• Journal of Educational Research in Mathematics
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• v.23 no.1
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• pp.17-35
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• 2013

The purpose of this study was to analyze the extendibility of division algorithm and Euclid algorithm for integers to algorithms for rational numbers based on word problems of fraction division. This study serviced to upgrade professional development of elementary and secondary mathematics teachers. In this paper, fractions were used as expressions of rational numbers, and they also represent rational numbers. According to discrete context and continuous context, and measurement division and partition division etc, divisibility was classified into two types; one is an abstract algebraic point of view and the other is a generalizing view which preserves division algorithms for integers. In the second view, we raised some contextual problems that can be used in school mathematics and then we discussed division algorithm, the greatest common divisor and the least common multiple, and Euclid algorithm for fractions.

• Education of Primary School Mathematics
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• v.22 no.2
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• pp.113-127
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• 2019

The purpose of this study is to justify the fraction division algorithm in elementary mathematics by applying the definition of natural number division to fraction division. First, we studied the contents which need to be taken into consideration in teaching fraction division in elementary mathematics and suggested the criteria. Based on this research, we examined whether the previous methods which are used to derive the standard algorithm are appropriate for the course of introducing the fraction division. Next, we defined division in fraction and suggested the unit-circle partition model and the square partition model which can visualize the definition. Finally, we confirmed that the standard algorithm of fraction division in both partition and measurement is naturally derived through these models.

• Proceedings of the IEEK Conference
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• 2003.07a
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• pp.109-112
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• 2003

본 논문에서는 확장 유클리드 알고리즘을 이용하여 VLSI 구현에 적합한 GF(2/sup m/)에서의 나눗셈 알고리즘을 제안하였다. 제안하는 나눗셈 알고리즘은 GF(2/sup m/)에서 2m-2번의 반복적인 비트 연산을 필요로 하며 입력 데이터에 의존적인 하드웨어 구조를 새로운 (m+1)-bit의 유한체 G와 H를 도입하여 간단하게 제어하도록 구현하였다. 본 논문에서 제안하는 알고리즘은 유한체 곱셈과 나눗셈이 요구되는 Error Correction Code와 암호 알고리즘에 효율적으로 적용이 가능하다. 현재 대표적으로 사용되는 기존 나눗셈 알고리즘과 비교해 볼 때 연산 시간은 비슷하지만 2-bit의 제어신호만을 필요로 하기 때문에 입력 데이터에 독립적인 O(1)의 complexity를 가짐으로 O(log₂(m+1))의 컨트롤을 갖는 다른 두 알고리즘에 비해 하드웨어 리소스 면에서 월등한 결과를 보인다.

• Proceedings of the Korean Information Science Society Conference
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• 1999.10c
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• pp.3-5
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• 1999

나눗셈 알고리즘은 다른 덧셈이나 곱셈 알고리즘에 비해 복잡하고, 수행 빈도수가 적다는 이유로 그동안 고속 나눗셈의 하드웨어 연구는 활발하지 않았다. 그러나 멀티미디어의 발전 및 고성능의 그래픽 랜더링을 위한 보다 빠른 부동소수점연산기(FPU)가 필요하게 되었으며, 이에 따라서 고속의 나눗셈 연산기의 필요성이 증가하게 되었다. 특히, 전체의 수행 시간 향상을 위해서라도 고속 나눗셈 연산기의 중용성은 더욱 부각되고 있다. 그러나 고속 나눗셈 연산기는 연산 속도와 크기라는 서로 상반되는 요소를 가지고 있다. 즉, 연산 속도가 빠르면 크기는 늘어나고, 크기를 줄이면 연산 속도는 늦어지게 된다. 본 논문은 높은 자릿수(Very-High Radix) 나눗셈 알고리즘에서 영역변환상수를 구하는 방법으로 연산이 아닌 검색테이블(Look-up Table)을 이용한다. 그리고 검색테이블의 크기를 줄이는 방법으로 영역변환상수의 범위 분석 및 캐리 저장형을 이용한 검색테이블 분할 방법을 이용하였다. 전체적으로는 영역변환상수를 구하는 연산주기가 필요없게 되므로 나눗셈 연산기의 영역 크기의 변화가 적으면서 연산 속도는 빨라졌음을 알 수 있다.

• Proceedings of the Korean Information Science Society Conference
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• 1998.10a
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• pp.44-47
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• 1998

나눗셈 알고리즘은 다른 덧셈이나 곱셈 알고리즘과 비교하여 복잡하고, 수행빈도수 적다는 이류로 그 동안 고속 나눗셈의 하드웨어 연구는 활발하지 않았다. 그러나 멀티미디어의 발전으로 고속 나눗셈의 필요성 및 전체적인 수행 시간 향상을 위해 고속 나눗셈 연산기의 중요성은 더욱 부각되고 있다. 그러나 칩의 크기는 제작 단가와 깊은 관련이 있기 때문에 고속 나눗셈 연산기를 칩으로 제작할 때 요구되는 성능과 비용을 만족하기 위한 적절한 분석이 필요하다. 본 논문은 자릿수 순환(Digt Recurrence) 알고리즘에서 속도가 빠른 높은 자릿수 이용(Very-High Radix) 알고리즘을 기반으로 최적화된 자릿수 (Radix) 범위를 제시하였다. 그리고 변환요소 (Scaling Factor)를 전처리(Pre-processing)하여 연산의 주기를 감소하고, 크기의 문제를 해결하기 위해서 상수표 대신 제어(Control)방법으로 값을 구하는 방법을 설계하였다.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.19 no.10
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• pp.2341-2349
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• 2015

The commonly used Goldschmidt's floating-point divider algorithm performs two multiplications in one iteration. In this paper, a tentative error corrected K'th Goldschmidt's floating-point number divider algorithm which performs K times multiplications in one iteration is proposed. Since the number of multiplications performed by the proposed algorithm is dependent on the input values, the average number of multiplications per an operation in single precision and double precision divider is derived from many reciprocal tables with varying sizes. In addition, an error correction algorithm, which consists of one multiplication and a decision, to get exact result in divider is proposed. Since the proposed algorithm only performs the multiplications until the error gets smaller than a given value, it can be used to improve the performance of a divider unit. Also, it can be used to construct optimized approximate reciprocal tables.

• Education of Primary School Mathematics
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• v.16 no.2
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• pp.93-106
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• 2013

A large part of students' difficulties with fractional division algorithms in the current algorithm textbooks, seem to be due to self-induction methods. Through concrete analysis of surveys and interviews, we confirmed the educational value of fractional algorithms used to elicit alternative ways of context of determination of a unit rate. In addition, we suggested alternative methods based on the results of the teaching methods and curriculum configuration.

• Education of Primary School Mathematics
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• v.20 no.1
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• pp.69-84
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• 2017

The purpose of this study is to review how to introduce a division algorithm in mathematics textbooks which were applied 2009 revised curriculum. As a result, the textbooks do not introduce the algorithm in the context of division by equal part. The standardized division algorithm was introduced apart from the stepwise division algorithms and there is no explanation in between them. And there is a lack connectivity between activities and algorithms. This study is expected to help new curriculum and textbook to introduce division algorithm in proper way.

• Education of Primary School Mathematics
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• v.24 no.2
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• pp.103-114
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• 2021

In this study I recognized the problems with the use of the terms 'quotient' and 'reminder' in the division of decimal and explored ways to improve them. The prior studies and current textbooks critically analyzed because each researcher has different views on the use of the terms 'quotient' and 'reminder' because of the same view of the values in the division calculation. As a result of this study, I proposed to view the result 'q' and 'r' of division of decimals by division algorithms b=a×q+r as 'quotient' and 'reminder', and the amount equal to or smaller to q the problem context as a final 'result value' and the residual value as 'remained value'. It was also proposed that the approximate value represented by rounding the quotient should not be referred to as 'quotient'.

• Journal of Elementary Mathematics Education in Korea
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• v.20 no.4
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• pp.521-539
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• 2016

The structures of partitive and quotitive division of fractions are dealt with differently, and this led to using partitive division context for helping develop invert-multiply algorithm and quotitive division for common denominator algorithm. This approach is unlikely to provide children with an opportunity to develop an understanding of common structure involved in solving different types of division. In this study, I propose two approaches, measurement approach and isomorphism approach, to develop a unifying understanding of fraction division. From each of two approaches of solving quotitive division based on proportional reasoning, I discuss an idea of constructing a measure space, unit of which is a quantity of divisor, and another idea of constructing an isomorphic relationship between the measure spaces of dividend and divisor. These ideas support invert-multiply algorithm for quotitive as well as partitive division and bring proportional reasoning into the context of fraction division. I also discuss some curriculum issues regarding fraction division and proportion in order to promote the proposed unifying understanding of partitive and quotitive division of fractions.