Analysis of Different Rational Number Definitions Used in the Literature

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ULUSOY Ç. A., Şahiner y.

Ankara Üniversitesi Eğitim Bilimleri Fakültesi Dergisi, vol.56, no.2, pp.879-927, 2023 (Peer-Reviewed Journal) identifier


A comprehensive search in the literature shows that there are two different rational number definitions: Definition 1 (D1) Q = {a/b : a and b integer, b≠0} and Definition 2 (D2) Q = {a/b : a and b is integer, b≠0, a and b are co-prime}. It is an important contradiction that the sets of Q (rational number) expressed by D1 and D2 are not the same set, and this contradiction is the source of motivation for this study. This article aims to analyze which definition is correct by examining the non-equivalent definitions of D1 and D2 within the framework of the construction and properties of rational numbers, and therefore to prevent the use of the incorrect definition from becoming widespread. This research was carried out using document analysis, one of the qualitative data analysis methods. As data collection tools, textbooks approved by the Ministry of National Education of Türkiye, national and international mathematics and mathematics education books, and documents consisting of academic articles (37 books and 15 articles ) containing the definitions of D1 and (or) D2 were used. With the analysis of the data, it has been determined for what reasons the condition of "being co-prime" is needed, which makes the definitions different from each other. As a result, it has been argued that there is no need for the condition "being co-prime" added to D2, and therefore D1 is a sufficient and correct definition to define rational numbers, and it does not create contradictions such the definition (D2). Additionally, when the definitions were evaluated according to the criteria of being a definition, it was concluded that D1 was suitable for the criteria.