Journal articles: 'Visualisation of rational numbers'

1

., Jyoti. "Rational Numbers." Journal of Advances and Scholarly Researches in Allied Education 15, no. 5 (July 1, 2018): 220–22. http://dx.doi.org/10.29070/15/57856.

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Scott Malcom, P. "Understanding Rational Numbers." Mathematics Teacher 80, no. 7 (October 1987): 518–21. http://dx.doi.org/10.5951/mt.80.7.0518.

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Understanding is sometimes an elusive goal in mathematics. Although we may believe we have a complete understanding of a concept, another approach to this same concept may bring us additional insight.

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Lennerstad, Håkan, and Lars Lundberg. "Decomposing rational numbers." Acta Arithmetica 145, no. 3 (2010): 213–20. http://dx.doi.org/10.4064/aa145-3-1.

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PEYTON JONES, SIMON. "12 Rational Numbers." Journal of Functional Programming 13, no. 1 (January 2003): 149–52. http://dx.doi.org/10.1017/s0956796803001412.

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Gong, Linming, Bo Yang, Tao Xue, Jinguang Chen, and Wei Wang. "Secure rational numbers equivalence test based on threshold cryptosystem with rational numbers." Information Sciences 466 (October 2018): 44–54. http://dx.doi.org/10.1016/j.ins.2018.07.046.

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Marcos, J. E. "Topological completions of the field of rational numbers which consist of Liouville numbers and rational numbers." Journal of Pure and Applied Algebra 138, no. 3 (May 1999): 251–77. http://dx.doi.org/10.1016/s0022-4049(98)00053-x.

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Hill, Rosemary Lucy. "The political potential of numbers: data visualisation in the abortion debate." Kvinder, Køn & Forskning, no. 1 (September 5, 2017): 83–96. http://dx.doi.org/10.7146/kkf.v26i1.109789.

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Data visualisation has been argued to have the power to ‘change the world’, implicitly for the better, but when it comes to abortion, both sides make moral claims to ‘good’. Visualisation conventions of clean lines and shapes simplify data, lending them a rhetoric of neutrality, as if the data is the whole story. It is imperative, therefore, to examine how data visualisations are used to shape women’s lives. This article draws on the findings of the Persuasive Data project . Google Image Scraper was used to locate abortion-related visualisations circulating online. The images, their web locations, and data use were social semiotically analysed to understand their visual rhetoric and political use. Anti-abortion groups are more likely to use data visualisation than pro-choice groups, thereby simplifying the issue and mobilising the rhetoric of neutrality. I argue that data visualisations are being used as a hindrance to women’s access to abortion, and that the critique of such visualisations needs to come from feminists. This article extends discussions of how data is often reified as objective, by showing how the rhetoric of objectivity within data visualisation conventions is harnessed to do work in the world that is potentially very damaging to women’s rights.

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Sándor, József. "On certain rational perfect numbers." Notes on Number Theory and Discrete Mathematics 28, no. 2 (May 12, 2022): 281–85. http://dx.doi.org/10.7546/nntdm.2022.28.2.281-285.

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Atanasiu, Dragu. "Laplace Integral on Rational Numbers." MATHEMATICA SCANDINAVICA 76 (December 1, 1995): 152. http://dx.doi.org/10.7146/math.scand.a-12531.

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Detorie, Rick. "Let's Be Rational about Numbers." Mathematics Teaching in the Middle School 20, no. 7 (March 2015): 394–97. http://dx.doi.org/10.5951/mathteacmiddscho.20.7.0394.

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Goetz, Melanie. "The irrationality of rational numbers." Journal - American Water Works Association 105, no. 7 (July 2013): 82–84. http://dx.doi.org/10.5942/jawwa.2013.105.0096.

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Almada, Carlos. "On counting the rational numbers." International Journal of Mathematical Education in Science and Technology 41, no. 8 (December 15, 2010): 1096–101. http://dx.doi.org/10.1080/0020739x.2010.500695.

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Smith III, John P. "Competent Reasoning With Rational Numbers." Cognition and Instruction 13, no. 1 (March 1995): 3–50. http://dx.doi.org/10.1207/s1532690xci1301_1.

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Bowker, Geoffrey C., and Susan Leigh Star. "Pure, Real and Rational Numbers." Social Studies of Science 31, no. 3 (June 2001): 422–25. http://dx.doi.org/10.1177/030631201031003006.

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15

Rowland, Eric, and Jeffrey Shallit. "Automatic Sets of Rational Numbers." International Journal of Foundations of Computer Science 26, no. 03 (April 2015): 343–65. http://dx.doi.org/10.1142/s0129054115500197.

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The notion of a [Formula: see text]-automatic set of integers is well-studied. We develop a new notion — the [Formula: see text]-automatic set of rational numbers — and prove basic properties of these sets, including closure properties and decidability.

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Vourdas, A. "Harmonic analysis on rational numbers." Journal of Mathematical Analysis and Applications 394, no. 1 (October 2012): 48–60. http://dx.doi.org/10.1016/j.jmaa.2012.04.059.

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Tasoev, B. G. "Rational approximations to certain numbers." Mathematical Notes 67, no. 6 (June 2000): 786–91. http://dx.doi.org/10.1007/bf02675633.

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., Norris Sookoo, and Ashok Sahai . "Partial Densities on the Rational Numbers." Journal of Applied Sciences 7, no. 6 (March 1, 2007): 830–34. http://dx.doi.org/10.3923/jas.2007.830.834.

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Sándor, József. "On certain rational perfect numbers, II." Notes on Number Theory and Discrete Mathematics 28, no. 3 (August 10, 2022): 525–32. http://dx.doi.org/10.7546/nntdm.2022.28.3.525-532.

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We continue the study from [1], by studying equations of type $\psi(n) = \dfrac{k+1}{k} \cdot \ n+a,$ $a\in \{0, 1, 2, 3\},$ and $\varphi(n) = \dfrac{k-1}{k} \cdot \ n-a,$ $a\in \{0, 1, 2, 3\}$ for $k > 1,$ where $\psi(n)$ and $\varphi(n)$ denote the Dedekind, respectively Euler's, arithmetical functions.

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20

Shulga, Nikita. "Rational approximations to two irrational numbers." Moscow Journal of Combinatorics and Number Theory 11, no. 1 (March 30, 2022): 1–10. http://dx.doi.org/10.2140/moscow.2022.11.1.

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21

MILLER, RUSSELL. "HTP-COMPLETE RINGS OF RATIONAL NUMBERS." Journal of Symbolic Logic 87, no. 1 (November 22, 2021): 252–72. http://dx.doi.org/10.1017/jsl.2021.96.

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AbstractFor a ring R, Hilbert’s Tenth Problem $HTP(R)$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP(\mathbb {Z}[W^{-1}])$ , which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$ . It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP(R_W)$ . In contrast, we show that every Turing degree contains a set W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP(\mathbb {Z}[W^{-1}])$ can succeed uniformly on a set of measure $1$ , and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$ .

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22

FURUTA, Koji. "A moment problem on rational numbers." Hokkaido Mathematical Journal 46, no. 2 (June 2017): 209–26. http://dx.doi.org/10.14492/hokmj/1498788018.

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23

Trushechkin, Anton S., and Igor V. Volovich. "Functional classical mechanics and rational numbers." P-Adic Numbers, Ultrametric Analysis, and Applications 1, no. 4 (November 15, 2009): 361–67. http://dx.doi.org/10.1134/s2070046609040086.

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24

Poonen, Bjorn. "Multivariable polynomial injections on rational numbers." Acta Arithmetica 145, no. 2 (2010): 123–27. http://dx.doi.org/10.4064/aa145-2-2.

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25

Bradley, Christopher J. "87.06 A theorem on rational numbers." Mathematical Gazette 87, no. 508 (March 2003): 107–11. http://dx.doi.org/10.1017/s0025557200172201.

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26

Du, Juan, Ron Goldman, and Xuhui Wang. "Rational curves over generalized complex numbers." Journal of Symbolic Computation 93 (July 2019): 56–84. http://dx.doi.org/10.1016/j.jsc.2018.04.010.

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27

Vallance, P. "Numbers alone cannot determine rational treatment." BMJ 310, no. 6975 (February 4, 1995): 330. http://dx.doi.org/10.1136/bmj.310.6975.330.

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Armstrong, Drew, Nicholas A. Loehr, and Gregory S. Warrington. "Rational Parking Functions and Catalan Numbers." Annals of Combinatorics 20, no. 1 (November 25, 2015): 21–58. http://dx.doi.org/10.1007/s00026-015-0293-6.

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29

Wimmer, Harald K. "Realizations of matrices of rational numbers." Journal of Number Theory 25, no. 2 (February 1987): 169–83. http://dx.doi.org/10.1016/0022-314x(87)90023-0.

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30

Gill, Judith. "Mathematics and gender: Beyond rational numbers?" Mathematics Education Research Journal 9, no. 3 (November 1997): 343–46. http://dx.doi.org/10.1007/bf03217323.

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31

BENIOFF, PAUL. "COMPLEX RATIONAL NUMBERS IN QUANTUM MECHANICS." International Journal of Modern Physics B 20, no. 11n13 (May 20, 2006): 1730–41. http://dx.doi.org/10.1142/s021797920603425x.

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A binary representation of complex rational numbers and their arithmetic is described that is not based on qubits. It takes account of the fact that 0s in a qubit string do not contribute to the value of a number. They serve only as place holders. The representation is based on the distribution of four types of systems, corresponding to +1, -1, +i, -i, along an integer lattice. Complex rational numbers correspond to arbitrary products of four types of creation operators acting on the vacuum state. An occupation number representation is given for both bosons and fermions.

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32

Félix, Yves, and Jean-Claude Thomas. "Rational Betti numbers of configuration spaces." Topology and its Applications 102, no. 2 (April 2000): 139–49. http://dx.doi.org/10.1016/s0166-8641(98)00148-5.

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33

Kennedy, Helen, and Rosemary Lucy Hill. "The Feeling of Numbers: Emotions in Everyday Engagements with Data and Their Visualisation." Sociology 52, no. 4 (February 14, 2017): 830–48. http://dx.doi.org/10.1177/0038038516674675.

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This article highlights the role that emotions play in engagements with data and their visualisation. To date, the relationship between data and emotions has rarely been noted, in part because data studies have not attended to everyday engagements with data. We draw on an empirical study to show a wide range of emotional engagements with diverse aspects of data and their visualisation, and so demonstrate the importance of emotions as vital components of making sense of data. We nuance the argument that regimes of datafication, in which numbers, metrics and statistics dominate, are characterised by a renewed faith in objectivity and rationality, arguing that in datafied times, it is not only numbers but also the feeling of numbers that is important. We build on the sociology of (a) emotions and (b) the everyday to do this, and in so doing, we contribute to the development of a sociology of data.

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34

O'Leary, C. O., and W. Drew. "Flow visualisation on rolling models using minitufts." Aeronautical Journal 91, no. 906 (July 1987): 269–74. http://dx.doi.org/10.1017/s0001924000021357.

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Summary Minitufts are increasingly used for flow visualisation on static wind tunnel models because they can be applied in larger numbers for increased detail and with less interference compared with conventional tufts. This Memorandum describes an extension of their use to continuously rotating models where the heavier conventional tufts are inadequate. Tests on two combat aircraft models in the 4 m x 2.7 m Low Speed Wind Tunnel are described. Measured variations of rolling moment are explained with the aid of the minituft photographs.

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35

Alkan, Emre. "Series representing transcendental numbers that are not U-numbers." International Journal of Number Theory 11, no. 03 (March 31, 2015): 869–92. http://dx.doi.org/10.1142/s1793042115500487.

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Using integral representations with carefully chosen rational functions as integrands, we find new families of transcendental numbers that are not U-numbers, according to Mahler's classification, represented by a series whose terms involve rising factorials and reciprocals of binomial coefficients analogous to Apéry type series. Explicit descriptions of these numbers are given as linear combinations with coefficients lying in a suitable real algebraic extension of rational numbers using elementary functions evaluated at arguments belonging to the same field. In this way, concrete examples of transcendental numbers which can be expressed as combinations of classical mathematical constants such as π and Baker periods are given together with upper bounds on their wn measures.

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36

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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37

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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38

Moss, Joan. "Research, Reflection, Practice: Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations." Teaching Children Mathematics 9, no. 6 (February 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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Rajab, Majed. "Visualisation Model Based on Phishing Features." Journal of Information & Knowledge Management 18, no. 01 (March 2019): 1950010. http://dx.doi.org/10.1142/s0219649219500102.

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The numbers of online purchases and electronic banking transactions have increased substantially in the era of electronic business and mobile commerce. These online financial activities have attracted a special web threat called “phishing” that targets Internet users by seeking their credentials in order to access their financial information. Phishing involves impersonating a legitimate website by creating a visually similar fake website to deceive users. In the last decade different solutions to fight phishing that are primarily based on educating users, user’s experience, search methods, machine learning and features similarity have been developed. This paper combines computational intelligence along with user’s experience approaches to develop an anti-phishing visualisation method. Our method employs effective features chosen following thorough analysis on features scores generated by Correlation Feature Set and Information Gain processing techniques. We validate our anti-phishing features using classification systems produced by rule induction data mining approach. False positives, false negatives and phishing detection rate are the basis of evaluating the classification systems to measure our anti-phishing methods features’ integrity.

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40

Cação, Isabel, Maria Irene Falcão, and Helmuth Malonek. "Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence." Complex Analysis and Operator Theory 11, no. 5 (February 25, 2017): 1059–76. http://dx.doi.org/10.1007/s11785-017-0649-5.

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41

Malcolmson, Peter, Frank Okoh, and Vasuvedan Srinivas. "Factorial Fermat curves over the rational numbers." Colloquium Mathematicum 142, no. 2 (2016): 285–300. http://dx.doi.org/10.4064/cm142-2-9.

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42

Morozov, A. S., and J. K. Truss. "On computable automorphisms of the rational numbers." Journal of Symbolic Logic 66, no. 3 (September 2001): 1458–70. http://dx.doi.org/10.2307/2695118.

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AbstractThe relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.

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43

Adamczewski, Boris, Christiane Frougny, Anne Siegel, and Wolfgang Steiner. "Rational numbers with purely periodic β -expansion." Bulletin of the London Mathematical Society 42, no. 3 (April 15, 2010): 538–52. http://dx.doi.org/10.1112/blms/bdq019.

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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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45

Shteinikov, Yurii N. "On the product sets of rational numbers." Proceedings of the Steklov Institute of Mathematics 296, no. 1 (January 2017): 243–50. http://dx.doi.org/10.1134/s0081543817010199.

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46

Frougny, Christiane, and Karel Klouda. "Rational base number systems forp-adic numbers." RAIRO - Theoretical Informatics and Applications 46, no. 1 (August 22, 2011): 87–106. http://dx.doi.org/10.1051/ita/2011114.

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47

Bennett, M. "Fractional parts of powers of rational numbers." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 2 (September 1993): 191–201. http://dx.doi.org/10.1017/s0305004100071528.

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AbstractThe author uses Padé approximation techniques and an elementary lemma on primes dividing binomial coefficients to sharpen a theorem of F. Beukers on fractional parts of powers of rationals. In particular, it is proven that ‖((N+ l)/N)k‖ > 3–k holds for all positive integers N and k satisfying 4 ≤ N ≤ k · 3k. Other results are described including an effective version of a theorem of K. Mahler for a restricted class of rationals.

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48

Abbott, John. "Fault-tolerant modular reconstruction of rational numbers." Journal of Symbolic Computation 80 (May 2017): 707–18. http://dx.doi.org/10.1016/j.jsc.2016.07.030.

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49

Bertot, Yves. "A simple canonical representation of rational numbers." Electronic Notes in Theoretical Computer Science 85, no. 7 (September 2003): 1–16. http://dx.doi.org/10.1016/s1571-0661(04)80754-0.

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50

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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Journal articles on the topic 'Visualisation of rational numbers'

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