• Journal for History of Mathematics
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• v.16 no.4
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• pp.79-90
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• 2003

This paper aims to consider the genesis of irrational numbers and to suggest a method for teaching the concept of irrational numbers. It is the notion of “incommensurability” in geometrical sense that makes Pythagoreans discover irrational numbers. According to the historica-genetic principle, the teaching method suggested in this paper is based on the very concept, incommensurability which the school mathematics lacks. The basic ideas are induced from Clairaut's and Arcavi's.

• School Mathematics
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• v.4 no.4
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• pp.643-655
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• 2002

In this paper we emphasize the introduction of ‘incommensurability’ on the teaching and learning the irrational number because we think of the origin of number as ‘ratio’. According to Greek classification of continuity as a ‘never ending’ divisibility, discrete number and continuous magnitude belong to another classes. That is, those components were dealt with respectively in category of arithmetic and that of geometry. But the comparison between magnitudes in terms of their ratios took the opportunity to relate ratios of magnitudes with numerical ratios. And at last Stevin coped with discrete and continuous quantity at the same time, using his instrumental decimal notation. We pay attention to the fact that Stevin constructed his number conception in reflecting the practice of measurement : He substituted ‘subdivision of units’ for ‘divisibility of quantities’. Number was the result of such a reflective abstraction. In other words, number was invented by regulation of measurement. Therefore, we suggest decimal representation from the point of measurement, considering the foregoing historical development of number. From the perspective that the conception of real number originated from measurement of ‘continuum’ and infinite decimals played a significant role in the ‘representation’ of measurement, decimal expression of real number should be introduced through contexts of measurement instead of being introduced as a result of algorithm.

• 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.

• Journal for History of Mathematics
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• v.21 no.3
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• pp.53-66
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• 2008

The concepts of ratio and proportion are familiar with students but have difficulties in use. The purpose of this paper is to identify the meanings of the concepts of ratio and proportion through investigating the historical development process of the meanings and representations of them. The early meanings of ratio and proportion were arithmetical meanings, however, geometrical meanings had taken the place of them because of the discovery of incommensurability. After the development of algebraic representation, the meanings of ratio and proportion have been growing into algebraic meanings including arithmetical and geometrical meanings. Through the historical development process of ratio and proportion, it is observable that the meanings of mathematical concepts affect development of symbols, and the development of symbols also affect the meanings of mathematical concepts.

• Journal of Educational Research in Mathematics
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• v.17 no.4
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• pp.437-452
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• 2007

Complementarity, complementary principle and complementary approach have been often used in school mathematics but its meaning has not been obvious. Thus this paper tries to make explicit the meaning by looking around complementary characteristic of mathematical knowledge. First of all, we examines the general meaning of complementarity and Investigate complementary characteristics of mathematical concepts through incommensurability and zeno's paradox. From this, complementary approach to school mathematics is studied. To understand and uncover complementary characteristics of mathematical concepts make it possible for student to have an insight. It is the most important thing that students can have an image of mathematics as a living system rather than as a mechanical application of rules and fragmentary in formations.

• 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.

• Proceedings of the Polymer Society of Korea Conference
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• 2006.10a
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• pp.294-294
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• 2006

Diblock copolymers whose morphology in the bulk is dictated by the volume fraction of the components and segmental interactions were confined within nanoscopic cylindrical pores. Since the confining geometry is nonplanar and nanoscopic, the extreme imposed curvature, comparable to molecular dimensions, places significant packing frustration on the chains. When incommensurability between the repeating period of diblock copolymers and the diameter of nanopore is coupled with the curvature, it causes the marked departures from bulk or even thin film behavior. The entropy penalty from the constraints and the curvature of the physical confinement determines unique nanostructures available only with this curved confinement.

• Journal for History of Mathematics
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• v.18 no.3
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• pp.55-66
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• 2005

In our school mathmatics, the irrational numbers and the real numbers are defined and instructed on the basis of decimal fractions. In relation to this fact, we identified the essences of the real number and the irrational number defined by the decimal fractions through the historical analysis. It is revealed that the formation of real numbers means the numerical measurements of all magnitudes and the formation of irrational numbers means the numerical measurements of incommensurable magnitudes. Finally, we suggest instructional plan for the meaninful understanding of the real number concept.

• Korean Journal of Crystallography
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• v.1 no.1
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• pp.1-7
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• 1990

It is shown that the symmetry of the incommensurate phase in Ba2NaNb5O15(BSH) belongs to the four dimensional superspace group C from an analysis of the systematic extinction or the rxtra reflections due to the incommensllrate structure. The resulting superstructure is characterizetl by the space group Ima2(C3l) and the origin of the incommensurability in BSN is briefly discussed. Especially. HREM image have shown the prrscnce of discommensurations. In addition. a group-theoretical consideration of structural phase transitions in BSN is suggested.

• School Mathematics
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• v.10 no.1
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• pp.123-138
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• 2008

The objective of this paper is to reveal the cause of failure of New Math in the field of the SMSG area education from the didactical point of view. At first, we analyzed Euclid's (Elements), De Morgan's (Elements of arithmetic), and Legendre's (Elements of geometry and trigonometry) in order to identify characteristics of the area conception in the SMSG. And by analyzing the controversy between Wittenberg(1963) and Moise(1963), we found that the elementariness and the mental object of the area concept are the key of the success of SMSG's approach. As a result, we conclude that SMSG's approach became separated from the mathematical contents of the similarity concept, the idea of same-area, incommensurability and so on. In this account, we disclosed that New Math gave rise to the lack of elementariness and geometrical mental object, which was the fundamental cause of failure of New Math.