Alex. Here's Looking at Euclid: A Surprising Excursion through the Astonishing World of Math. Free Press, 2010.

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Book chapters on the topic "Operations on rational numbers"

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Krasa, Nancy, Karen Tzanetopoulos, and Colleen Maas. "Rational Numbers and Operations." In How Children Learn Math. Routledge, 2022. http://dx.doi.org/10.4324/9781003157656-15.

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Lehrer, Richard, and Leona Schauble. "Measurement Models of Arithmetic Operations and Rational Number." In Measuring and Visualizing Space in Elementary Mathematics Learning. Routledge, 2023. http://dx.doi.org/10.4324/9781003287476-7.

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Jakobsen, Peter Viggo. "Deterrence in Peace Operations: Look Beyond the Battlefield and Expand the Number of Targets and Influence Mechanisms." In NL ARMS. T.M.C. Asser Press, 2020. http://dx.doi.org/10.1007/978-94-6265-419-8_17.

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AbstractThe peace operations literature suffers from a narrow focus on battlefield deterrence. It ignores the need to deter actors beyond the battlefield from supporting the combatants using force, and analyses the use of military threats and force in peace operations in a vacuum without taking into account the other instruments that deterring actors employ simultaneously to influence the combatants, combatant allies, combatant supporters and bystanders that undermine deterrence in peace operations. Since most peace operation forces lack the capacity and willingness to threaten and use force in accordance with the requirements stipulated by rational deterrence theory, influencing actors beyond the battlefield is more important with respect to deterring violence than the military efforts undertaken by peace operation forces to deter combatants from using force or to compel them to stop doing so. Accordingly, this chapter develops a new analytical framework that will enable peace operation theorists and practitioners to target all the actors that undermine deterrence on the battlefield and beyond with all the tools at their disposal—persuasion, inducement and coercion. The framework will improve both theory and practice by providing a better understanding of the conditions under which peace operations can contribute to deterring and, if need be, compelling combatants from using force as well as identifying the tools that practitioners can employ to this end. It highlights that peace operations merely constitute the top of the deterrence iceberg, and that peace operation forces must be supported by other actors and tools to succeed with respect to deterring violence and facilitating conflict resolution.
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Ye, Weijia, Rubiao Han, Peng Gao, and Ning He. "Research on Airport Communication and Navigation Equipment Inspection Assistance System Based on AR Technology." In Lecture Notes in Electrical Engineering. Springer Nature Singapore, 2025. https://doi.org/10.1007/978-981-96-2409-6_6.

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Abstract Aviation transportation is an important component of our country’s transportation system. With the rapid improvement of the national economy, the civil aviation industry in our country has experienced rapid development, and the number and scale of airports are also constantly increasing. Communication and navigation systems play an important role in the transportation of aircraft and the scale operation of airports. Efficient and accurate communication and navigation system can not only effectively ensure air transportation safety but also improve overall operational efficiency, making airport operations more rational and secure. This paper combines augmented reality technology with VSLAM (Visual Simultaneous Localization and Mapping) technology to assist personnel in daily communication and navigation equipment inspection tasks. By providing some functions to improve the efficiency of airport personnel, the function such as positioning, some auxiliary information, and remote assistances.
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Eriksson, Kenneth, Donald Estep, and Claes Johnson. "Rational Numbers." In Applied Mathematics: Body and Soul. Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-05796-4_7.

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Bhattacharjee, Meenaxi, Rögnvaldur G. Möller, Dugald Macpherson, and Peter M. Neumann. "Rational Numbers." In Notes on Infinite Permutation Groups. Hindustan Book Agency, 1997. http://dx.doi.org/10.1007/978-93-80250-91-5_9.

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Bhattacharjee, Meenaxi, Dugald Macpherson, Rögnvaldur G. Möller, and Peter M. Neumann. "Rational numbers." In Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0092559.

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Shah, Nita H., and Vishnuprasad D. Thakkar. "Rational Numbers." In Journey from Natural Numbers to Complex Numbers. CRC Press, 2020. http://dx.doi.org/10.1201/9781003105244-3.

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Noël, Marie-Pascale, and Giannis Karagiannakis. "Rational numbers." In Effective Teaching Strategies for Dyscalculia and Learning Difficulties in Mathematics. Routledge, 2022. http://dx.doi.org/10.4324/b22795-6.

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Ovchinnikov, Sergei. "Rational Numbers." In Real Analysis: Foundations. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64701-8_1.

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Conference papers on the topic "Operations on rational numbers"

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Chawdhry, Herschel. "Interpolation of rational functions in loop calculations by using p-adic numbers." In Loops and Legs in Quantum Field Theory. Sissa Medialab, 2024. http://dx.doi.org/10.22323/1.467.0037.

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Roti, J. S., and M. M. Mullins. "Operating Experiences in Treating Alkaline, No pH Control Cooling Water Systems." In CORROSION 1989. NACE International, 1989. https://doi.org/10.5006/c1989-89158.

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Abstract Alkaline, no-pH control cooling water treatment applications represent a growing market in today's industrial environment. Many plants prefer to not feed acid to their open recirculating cooling systems to destroy alkalinity and lower pH because of the inherent handling and safety considerations of this material. Other plants view the use of acid as just another source of system upset should this material be over or under fed. This paper reviews the rationale of why more facilities are moving to this technology, outlines a number of treatment programs which are available to effectively handle these environments and offers laboratory and field data to support when and under what conditions these programs function effectively.
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Manouchehri, Soheil. "Subsea Pipelines and Flowlines Decommissioning: What We Should Know for a Rational Approach." In ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/omae2017-61239.

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Offshore and subsea decommissioning will increase in the next five years or so as many producing fields are matured and cease production while the oil price continues to remain low. This emphasizes the need for a thorough decommissioning plan to ensure a safe and technically feasible solution while it is economically viable and safeguards the environment. Offshore and subsea decommissioning is commonly considered on a case-by-case basis using the Comparative Assessment (CA) process in which the best decommissioning solution is obtained. Health, Safety and Environmental (HSE) considerations are always paramount in any decommissioning process. The aim is to significantly reduce the long term risks to other benefactors of the sea while the associated short term risks to those responsible for decommissioning operations are minimized. A major part of any decommissioning project is subsea pipelines decommissioning (by “pipelines”, it is meant to include flowlines, trunklines and flexible too). There are a number of techniques available for decommissioning of subsea pipelines ranging from preservation for potential future use to full recovery or leaving in-situ. However, each subsea pipeline decommissioning technique should be considered on its own merit. Selection of each decommissioning technique depends on many parameters, inter alia, size of pipeline, type of pipeline (e.g. single pipe, pipe-in-pipe, piggyback), type of conveying fluid, operational environment (location), production history, Inspection, Repair and Maintenance (IRM) records, HSE considerations, connection to other facilities, technical feasibility (including potential use of advanced technologies), regulatory authorities requirements and socio-economic considerations. This paper will look at specifics of subsea pipelines decommissioning. It will examine the procedures to be undertaken from desk top activities (e.g. planning and CA) up to operational activities (e.g. pigging, flushing, cleaning, removal or leaving in-situ). Different scenarios are discussed and potential advantages and disadvantages of each scenario are presented. In addition, a guide is proposed for future pipelines decommissioning projects to follow a rational approach.
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Cantamessa, Marco, and Francesca Montagna. "Beyond Lean Manufacturing: Developing an Integrated Methodology to Design Effective Manufacturing Systems." In ASME 2008 9th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2008. http://dx.doi.org/10.1115/esda2008-59223.

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Global competition forces manufacturing companies to operate deep changes in their production systems, involving physical resources, operating procedures and the organization. In this context, the Lean Manufacturing (LM) paradigm is quite popular and lean practice efforts have crossed from the automotive sector into other industries. Most academic contributions to literature concerning LM are on specific aspects of LM, while there is little discussion on the overall implementation of LM. This is despite the fact that lean principles encompass all aspects of manufacturing operations and the need of a systemic approach is clearly shown in practitioner-oriented literature. A number of factors (i.e. cultural, technological and industrial differences) may influence or inhibit the implementation of LM. This paper focuses on some of the critical aspects of LM and analyzes the reason why LM principles are difficult to implement. In particular it suggests that, by formulating a solid scientific basis, the application of LM in industry could become easier and more rigorous, and not only based on past experience. At the same time, the paper suggests that when theory-based tools are used in an isolated way, this carries the risk of not effectively coping with the systemic nature of manufacturing systems. This paper proposes the idea of an innovative methodology able to lead beyond the usual concept of LM, i.e. to adopt its basic principles and systemic perspective, but following a rational and deductive approach that explicitly considers company specific features. The methodology is based on the “Systems of Systems” approach currently being used in the context of complex military initiatives and on the integration of different tools, each focused on specific aspects of the manufacturing system. The paper makes a preliminary attempt to describe how the main aspects of manufacturing systems (resources, quality management systems, production planning and control procedures, etc.) can be represented within the SoS framework and how SoS can support the rational definition of the path leading from corporate strategy to system redesign.
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ȚARĂLUNGĂ, Boris. "Rational numbers that are not represented as the sum of unit rational fractions." In "Ştiință și educație: noi abordări și perspective", conferinţă ştiinţifică internaţională. Ion Creangă Pedagogical State University, 2024. https://doi.org/10.46727/c.v4.21-22-03-2024.p68-72.

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Tsakatikas, D., and M. Sfantsikopoulos. "Development of a Decision Support System for Unscheduled Maintenance MRO Component Safety Stock." In ASME 2007 International Manufacturing Science and Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/msec2007-31115.

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The paper presents a methodology for the establishment of the MRO (Maintenance, Repair and Operation) material criticality that is focused on the Industrial Unplanned Maintenance needs. The thus obtained criticality is used to rationalize the efficiency of the Plant MRO-material Inventory. Through an appropriately adapted Top-Down FMECA (Failure Modes Effects and Criticality Analysis) the concept of the Component Dynamic Criticality is introduced and calculated for all the functional components of an industrial production facility. The components are assigned with characteristic code numbers. MRO-material Inventory control is then carried out within the frame of the developed Decision Support System. A case study demonstrates the method. The use of the developed Decision Support System considerably increases not only the efficiency of the Plant MRO-Material Inventory and control of the Unplanned Maintenance Downtime, but also its continuously updated and highly constructive System Dynamic Database of Component Failure Modes and their Effects benefits the Conventional Maintenance Schemes, e.g. Scheduled and/or Predictive Maintenance, as well. MRO-Material Inventory control for rational Unplanned Maintenance Downtime has not been addressed to this date in a systematic way. By addressing this issue the present approach contributes to an increased availability of industrial production facilities and to a better performance of CMMS and/or ERP tools.
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Vălcan, Teodor-Dumitru. "Structures of Fields of Rational Numbers, Isomorphic Between Them." In 10th International Conference Education, Reflection, Development. European Publisher, 2023. http://dx.doi.org/10.15405/epes.23056.8.

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Pion, Sylvain, and Chee K. Yap. "Constructive root bound for k-ary rational input numbers." In the nineteenth conference. ACM Press, 2003. http://dx.doi.org/10.1145/777792.777831.

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Adishamita, Rasi, Mashadi Mashadi, and Muliana Muliana. "New Formula for Interval Numbers with Powers of Positive Rational Numbers and Its Properties." In Proceedings of the 3rd Sriwijaya International Conference on Basic and Applied Sciences, SICBAS 2023, November 3, 2023, Palembang, Indonesia. EAI, 2024. http://dx.doi.org/10.4108/eai.3-11-2023.2347908.

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Yokoyama, Ryohei, and Koichi Ito. "Evaluation of Operational Performance of Gas Turbine Cogeneration Plants Using an Optimization Tool: OPS-Operation." In ASME Turbo Expo 2003, collocated with the 2003 International Joint Power Generation Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/gt2003-38381.

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To attain the highest performance of gas turbine cogeneration plants, it is necessary to rationally select the numbers and capacities of gas turbines and auxiliary equipment in consideration of their operational strategies corresponding to energy demands which change with season and time. It is also important to rationally select the options such as the variable heat to power by the steam injection or combined cycle, and the inlet air cooling by the ice storage combined with electric compression refrigeration or steam absorption refrigeration. The evaluation of the effects of these alternatives on the performance is an important work for designers. However, it takes much time to conduct the work thoroughly. The authors have developed an optimization tool named “OPS-Operation” to assess the operational strategies for given configurations and specifications of energy supply plants. This tool has a user-friendly interface for the functions of data registration, graphical flowsheet editing, automatic programming and optimization calculation, and graphical representation of results. In this paper, the effects of the aforementioned alternatives on the operational performance of gas turbine cogeneration plants are evaluated using the optimization tool in terms of many criteria including operational cost, energy consumption, and CO2 emission. It is demonstrated that the tool is very effective to evaluate the performance rationally, flexibly, and easily.
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Reports on the topic "Operations on rational numbers"

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Rosenfeld. L51741 Development of a Model for Fatigue Rating Shallow Unrestrained Dents. Pipeline Research Council International, Inc. (PRCI), 1997. http://dx.doi.org/10.55274/r0010337.

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The ability to fatigue-rate dents in line pipe is important for three reasons. First, field failures give evidence that fatigue crack growth occurs in dents under certain circumstances in spite of the fact that plain dents are generally thought to have little effect on the integrity of natural gas pipelines based on the results of single-cycle burst tests. Second, many operators are discovering large numbers of dents on the bottom quadrants of their pipe associated with rocks and backfill loads. Guidance is needed for discerning dents for which excavation and inspection is economically wasteful and counterproductive to pipeline safety from those dents for which further action would be beneficial. Third, fatigue life may be a more rational basis for rating the severity of a dent than present criteria which rely solely on dent depth with a maximum depth of 6 percent of the pipe diameter as a generally accepted limit. In some cases deeper dents might be permitted to remain in service, while in other cases, shallower dents should be repaired. The goal of this project was to develop guidelines for pipeline operators to assess the severity of dents on the basis of their fatigue life in-service. The assessment uses pipeline operating pressures and simple geometric measurements of the dent.
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NATIONAL WAR COLL WASHINGTON DC. Fixing US Special Operations. Rational Actors Not Allowed. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada441771.

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Satogata, T. RHIC RF Harmonic Numbers for Low Energy Operations. Office of Scientific and Technical Information (OSTI), 2008. http://dx.doi.org/10.2172/939977.

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Satogata T. RHIC RF Harmonic Numbers For Low Energy Operations. Office of Scientific and Technical Information (OSTI), 2008. http://dx.doi.org/10.2172/1061899.

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

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Smit, Timo, Sofia Sacks Ferrari, and Jaïr van der Lijn. Trends in Multilateral Peace Operations, 2019. Stockholm International Peace Research Institute, 2020. http://dx.doi.org/10.55163/ixjs4170.

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Despite evidence of their positive impacts, United Nations peacekeeping operations continue to face budget cuts, cynicism in the political arena and concern over personnel physical safety. This context underpins the global and regional trends in multilateral peace operations in 2019. This SIPRI Fact Sheet gives a snapshot of multilateral peace operations in 2019, with statistics on personnel, country contributions and fatalities for operations conducted by the UN, regional organizations or alliances, and ad hoc coalitions of states. Global and regional trends in 2019 follow developments from recent years, including the downward trends associated with the reductions and closures of many UN peace operations since 2015. Sub-Saharan Africa continues to host the majority of operations and personnel, although these numbers have decreased, while the Middle East and North Africa is drawing attention for increasing numbers of operations and personnel. Hostile death rates for 2019 are largely attributed to the UN Multidimensional Integrated Stabilization Mission in Mali, the deadliest operation since its establishment in 2013, with all other operations demonstrating relatively low numbers of fatalities.
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Lutz, Carsten. Adding Numbers to the SHIQ Description Logic - First Results. Aachen University of Technology, 2001. http://dx.doi.org/10.25368/2022.117.

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Recently, the Description Logic (DL) SHIQ has found a large number of applications. This success is due to the fact that SHIQ combines a rich expressivity with efficient reasoning, as is demonstrated by its implementation in DL systems such as FaCT and RACER. One weakness of SHIQ, however, limits its usability in several application areas: numerical knowledge such as knowledge about the age, weight, or temperature of real-world entities cannot be adequately represented. In this paper, we propose an extension of SHIQ that aims at closing this gap. The new Description Logic Q-SHIQ, which augments SHIQ by additional, 'concrete domain' style concept constructors, allows to refer to rational numbers in concept descriptions, and also to define concepts based on the comparison of numbers via predicates such as &lt; or =. We argue that this kind of expressivity is needed in many application areas such as reasoning about the semantic web. We prove reasoning with Q-SHIQ to be EXPTIME-complete (thus not harder than reasoning with SHIQ) by devising an automata-based decision procedure.
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Soder, Kirsten. Multilateral Peace Operations: Personnel, 2009. Stockholm International Peace Research Institute, 2010. http://dx.doi.org/10.55163/anmf8652.

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This SIPRI Fact Sheet presents data from the SIPRI Multilateral Peace Operations Database to illuminate some of the most significant developments in peace operations personnel in 2009 as well as trends over the decade since 2000. It includes statistical data on numbers and type of peacekeeping personnel, the organizations and countries mounting operations and deploying personnel.
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Soder, Kirsten. Multilateral Peace Operations: Europe, 2009. Stockholm International Peace Research Institute, 2010. http://dx.doi.org/10.55163/wkwc3096.

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This SIPRI Fact Sheet presents data from the SIPRI Multilateral Peace Operations Database to illuminate some of the most significant developments in peace operations in Europe in 2009 as well as trends over the decade since 2000. It includes statistical data on numbers and type of peacekeeping personnel, the organizations and countries mounting operations and deploying personnel, along with maps of peace operations that were active during 2009.
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Soder, Kirsten. Multilateral Peace Operations: Asia, 2009. Stockholm International Peace Research Institute, 2010. http://dx.doi.org/10.55163/zsky7548.

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This SIPRI Fact Sheet presents data from the SIPRI Multilateral Peace Operations Database to illuminate some of the most significant developments in peace operations in Asia in 2009 as well as trends over the decade since 2000. It includes statistical data on numbers and type of peacekeeping personnel, the organizations and countries mounting operations and deploying personnel, along with maps of peace operations that were active during 2009.
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