• Journal for History of Mathematics
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• v.23 no.3
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• pp.57-73
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• 2010

Transcendental numbers are important in the history of mathematics because their study provided that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was insoluble. Liouville established in 1844 that transcendental numbers exist. In 1874, Cantor published his first proof of the existence of transcendentals in article [10]. Louville's theorem basically can be used to prove the existence of Transcendental number as well as produce a class of transcendental numbers. The number e was proved to be transcendental by Hermite in 1873, and $\pi$ by Lindemann in 1882. In 1934, Gelfond published a complete solution to the entire seventh problem of Hilbert. Within six weeks, Schneider found another independent solution. In 1966, A. Baker established the generalization of the Gelfond-Schneider theorem. He proved that any non-vanishing linear combination of logarithms of algebraic numbers with algebraic coefficients is transcendental. This study aims to examine the concept and development of transcendental numbers and to present students with its open problems promoting a research on it any further.

• Journal of Educational Research in Mathematics
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• v.16 no.4
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• pp.297-312
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• 2006

As research on the instruction method of the concept of irrational numbers, this thesis is theoretically based on the Freudenthal's Mathematising Instruction Theory and a conducted case study in order to find an introduction method of irrational numbers. The purpose of this research is to provide practical information about the instruction method ?f irrational numbers. For this, research questions have been chosen as follows: 1. What is the introducing method of irrational numbers based on the Freudenthal's Mathematising Instruction Theory? 2 What are the Characteristics of the teaming process shown in class using introducing instruction of irrational numbers based on the Freudenthal's Mathematising Instruction? For questions 1 and 2, we conducted literature review and case study respectively For the case study, we, as participant observers, videotaped and transcribed the course of classes, collected data such as reports of students' learning activities, information gathered through interviews, and field notes. The result was analyzed from three viewpoints such as the characteristics of problems, the application of mathematical means, and the development levels of irrational numbers concept.

• Journal of Animal Environmental Science
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• v.7 no.3
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• pp.183-190
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• 2001

The objective of this research was to design and construct an image processing system to measure easily and accurately cow's weight. The image processing system was built for a dairy cattle to be measured and estimated it's weight using camera and personal computer. The pixel numbers, which was derived from the image processing system, were counted to estimate the weight of a dairy cattle. They were utilized various was for finding the relationships between pixel numbers and it's real weight. Based on the results of this research the following conclusions were made: 1. It's weight could be estimated by using pixel numbers, which was captured from top and side cameras to measure it. The correlations with tea-view pixel numbers, side-view pixel numbers, superficial area pixel numbers and the volume pixel numbers were 0.909, 0.939, 0.944 and 0.965. 2. 50 cattle was used to execute an experiment with the image processing system, but average errors were big to make out the good relationship between cow's weight and pixel numbers. In order measure accurately a cattle weight, cattle weight, cattle groups would be divided by the age of cattle and further study should be carried out to be based on the results of this research. 3. The average time it took to perform the image processing to be measure it was 10 seconds, but it took 10 minutes for cattle to enter for measuring it's weight into the weighting system.

• Korean Journal of Logic
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• v.14 no.3
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• pp.137-176
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• 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 for History of Mathematics
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• v.25 no.4
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• pp.1-10
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• 2012

In Chinese mathematics, there have been two numeral systems, namely one in spoken language for recording and the other by counting rods for computations. They concerned with problems dealing with practical applications, numbers in them are concrete numbers except in the process of basic operations. Thus they could hardly develop a pure theory of numbers. In Song dynasty, 0 and TianYuanShu were introduced, where the coefficients were denoted by counting rods. We show that in this process, counting rods took over the role of the numeral system in spoken language and hence counting rod numeral system plays the role of that for abstract numbers together with the tool for calculations. Decimal fractions were also understood as denominate numbers but using the notions by counting rods, decimals were also admitted as abstract numbers. Noting that abacus replaced counting rods and TianYuanShu were lost in Ming dynasty, abstract numbers disappeared in Chinese mathematics. Investigating JianJie YiMing SuanFa(簡捷易明算法) written by Shen ShiGui(沈士桂) around 1704, we conclude that Shen noticed repeating decimals and their operations, and also used various rounding methods.

• Bulletin of the Korean Mathematical Society
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• v.42 no.3
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• pp.617-622
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• 2005

In this paper, using Gauss multiplication formula, a recurrence relation for Bernoulli numbers, generalizing Namias' results, is given.

• Korean Journal of Mathematics
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• v.9 no.2
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• pp.83-98
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• 2001

We characterize ($r,w$)-nuclear operators acting from a Banach space whose dual has weak type $q$ into a Banach space of weak type $p$ by the asymptotic behaviour of their entropy numbers.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.39-51
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• 2015

The kt + r subscripted tribonacci numbers will be expressed by three k step apart tribonacci numbers for any 0 < r < $k{\in}\mathbb{Z}$.

• Journal of the Korean Data and Information Science Society
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• v.9 no.2
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• pp.299-303
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• 1998

In this paper weak laws of large numbers are obtained for random variables in $L^{1}(R)$ which satisfy a compact uniform integrability condition.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.443-450
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• 2016

In this paper we give some properties, explicit formulas, several identities, a connection with tangent numbers and polynomials, and some integral formulas.