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Journal of Educational Research in Mathematics -
Vol. 31 ,
No. 1

[ Article ] | |

Journal of Educational Research in MathematicsVol. 30, No. 4, pp.575-599 | |

Abbreviation: JERM | |

ISSN: 2288-7733 (Print) 2288-8357 (Online) | |

Print publication date 30 Nov 2020 | |

Received 01 Sep 2020 Revised 06 Nov 2020 Accepted 13 Nov 2020 | |

DOI: https://doi.org/10.29275/jerm.2020.11.30.4.575 | |

A Historical and Mathematical Study on Continuum : Focusing on the Composition and Infinite Division | |

Baek, Seungju ^{*} ; Choi, Younggi^{**}^{, †}
| |

*Teacher, Sejong Science High School, South Korea (seung07@snu.ac.kr) | |

**Professor, Seoul National University, South Korea (yochoi@snu.ac.kr) | |

Correspondence to : ^{†}Professor, Seoul National University, South Korea, yochoi@snu.ac.kr | |

Abstract

This study conducted historical and mathematical analyses of the composition of continuum and infinite division. Regarding the results of the historical analysis, first, it can be seen that mathematicians struggled with the problems related to Zeno’s “paradox of plurality” in the composition and infinite division of the continuum. Second, historically, infinite small appeared frequently. Third, the problem of continuum, which was a geometric problem, was not explained by geometry alone, and mathematical answers to this problem became possible after the straight line became homomorphic with the real set, following the arithmetization of mathematics. Through mathematical analysis, Zeno's “paradox of plurality”, which includes questions about the composition and infinite division of a continuum, could be explained depending on the uncountability of real numbers and the countable additivity of Lebesgue measure. In addition, it was confirmed that it is inappropriate to divide a straight line to a point. The historical and mathematical analyses of this study suggest that students may have cognitive difficulties when dealing with the composition and infinite division of a continuum in relation to area, volume, and integral in school mathematics. The continuum is a concept closely related to the limit and calculus of school mathematics, and the historicalㆍmathematical analysis of this study could serve as a foundation for teaching and learning plans of limits and calculus.

Keywords: Continuum, Zeno, Paradox of Plurality, Infinitely Divisible, Uncountability of Real Number Set, Lebesgue Measure |

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Journal of Educational Research in Mathematics, #1305 Daewoo The'O Ville, 115 Hangang-daero, Yongsan-gu, Seoul, 04376, Republic of Korea (Tel: +82‐2‐797‐7780, Fax: +82‐2‐797‐7750).

Submitting manuscripts for peer review as well as other correspondences can either be made via online by an email (ksesm@daum.net) or sending a mail at Secretariat of

Journal of Educational Research in Mathematics, #1305 Daewoo The'O Ville, 115 Hangang-daero, Yongsan-gu, Seoul, 04376, Republic of Korea (Tel: +82‐2‐797‐7780, Fax: +82‐2‐797‐7750).