• 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 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.

• School Mathematics
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• v.10 no.3
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• pp.357-374
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• 2008

The complex number system is supposed to introduce first chapter in the first grade of high school. When number system is expanded to complex numbers, the main aim is to understand preservation of algebraic structure with regard to the flow of curriculum and textbook. This research reviewed overall alternative articulation and treatment of textbooks from a logical viewpoint. Two research questions are developed below. First, in the structure of the current curriculum, when we consider student's 'level', how are the alternative articulation and treatment of textbooks in complex unit on a logical point of view? Second, What are more logical alternative articulation and treatment? What alternative articulation and treatment are suitable for a running goal? and what are the improvement which is definitive?

• The Mathematical Education
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• v.14 no.1
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• pp.19-21
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• 1975

약 위상과 강위상이 주어진 선형공간 E에서 수열의 약수렴과 강수렴에 관한 성질을 보조정리로하여 두 Banach공간 E와 F사이에 정의된 연속작용소 공간 L(E.F)에서 작용소열 { $T_{n}$ }이 T에 강수렴하기 위한 필요충분조건은 F가 Hilbert 공간일때는 { $T_{n}$ }이 T에 약수렴함과 동시에 실수열 (equation omitted)$T_{\chi}$ $n^{\chi}$(equation omitted)가 실수 (equation omitted)T$\chi$(equation omitted)에 수렴할 때이다.

• The Mathematical Education
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• v.15 no.1
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• pp.5-7
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• 1976

Euclid 기하학이 성립하는 공간은 우리들과 가장 밀접한 공간이다. Descartes의 해석기하학은 Euclid의 3차원공간에서 성립한다. 이 경우 점이라 해도 그것은 3개의 실수의 순서쌍(x, y, z)에 의해 표현되는 것으로 생각해도 좋다. 일반의 n차원 Euclid 공간 R$^n$에 대해서도 같은 생각으로 정의할 수 있다. 이 경우 n=1은 수치선, n=2는 평면, n=3은 소위 3차원의 공간으로서 직관적으로 상상할 수 있으나 n(equation omitted)4인 경우는 상상하기 어렵다. 여기서는 거리의 성질과 추상공간을 논하고 Euclid 공간의 거리에서 출발하여 그 성질중 삼각부등식을 계산을 통하여 증명하므로서 공간의 확장화가 이루워짐을 보였다.

• Proceedings of the Korean Geotechical Society Conference
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• 2005.03a
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• pp.499-506
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• 2005

지반오염조사에 대한 유전상수 측정기법의 적용성을 평가하기 위하여 중금속 오염도에 따른 흙의 유전특성 변화를 분석하였다. 유전상수의 실수부와 허수부 모두 체적함수비에 따른 증가경향을 나타내었으며, 특히 MHz 범위에서 유전상수 실수부는 쌍극자모멘트에 비례하기 때문에 흙의 유전상수는 체적함수비에 따른 선형적인 증가경향을 나타내었다. 중금속 용액은 50kHz 이하의 저주파영역에서 전극 분극효과에 의해 농도 증가에 따라 유전상수 실수부가 증가하는 경향을 나타내었으나, 고주파 영역에서는 이온의 수화작용에 의한 물분자의 배향분극 발현 감소로 인하여 유전상수 실수부가 감소하였다. 유전상수 허수부의 경우에는 모든 주파수 영역에서 중금속 농도 증가에 따른 전도손실에 의하여 증가하는 경향을 나타내었다. 흙과 중금속 혼합시료의 경우 함수비가 큰 시료에서는 중금속 용액 자체의 유전특성이 그대로 발현되었으나, 함수비가 작은 시료에서는 공간전하분극의 영향이 우세하여 유전상수 실수부가 10-20%정도 증가하는 경향을 나타내었다. 유전상수 허수부의 경우에는 중금속 농도 증가에 따른 뚜렷한 증가경향을 확인할 수 있었다. 본 연구의 결과에 의하면 중금속의 오염감지에 대해서는 유전상수 실수부보다는 허수부의 적용성이 높은 것으로 나타났으며, 현장에서의 정확한 오염도 평가를 위해서는 함수비에 대한 평가가 선행되어야 할 것으로 판단된다.

• Journal for History of Mathematics
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• v.22 no.3
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• pp.1-30
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• 2009

Frege's logicism has been frequently regarded as a development in number theory which succeeded to the so called arithmetization of analysis in the late 19th century. But it is not easy for us to accept this opinion if we carefully examine his actual works on real analysis. So it has been often argued that his logicism was just a philosophical program which had not contact with any contemporary mathematical practices. In this paper I will show that these two opinions are all ill-founded ones which are due to the misunderstanding of the theoretical place of Frege's logicism in the context of contemporary mathematical practices. Firstly, I will carefully examine Cantorian definition of real numbers and Frege's critiques of it. On the basis of this, I will show that Frege's aim was to produce the purely logical definition of ratios of quantities. Secondly, I will consider the mathematical background of Frege's logicism. On the basis of this, I will show that his standpoint in real analysis was much subtler than what we used to expect. On the one hand, unlike Weierstrass and Cantor, Frege wanted to get such real analysis that could be universally applicable. On the other hand, unlike most mathematicians who insisted on the traditional conceptions, he would not depend upon any geometrical considerations in establishing real analysis. Thirdly, I will argue that Frege regarded these two aspects - the independence from geometry and the universal applicability - as those which characterized logic itself and, by logicism, arithmetic itself. And I will show that his conception of real numbers as ratios of quantities stemmed from his methodological maxim according to which the nature of numbers should be explained by the common roles they played in various contexts to which they applied, and that he thought that the universal applicability of numbers could not be adequately explicated without such an explanation.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2002.05a
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• pp.114-117
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• 2002

현재 개발 중에 있는 비파괴 검사법인 Tapping Sound Analysis 의 실험적 검증을 위한 연구를 수행하였다. 손상이 없는 복합재료 구조물과 손상이 있는 복합재료 구조물에 대한 타격 실험을 통해 타격음과 타격력을 측정하여 비교하였다. Wavelet packet transform에 근거한 특성 추출법을 이용하여 타격음으로부터 손상 판단을 위한 특성을 추출하였다. 손상이 없는 구조물과 손상이 있는 구조물의 특성을 비교하기 위해, 특성 지수를 정의하였다. 정의된 특성 지수를 이용하여 손상이 없는 구조물과 손상이 있는 구조물의 타격음의 차이를 하나의 실수로 표현하였다.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.46 no.3
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• pp.10-19
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• 2009

In this paper, an efficient optical watermark method using multiple phase wrapping and real-valued decoding key is proposed. In the embedding process, two zero-padded original images placed in two quadrants on input plane are multiplied with two statistically independent random phase patterns and are Fourier transformed, respectively. Two encoded images are obtained by taking the real-valued data from these Fourier transformed images. And then two phase-encoded patterns, used as a hidden image and a decoding key, are generated by the use of multiple phase wrapping from each of the encoded images. A transmitted image is made from the linear superposition of the weighted hidden images and a cover image. In reconstruction process, the mirror reconstructed images can be obtained at all quadrants by the inverse-Fourier transform of the product of the transmitted image and the decoding key. Computer simulation and optical experiment are demonstrated in order to confirm the proposed method.

• Journal of the Korean Magnetics Society
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• v.4 no.3
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• pp.208-213
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• 1994

Wide-band absorbency and small layer thickness are required in the high- performance microwave absorber. In order to reduce the thickness of the absorber, ferrite-iron-rubber composites were proposed and its properties were investigated. The increase in both real part of perrreability and permittivity of the composites were observed with addition of -Fe powder and this behavior results in the decrease of fm dm term. The matching thicknesses of ferrite-iron-ruber composites were relatively small(0.5~2.0 mm) in the frequency range of 3~17 GHz as comIBfed to than that of ferrite-rubbr composit.