再论《九章算术》通分术

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	再论《九章算术》通分术
	朱一文
刊名	自然科学史研究
	2009-07
卷号	28 期号:3 页码:290-301
关键词	《算数书》《九章算术》通分术
中文摘要	前人把《九章算术》通分术理解为分子、分母扩大相同倍数而分数值保持不变的算法,实际上等价于现今的算法。其实《九章算术》中的通分是指用一个正整数乘以分数以化分数为整数的算法;前人对此理解有误,因此导致了前人对于《九章算术》经分术"同而通之"和少广术"通而同之"的错误理解,以及对于《九章算术》环田题密率术的校勘争论。依据对通分术新的理解,对以上问题进行了辨析,进而认为:古今通分术计算结果殊途同归是产生对通分术普遍误解的算理基础;忽视古代数学术语和算法的统一性,忽视古代数学筹算的表达形式带来的实用性、经济性的要求,以及大量的以今释古是对通分术产生误解的原因。事实上,《九章算术》通分术,在现今数学中并无与之精确对应的算法;前人对通分的误解实际上抹杀了通分术演进的历史——从《九章算术》少广术总术和分术的不一致,可以看出从先秦《算数书》到西汉《九章算术》编撰,通分术历史演进的痕迹。更多还原
英文摘要	As regards the history of Chinese mathematics, our predecessors have a common understanding about the reduction of fractions to a common denominator in Nine Chapters on the Mathe-matical Art, which is actually the same as the rule of today. This article argues that the rule in Nine Chapters on the Mathematical Art is different from the rule of today, which uses a Whole number to multiply a fraction so as to get an integer. Moreover, because of misapprehension, there is a misun- derstanding between ＂finding a common denominator to reduce fractions＂ in Jingfen rule （经分术） of Nine Chapters on the Mathematical Art and ＂reducing fractions to find a common denominator＂ in Shaoguang rule（少广术）, and there is dispute on collation of the precise rule of annular field prob- lem. Based on new understanding of the reduction of fractions to a common denominator, the article reviews the problems above and reaches the conclusions as follows ：＂finding a common denominator to reduce fractions＂ and ＂reducing fractions to find a common denominator＂ are different in arithmetic procedure, and the main purpose of the precise radio rule of annular field problem is to introduce the concept of＂tongfen neizi＂（通分内子）. Suanshushu（算数书） and the main body of Nine Chapters on the Mathematical Art are both accomplished in the pre-Qin period. It is hard to decide which is earlier and which is later. However, there must be some common rules. Actually, on the reduction of fractions to a common denominator, the two books are the same. And the result enhances the article＇s argument. The article argues further that the basic reason is that ancient rule and today＇s rule are equivalent in mathematical principle, and the main cause for misapprehension not only roots in overlooking the unification of the ancient mathematical terms and mathematical rules, but also roots in overlooking the economical and applied demands for using counting rods, and also roots in largely using nowadays knowledge to explain ancient knowledge. In fact, there is no present mathematics nile corresponding to the rule in Nine Chapters on the Mathematical Art. The misapprehension of the past obliterates the history of the reduction of fractions to a common denominator. From the disagreement between the main Shaoguang rule （少广术） and its secondary rules, a historical trace of Suanshushu in pre-Qin period to Nine Chapters on the Mathematical Art composed in the Western Han Dynasty can be found.
学科主题	数学史
语种	中文
公开日期	2013-02-04
内容类型	期刊论文
源URL	[http://ir.ihns.ac.cn/handle/311051/2012]
专题	自然科学史研究所_科学技术史_中国古代科技史
推荐引用方式 GB/T 7714	朱一文. 再论《九章算术》通分术[J]. 自然科学史研究,2009,28(3):290-301.
APA	朱一文.(2009).再论《九章算术》通分术.自然科学史研究,28(3),290-301.
MLA	朱一文."再论《九章算术》通分术".自然科学史研究 28.3(2009):290-301.