• Title/Summary/Keyword: 합동공리

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A Study on the Comparison of Triangle Congruence in Euclidean Geometry (유클리드 기하학에서 삼각형의 합동조건의 도입 비교)

  • Kang, Mee-Kwang
    • The Mathematical Education
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    • v.49 no.1
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    • pp.53-65
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    • 2010
  • The congruent conditions of triangles' plays an important role to connect intuitive geometry with deductive geometry in school mathematics. It is induced by 'three determining conditions of triangles' which is justified by classical geometric construction. In this paper, we analyze the essential meaning and geometric position of 'congruent conditions of triangles in Euclidean Geometry and investigate introducing processes for them in the Elements of Euclid, Hilbert congruent axioms, Russian textbook and Korean textbook, respectively. Also, we give justifications of construction methods for triangle having three segments with fixed lengths and angle equivalent to given angle suggested in Korean textbooks, are discussed, which can be directly applicable to teaching geometric construction meaningfully.

Proof of the Pythagorean Theorem from the Viewpoint of the Mathematical History (수학사적 관점에서 본 피타고라스 정리의 증명)

  • Choi, Young-Gi;Lee, Ji-Hyun
    • School Mathematics
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    • v.9 no.4
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    • pp.523-533
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    • 2007
  • This article focused the meaning of Pythagoras' and Euclid's proof about the Pythagorean theorem in a historical and mathematical perspective. Pythagoras' proof using similarity is based on the arithmetic assumption about commensurability. However, Euclid proved the Pythagorean theorem again only using the concept of dissection-rearrangement that is purely geometric so that it does not need commensurability. Pythagoras' and Euclid's different approaches to geometry have to do with Birkhoff's axiom system and Hilbert's axiom system in the school geometry Birkhoff proposed the new axioms for plane geometry accepting real number that is strictly defined. Thus Birkhoff's metrical approach can be defined as a Pythagorean approach that developed geometry based on number. On the other hand, Hilbert succeeded Euclid who had pursued pure geometry that did not depend on number. The difference between the proof using similarity and dissection-rearrangement is related to the unsolved problem in the geometry curriculum that is conflict of Euclid's conventional synthetical approach and modern mathematical approach to geometry.

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An axiomatic analysis on contents about the area of plane figures in the elementary school mathematics (초등학교 수학에서의 넓이 지도 내용에 대한 공리적 해석)

  • Do, Jong Hoon;Park, Yun Beom
    • Education of Primary School Mathematics
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    • v.17 no.3
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    • pp.253-263
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    • 2014
  • In this paper we review an axiomatic definition of the area of plane figures with area axioms, discuss what the area axioms mean, and analyze the contents about the area of plane figures in elementary school mathematics from the view point of area axioms. So we evaluate which aspects of the concept of area are emphasized or deemphasized in the current elementary school mathematics textbook.

A historical study of de Zolt's axiom (졸트 공리의 역사적 고찰)

  • Jo, Kyeonghee
    • Journal for History of Mathematics
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    • v.30 no.5
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    • pp.261-287
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    • 2017
  • De Zolt's axiom which is a precise formulation of Euclid's Common Notion 5, "the whole is greater than the part", for the notion of 'content' holds in any Hilbert plane. In this article, we study the history of de Zolt's axiom which has its origin in Euclid's Common Notions, and introduce an example of a plane geometry in which de Zolt's axiom does not hold. We show that there is no area function in this geometry and every square is equidecomposable with a square which is properly contained in the first one. From this we also show that there are two equidecomposable rectangles which have the same base and do not have the same altitude, and there is a rectangle which is equicomplementable with an emptyset.