• The Mathematical Education
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• v.56 no.2
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• pp.131-145
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• 2017

The representation of irrational numbers has a key role in the learning of irrational numbers. However, transparent and finite representation of irrational numbers does not exist in school mathematics context. Therefore, many students have difficulties in understanding irrational numbers as an 'Object'. For this reason, this research explored how mathematics textbooks affected to students' understanding of irrational numbers in the view of process and object. Specifically we analyzed eight textbooks based on current curriculum and used framework based on previous research. In order to supplement the result derived from textbook analysis, we conducted questionnaires on 42 middle school students. The questions in the questionnaires were related to the representation and calculation of irrational numbers. As a result of this study, we found that mathematics textbooks develop contents in order of process-object, and using 'non repeating decimal', 'numbers cannot be represented as a quotient', 'numbers with the radical sign', 'number line' representation for irrational numbers. Students usually used a representation of non-repeating decimal, although, they used a representation of numbers with the radical sign when they operate irrational numbers. Consequently, we found that mathematics textbooks affect students to understand irrational numbers as a non-repeating irrational numbers, but mathematics textbooks have a limitation to conduce understanding of irrational numbers as an object.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.487-498
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• 2007

In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.987-998
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• 2015

In the paper, the authors discover an integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind.

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.571-591
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• 2012

To the best of our knowledge, there is no method, in the literature, to find the fuzzy optimal solution of fully fuzzy shortest path (FFSP) problems i.e., shortest path (SP) problems in which all the parameters are represented by fuzzy numbers. In this paper, a new method is proposed to find the fuzzy optimal solution of FFSP problems. Kumar and Kaur [Methods for solving unbalanced fuzzy transportation problems, Operational Research-An International Journal, 2010 (DOI 10.1007/s 12351-010-0101-3)] proposed a new method with new representation, named as JMD representation, of trapezoidal fuzzy numbers for solving fully fuzzy transportation problems and shown that it is better to solve fully fuzzy transportation problems by using proposed method with JMD representation as compare to proposed method with the existing representation. On the same direction in this paper a new method is proposed to find the solution of FFSP problems and it is shown that it is also better to solve FFSP problems with JMD representation as compare to existing representation. To show the advantages of proposed method with this representation over proposed method with other existing representations. A FFSP problem solved by using proposed method with JMD representation as well as proposed method with other existing representations and the obtained results are compared.

• Communications of Mathematical Education
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• v.28 no.3
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• pp.375-394
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• 2014

The purpose of this study is to examine the transition of mathematical representation in Joseon mathematics, which is focused on numbers and operations, letters and expressions. In Joseon mathematics, there had been two numeral systems, one by chinese character and the other by counting rods. These systems were changed into the decimal notation which used Indian-Arabic numerals in the late 19th century passing the stage of positional notation by Chinese character. The transition of the representation of operation and expressions was analogous to that of representation of numbers. In particular, Joseon mathematics represented the polynomials and equations by denoting the coefficients with counting rods. But the representation of European algebra was introduced in late Joseon Dynasty passing the transitional representation which used Chinese character. In conclusion, Joseon mathematics had the indigenous representation of numbers and mathematical symbols on our own. The transitional representation was found before the acceptance of European mathematical representations.

• School Mathematics
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• v.18 no.3
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• pp.647-666
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• 2016

This study investigates pre-service teacher's understanding of the concept and representations of irrational numbers. We classified the representations of irrational numbers into six categories; non-fraction, decimal, symbolic, geometric, point on a number line, approximation representation. The results of this study are as follows. First, pre-service teachers couldn't relate non-fractional definition and incommensurability of irrational numbers. Secondly, we observed the centralization tendency on symbolic representation and the little attention to other representations. Thirdly, pre-service teachers had more difficulty moving between symbolic representation and point on a number line representation of ${\pi}$ than that of $\sqrt{5}$ We suggested the concept of irrational numbers should be learned in relation to various representations of irrational numbers.

• Journal of Educational Research in Mathematics
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• v.24 no.4
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• pp.595-605
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• 2014

Infinite decimals have an infinite number of digits, chosen arbitrary and independently, to the right side of the decimal point. Since infinite decimals are ambiguous numbers impossible to write them down completely, the infinite decimal representation accompanies unavoidable opaqueness. This article focused the transparent aspect of infinite decimal representation with respect to the completeness axiom of real numbers. Long before the formalization of real number concept in $19^{th}$ century, many mathematicians were able to deal with real numbers relying on this transparency of infinite decimal representations. This analysis will contribute to overcome the double discontinuity caused by the different conceptualizations of real numbers in school and academic mathematics.

• Journal of applied mathematics & informatics
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• v.29 no.3_4
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• pp.831-846
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• 2011

There are several methods, in the literature, for finding the fuzzy optimal solution of fully fuzzy transportation problems (transportation problems in which all the parameters are represented by fuzzy numbers). In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings, a new method (based on fuzzy linear programming formulation) is proposed to find the fuzzy optimal solution of unbalanced fuzzy transportation problems with a new representation of trapezoidal fuzzy numbers. The advantages of the proposed method over existing method are discussed. Also, it is shown that it is better to use the proposed representation of trapezoidal fuzzy numbers instead of existing representation of trapezoidal fuzzy numbers for finding the fuzzy optimal solution of fuzzy transportation problems. To illustrate the proposed method a fuzzy transportation problem (FTP) is solved by using the proposed method and the obtained results are discussed. The proposed method is easy to understand and to apply for finding the fuzzy optimal solution of fuzzy transportation problems occurring in real life situations.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.345-354
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• 2006

In this paper first, we find a canonical symmetrical trapezoidal(triangular) for the solution of the fuzzy linear system $A\tilde{x}=\tilde{b}$, where the elements in A and $\tilde{b}$ are crisp and arbitrary fuzzy numbers, respectively. Then, a model for fuzzy linear programming problem with fuzzy variables (FLPFV), in which, the right hand side of constraints are arbitrary numbers, and coefficients of the objective function and constraint matrix are regarded as crisp numbers, is discussed. A numerical procedure for calculating a canonical symmetrical trapezoidal representation for the solution of fuzzy linear system and the optimal solution of FLPFV, (if there exist) is proposed. Several examples illustrate these ideas.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.519-527
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• 2011

Recently, the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$ are introduced in [3]: In this paper we give some interesting p-adic integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials related to the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$. From those integral representation on $\mathbb{Z}_p$ of the weighted q-Bernstein polynomials, we can derive some identities on the modified q-Bernoulli numbers and polynomials with weight ${\alpha}$.