• Journal of the Korean School Mathematics Society
• /
• v.20 no.4
• /
• pp.495-519
• /
• 2017

The purpose of this study is to analyze the concept of the real number e and to investigate the understanding of pre-service teachers about the real number e. 28 pre-service teachers were asked to take a test based on the various ideas of the real number e and 8 pre-service teachers were interviewed. The results of this study are as follows. First, a large number of pre-service teachers couldn't recognize relation between the formal definition and the representations of the real number e. Secondly, pre-service teachers judged appropriately for the irrationality and the construction impossibility of the real number e, but they couldn't provide reasonable evidence. Lastly, pre-service teachers understood the continuous compounding context and exponential function context of the real number e, but they had a difficulty in understanding the geometric context and natural logarithm context of the real number e.

• Nuclear Engineering and Technology
• /
• v.23 no.2
• /
• pp.174-182
• /
• 1991

Effects of human error relevant to the periodic test are incorporated in the evaluations of the unavailability and optimal test interval of a safety system. Two types of possible human error with respect to the test and maintenance are considered. One is the possibility that a good safety system is inadvertently left in a bad state after test（Type A human error) and the other is the possibility that a bad safety system is undetected upon the test（Type B human error). An event tree model is developed for the steady-state unavailability of a safety system in order to determine the effects of human errors on the system unavailability and the optimal test interval. A reliability analysis of the Safety Injection System (SIS) was peformed to evaluate the effects of human error on the SIS unavailability. Results of various sensitivity analyses show that ; (1) the steady-state unavailability of the safety system increases as the probabilities of both types of human error increase and it is far more sensitive to Type A human error, (2) the optimal test interval increases slightly as the probability of Type A human error increases but it decreases as the probability of Type B human error increases, and (3) provided that the test interval of the safety injction pump is kept unchanged, the unavailability of SIS increases significantly as the probability of Type A human error increases but slightly as the probability of Type B human error increases. Therefore, to obtain the realistic result of reliability analysis, one should take shorter test interval (not optimal test interval) so that the unavailability of SIS can be maintained at the same level irrespective of human error. Since Type A human error during test & maintenance influeces greatly on the system unavailability, special efforts to reduce the possibility of Type A human error are essential in the course of test & maintenance.

• Communications of Mathematical Education
• /
• v.19 no.1 s.21
• /
• pp.45-55
• /
• 2005

대학수학 1학년 과정(미분적분학)에서 정리, 정의 등 개념의 이해를 도와주기 위해 학생들이 갖는 어려움을 그들이 자주 겪는 실수를 통해 찾아내어 분석하고 올바른 이해의 길로 안내한다. 실수를 탓하기보다 학생의 편에 서서 이해하고 도움을 주도록 한다. 흔히 부딪칠 수 있는 예제 문제를 풀어보게 하고 공통으로 저지르는 실수를 제시하여 개념의 이해나 문제풀이를 바르게 하도록 이끌어 준다.

• Journal for History of Mathematics
• /
• v.18 no.3
• /
• pp.55-66
• /
• 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 Logic
• /
• v.14 no.3
• /
• pp.137-176
• /
• 2011

Bob Hale recently proposed a theory of real numbers based on abstraction principles. In his theory, real numbers are regarded as ratios of quantities and the criteria of identities of ratios of quantities are given by an Eudoxan ratio principle. The reason why Hale defines real numbers as ratios of quantities is that he wants to satisfy Frege's requirement that arithmetical concepts should be defined to be adequate for their universal applicability. In this paper I show why Hale's explanation of applications of real numbers fails to satisfy Frege's requirement, and I propose an alternative explanation. At first I show that there are some gaps between his explanation of the concept of quantity and his stipulation of domains of quantities, and that those gaps give rise to some difficulties in his explanation of applications of real numbers. Secondly I introduce a new ratio principle which can be applied to any kinds of quantities, and I argue that it allows us an adequate explanation of the reason why real numbers as ratios of quantities can be universally applicable. Finally I enquire into some procedures of the measurement of quantities, and I propose some principles which we should presuppose in order to successfully apply real numbers to the measurement of quantities.

• Journal of Korea Society of Industrial Information Systems
• /
• v.18 no.6
• /
• pp.31-37
• /
• 2013

Modern graphic processors, multimedia processors and audio processors mostly use floating-point number. High-level language such as C and Java use both single precision and double precision floating-point number. In this paper, an algorithm which computes the reciprocal of double precision floating-point number using a 32 bit multiplier is proposed. It divides the mantissa of double precision floating-point number to upper part and lower part, and calculates the reciprocal of the upper part with Newton-Raphson algorithm. And it computes the reciprocal of double precision floating-point number with calculated upper part reciprocal as the initial value. Since the number of multiplications performed by the proposed algorithm is dependent on the mantissa of floating-point number, the average number of multiplications per an operation is derived from some reciprocal tables with varying sizes.

• Proceedings of the Korean Institute of Intelligent Systems Conference
• /
• 1996.10a
• /
• pp.115-118
• /
• 1996

연관규칙(Association Rule)은 데이터 베이스에 존재하는 속성들 사이의 관계를 기술하는 것으로, 간단하면서도 사용자에게 많은 정보를 줄 수 있다. 그러나, 지금까지는 이진 데이터베이스에 존재하는 연관규칙의 발견에 대해서 주로 연구되어 왔으며, 실수값 속성을 갖는 데이터에 관한 연구는 미비하였다. 본 논문에서는 퍼지집합을 이용하여 실수값 사이에 존재하는 연관규칙을 기술하고, 그것을 찾아내는 방법을 제시한다. 제시하는 방법은 사용자에 의해서 정의된 언어항을 이용하여, 실수값 속성을 가진 데이터를 이진 데이터로 재구성한다. 그리고 재구성된 이진 데이터에 기존의 연관규칙 발견 방법을 이용하여 연관규칙을 찾아내고, 찾아진 연관규칙을 정의된 언어항을 이용하여 다시 기술한다.

• Journal of Educational Research in Mathematics
• /
• v.25 no.3
• /
• pp.263-279
• /
• 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.

• Journal for History of Mathematics
• /
• v.25 no.3
• /
• pp.53-75
• /
• 2012

When imaginary numbers were first encountered in the 16th century, mathematicians were able to calculate the imaginary numbers the same as they are today. However, it required 200 years to mathematically acknowledge the existence of imaginary numbers. The new mathematical situation that arose with a development in mathematics required a harmony of real numbers and imaginary numbers. As a result, the concept of complex number became clear. A history behind the development of complex numbers involved a process of determining a comprehensive perspective that ties real numbers and imaginary numbers in a single category, complex numbers. This came after a resolution of conflict between real numbers and imaginary numbers. This study identified the new perspective and way of mathematical thinking emerging from resolving the conflicts. Also educational implications of the analysis were discussed.

• Computational Structural Engineering
• /
• v.8 no.2
• /
• pp.123-132
• /
• 1995

Genetic Algorithms(GA), which are based on the theory of natural evolution, have been evaluated highly for their robust performances. Traditional GA has mostly used binary code for representing design variable. The binary code GA has many difficulties to solve optimization problems with continuous design variables because of its large computer core memory size, inefficiency of its computing time, and its bad performance on local search. In this paper, a real code GA is proposed for dealing with the above problems. So, new crossover and mutation processes of GA are developed to use continuous design variables directly. The results of read code GA are compared with those of binary code GA for several single and multiple objective optimization problems. As a result of comparisons, it is found that the performance of the real code GA is better than that of the binary code GA, and concluded that the real code GA developed here can be used for the general optimization problem.