• 한국수학교육학회지시리즈A:수학교육
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• 제41권2호
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• pp.227-231
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• 2002

The Pythagorean theorem and Pythagorean triple are well known. We know some Pythagorean triples, however we don't Cow well that every natural number can belong to some Pythagorean triple. In this paper, we show that every natural number, which is not less than 2, can be a length of a leg(a side opposite the acute angle in a right triangle) in some right triangle, and list some Pythagorean triples.

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.333-351
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• 2024

In this paper, the concept of Pythagorean fuzzy sets are used to describe in semigroups. Then, some characterizations of regular (resp., intra-regular) semigroups by means of Pythagorean fuzzy left (resp., right) ideals and Pythagorean fuzzy (resp., generalized) bi-ideals of semigroups are investigated. Furthermore, the class of both regular and intra-regular semigroups by the properties of many kinds of their Pythagorean fuzzy ideals also being studied.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.521-528
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• 2007

We present a new proof of the characterization theorem for Minkowski Pythagorean-hodograph curves in the Minkowski spaces $\mathbf{R}^{n+1,m}$. For an polynomial curves $\mathbf{s}(t)=(x_1(t),...,\;x_{n+m}(t))$, we also find Minkowski Pythagorean-hodograph curves $\mathbf{r}(t)=(x_0(t),\;x_1(t),...,\;x_{n+m}(t))$. In case m=0, Minkowski Pythagorean-hodograph curves become Pythagorean-hodograph curves in the Euclidean spaces $\mathbf{R}^{n+1}$ and Theorems in this paper hold for these Pythagorean-hodograph curves.

• East Asian mathematical journal
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• 제28권1호
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• pp.13-23
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• 2012

In this paper, we present the geometric Hermite interpolation for planar Pythagorean-hodograph cubics for some general Hermite data.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제25권1호
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• pp.147-166
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• 2011

본 고에서는 중학교 3학년 영재학생 5명을 대상으로 한 실험 수업 결과를 바탕으로 스프레드시트 환경에서 학생들이 생성수의 조건 변화에 따른 피타고라스 삼원수의 다양한 성질을 탐구할 수 있었음을 논하였고, 스프레드시트 환경이 스프레드시트의 구체적 수치로부터 학생이 발견하고 증명한 성질을 보다 일반적인 경우로 확장할 수 있는 기회를 줄 수 있었음을 논하였다. 또한 피타고라스 상원수의 다양한 성질을 탐구함에 있어서 교사의 적절한 안내가 필요함을 함께 논하였다.

• 한국수학교육학회지시리즈A:수학교육
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• 제57권2호
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• pp.93-110
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• 2018

One of the biggest changes in the 2015 revised mathematics curriculum is shifting to the second year of middle school in Pythagorean theorem. In this study, the following subjects were studied. First, Pythagoras theorem analyzed the expected problems caused by the shift to the second year middle school. Secondly, we have researched the reconstruction method to solve these problems. The results of this study are as follows. First, there are many different ways to deal with Pythagorean theorem in many countries around the world. In most countries, it was dealt with in 7th grade, but Japan was dealing with 9th grade, and the United States was dealing with 7th, 8th and 9th grade. Second, we derived meaningful implications for the curriculum of Korea from various cases of various countries. The first implication is that the Pythagorean theorem is a content element that can be learned anywhere in the 7th, 8th, and 9th grade. Second, there is one prerequisite before learning Pythagorean theorem, which is learning about the square root. Third, the square roots must be learned before learning Pythagorean theorem. Optimal positions are to be placed in the eighth grade 'rational and cyclic minority' unit. Third, Pythagorean theorem itself is important, but its use is more important. The achievement criteria for the use of Pythagorean theorem should not be erased. In the 9th grade 'Numbers and Calculations' unit, after learning arithmetic calculations including square roots, we propose to reconstruct the square root and the utilization subfields of Pythagorean theorem.

• 대한수학교육학회지:학교수학
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• 제4권3호
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• pp.347-360
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• 2002

This study analyzed the mathematical and didactical contexts of the Euclid's proof about Pythagorean theorem and compared with the teaching methods about Pythagorean theorem in school mathematics. Euclid's proof about Pythagorean theorem which does not use the algebraic methods provide students with the spatial intuition and the geometric thinking in school mathematics. Furthermore, it relates to various mathematical concepts including the cosine rule, the rotation, and the transfor-mation which preserve the area, and so forth. Visual demonstrations can help students analyze and explain mathematical relationship. Compared with Euclid's proof, Algebraic proof about Pythagorean theorem is very simple and it supplies the typical example which can give the relationship between algebraic and geometric representation. However since it does not include various spatial contexts, it forbid many students to understand Pythagorean theorem intuitively. Since both approaches have positive and negative aspects, reciprocal complementary role is required in pedagogical aspects.

• 한국수학사학회지
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• 제33권3호
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• pp.155-165
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• 2020

Pythagorean theorem is a proposition on the relationship between the lengths of three sides of a right triangle. It is well known that Pythagorean theorem for Euclidean geometry deforms into an interesting form in non-Euclidean geometry. In this paper, we investigate a new perspective that replaces right triangles with 'proper triangles' so that Pythagorean theorem extends to non-Euclidean geometries without any modification. This is seen from the perspective that a rectangle is an equiangular quadrilateral, and a right triangle is a half of a rectangle. Surprisingly, a proper triangle (defined by Paolo Maraner), which is a half of an equiangular quadrilateral, satisfies Pythagorean theorem in many geometries, including hyperbolic geometry and spherical geometry.

• Journal of applied mathematics & informatics
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• 제42권2호
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• pp.433-444
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• 2024

Our study explores the connection between the Pythagorean theorem and the Fixed-point theorem in metric spaces. Both of which center around the concepts of distance transformations and point relationships. The Pythagorean theorem deals with right triangles in Euclidean space, emphasizing distances between points. In contrast, fixed-point theorems pertain to the points that remain unchanged under specific transformations thereby preserving distances. The article delves into the intrinsic correlation between these concepts and presents a novel study in Euclidean metric spaces, examining the relationship between contraction mapping and Pythagorean Right Triangles. Practical applications are also discussed particularly in the context of image compression. Here, the integration of the Pythagorean right triangle paradigm with contraction mappings results in efficient data representation and the preservation of visual data relation-ships. This illustrates the practical utility of seemingly abstract theories in addressing real-world challenges.

• East Asian mathematical journal
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• 제29권1호
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• pp.53-68
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• 2013

We solve the geometric Hermite interpolation problem with planar Pythagorean-hodograph cubics. For every Hermite data, we determine the exact number of the geometric Hermite interpolants and represent the interpolants explicitly. We also present a simple criterion for determining whether the interpolants have a loop or not.