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Статті в журналах з теми "Numbers, Rational"

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Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 30 січня 2023

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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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2

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.

3

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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4

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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5

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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6

Roy, Damien, and Johannes Schleischitz. "Numbers with Almost all Convergents in a Cantor Set." Canadian Mathematical Bulletin 62, no. 4 (December 3, 2018): 869–75. http://dx.doi.org/10.4153/s0008439518000450.

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AbstractIn 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

7

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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8

Korhonen, Risto. "Approximation of real numbers with rational number sequences." Proceedings of the American Mathematical Society 137, no. 01 (August 14, 2008): 107–13. http://dx.doi.org/10.1090/s0002-9939-08-09479-3.

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9

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.

10

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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11

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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12

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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13

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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14

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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15

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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16

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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17

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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18

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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Анотація:

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.

19

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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20

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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21

Trespalacios, Jesús, and Barbara Chamberline. "Pearl diver: Identifying numbers on a number line." Teaching Children Mathematics 18, no. 7 (March 2012): 446–47. http://dx.doi.org/10.5951/teacchilmath.18.7.0446.

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22

Elias, Henrique Rizek, Alessandro Jacques Ribeiro, and Angela Marta Pereira das Dores Savioli. "Epistemological Matrix of Rational Number: a Look at the Different Meanings of Rational Numbers." International Journal of Science and Mathematics Education 18, no. 2 (March 19, 2019): 357–76. http://dx.doi.org/10.1007/s10763-019-09965-4.

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23

BERGER, ARNO. "On linear independence of trigonometric numbers." Carpathian Journal of Mathematics 34, no. 2 (2018): 157–66. http://dx.doi.org/10.37193/cjm.2018.02.04.

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A necessary and sufficient condition is established for 1, cos(πr1), and cos(πr2) to be rationally independent, where r1, r2 are rational numbers. The elementary computational argument yields linear independence over larger number fields as well.

24

Koo, Reginald. "95.03 Geometric enumeration of the rationals between any two rational numbers." Mathematical Gazette 95, no. 532 (March 2011): 63–66. http://dx.doi.org/10.1017/s0025557200002357.

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25

Hurst, Michelle, and Sara Cordes. "Rational-number comparison across notation: Fractions, decimals, and whole numbers." Journal of Experimental Psychology: Human Perception and Performance 42, no. 2 (2016): 281–93. http://dx.doi.org/10.1037/xhp0000140.

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26

Mueller, Julia, and W. M. Schmidt. "On the number of good rational approximations to algebraic numbers." Proceedings of the American Mathematical Society 106, no. 4 (April 1, 1989): 859. http://dx.doi.org/10.1090/s0002-9939-1989-0961415-1.

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27

Van Hoof, Jo, Lieven Verschaffel, and Wim Van Dooren. "Number sense in the transition from natural to rational numbers." British Journal of Educational Psychology 87, no. 1 (October 31, 2016): 43–56. http://dx.doi.org/10.1111/bjep.12134.

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28

., 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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29

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.

30

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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31

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 $\mathbb {Q}$ .

32

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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33

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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34

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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35

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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36

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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37

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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38

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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39

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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40

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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41

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.

42

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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43

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.

44

Cilleruelo, J., D. S. Ramana, and O. Ramaré. "The number of rational numbers determined by large sets of integers." Bulletin of the London Mathematical Society 42, no. 3 (April 18, 2010): 517–26. http://dx.doi.org/10.1112/blms/bdq021.

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45

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.

46

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.”

47

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.

48

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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49

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.

50

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

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