In this paper, I investigate the contents of the important texts by authors in the Seki school in the Japan’s Edo period and analyze the symbolic representations of fractions that Wasan mathematicians used in the process of the development and transition from Boshoho (傍書法) to tenzan (點竄), and their related conceptual meanings. Through the analysis, I find that there are five different ways used to represent fractions in the process of the development of symbolic representations: 1.recording the division operation by words; 2.recording the divisor and the dividend separately; 3. recording the division operation on the right hand side of ‘|’; 4. recording the division operation on the left hand side of ‘|’; 5. the left hand side and right hand side of ‘|’ being both numbers or rod expressions, where the left hand side indicates the denominator and the right the numerator. By this way, the fraction becomes a mathematical object which can be operated and manipulated. The symbolic representations of the fraction are developed and transited, from the rhetorical mathematical expressions and the kind of division with procedural and operational features, to the structural mathematical object “fraction”, and thus the operational process precedes the structural object. In particular, the development of the symbols of the fraction is not linear, and multiple representations emerged during the process. The symbols of the fraction that Wasan mathematicians used reveal the duality of mathematical conceptions. Finally, the clarification of the development and the shift about the symbols of the fraction used by the Wasan mathematicians turns into an important evidence for modern scholars to determine the author, scriber and the date of a text.
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The Development, Transition and Conceptual Meanings of the Symbolic Representations about Fractions in the Seki School of Wasan
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