• Title, Summary, Keyword: rational numbers

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A Study on the Theoretical Background of the Multiplication of Rational Numbers as Composition of Operators (두 조작의 합성으로서의 유리수 곱의 이론적 배경 고찰)

  • Choi, Keunbae
    • East Asian mathematical journal
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    • v.33 no.2
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    • pp.199-216
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    • 2017
  • A rational number as operator is eventually that it is considered a mapping. Depending on how selecting domain (the target of operation by rational number) and codomain (including the results of operations by rational number), it is possible to see the rational in two aspects. First, rational numbers can be deal with functions if we choose the target of operation by rational number as a number field containing rationals. On the other hand, if we choose the target of operation by rational number as integral domain $\mathbb{Z}$, then rational numbers can be regarded as partial functions on $\mathbb{Z}$. In this paper, we regard the rational numbers with a view of partial functions, we investigate the theoretical background of the relationship between the multiplication of rational numbers and the composition of rational numbers as operators.

On Explaining Rational Numbers for Extending the Number system to Real Numbers (실수로의 수 체계 확장을 위한 유리수의 재해석에 대하여)

  • Shin, Bo-Mi
    • Journal of the Korean School Mathematics Society
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    • v.11 no.2
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    • pp.285-298
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    • 2008
  • According to the 7th curriculum, irrational numbers should be introduced using infinite decimals in 9th grade. To do so, the relation between rational numbers and decimals should be explained in 8th grade. Preceding studies remarked that middle school students could understand the relation between rational numbers and decimals through the division appropriately. From the point of view with the arithmetic handling activity, I analyzed that the integers and terminating decimals was explained as decimals with repeating 0s or 9s. And, I reviewed the equivalent relations between irrational numbers and non-repeating decimals, rational numbers and repeating decimals. Furthermore, I suggested an alternative method of introducing irrational numbers.

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연산자로서의 유리수 체계의 구성에 관한 연구

  • Chung, Young-Woo;Kim, Boo-Yoon
    • East Asian mathematical journal
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    • v.28 no.2
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    • pp.135-158
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    • 2012
  • The ideals of the rings of integers are used to induce rational number system as operators(=group homomorphisms). We modify this inducing method to be effective in teaching rational numbers in secondary school. Indeed, this modification provides a nice model for explaining the equality property to define addition and multiplication of rational numbers. Also this will give some explicit ideas for students to understand the concept of 'field' efficiently comparing with the integer number system.

Beyond the Union of Rational and Irrational Numbers: How Pre-Service Teachers Can Break the Illusion of Transparency about Real Numbers? (유리수와 무리수의 합집합을 넘어서: 실수가 자명하다는 착각으로부터 어떻게 벗어날 수 있는가?)

  • Lee, Jihyun
    • Journal of Educational Research in Mathematics
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    • v.25 no.3
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    • pp.263-279
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    • 2015
  • The introduction of real numbers is one of the most difficult steps in the teaching of school mathematics since the mathematical justification of the extension from rational to real numbers requires the completeness property. The author elucidated what questions about real numbers can be unanswered as the "institutional didactic void" in school mathematics defining real numbers as the union of the rational and irrational numbers. The pre-service teachers' explanations on the extension from rational to real numbers and the raison d'$\hat{e}$tre of arbitrary non-recurring decimals showed the superficial and fragmentary understanding of real numbers. Connecting school mathematics to university mathematics via the didactic void, the author discussed how pre-service teachers could break the illusion of transparency about the real number.

Inducing Irrational Numbers in Junior High School (중학교에서의 무리수 지도에 관하여)

  • Kim, Boo-Yoon;Chung, Young-Woo
    • Journal for History of Mathematics
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    • v.21 no.1
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    • pp.139-156
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    • 2008
  • We investigate the inducing method of irrational numbers in junior high school, under algebraic as well as geometric point of view. Also we study the treatment of irrational numbers in the 7th national curriculum. In fact, we discover that i) incommensurability as essential factor of concept of irrational numbers is not treated, and ii) the concept of irrational numbers is not smoothly interconnected to that of rational numbers. In order to understand relationally the incommensurability, we suggest the method for inducing irrational numbers using construction in junior high school.

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ON THE SETS OF LENGTHS OF PUISEUX MONOIDS GENERATED BY MULTIPLE GEOMETRIC SEQUENCES

  • Polo, Harold
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1057-1073
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    • 2020
  • In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids M that are hereditarily atomic (i.e., every submonoid of M is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we realize arithmetic progressions as the sets of distances of some additive submonoids of the nonnegative rational numbers.

CORRELATION DIMENSIONS OF QUASI-PERIODIC ORBITS WITH FREQUENCIES CIVEN BY QUASI ROTH NUMBERS

  • Naito, Koichiro
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.857-870
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    • 2000
  • In this paper, we estimate correlation dimensions of discrete quasi periodic ordits with frequencies, irrational numbers, which are called quasi Roth numbers. We specify the lower estimate valuse of the dimensions by using the parameters which are derived the rational approximable properties of the quasi Roth numbers.

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CONTINUED FRACTIONS AND THE DENSITY OF GRAPHS OF SOME FUNCTIONS

  • Chae, Hi-joon;Jun, Byungheup;Lee, Jungyun
    • Korean Journal of Mathematics
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    • v.25 no.2
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    • pp.137-145
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    • 2017
  • We consider some simple periodic functions on the field of rational numbers with values in ${\mathbb{Q}}/{\mathbb{Z}}$ which are defined in terms of lowest-term-expression of rational numbers. We prove the density of graphs of these functions by constructing explicitly points on the graphs close to a given point using continued fractions.