从定位法看中国传统乘除运算的变迁

The Multiplication and Division Operations in Traditional Chinese Mathematics from the Perspective of Ding-Wei

乘除定位法是解决数学问题中积或商每一位的数量级和单位的方法,目前发现最早的记录出现于南宋杨辉的《算法通变本末》。时至明代,筹算逐渐为珠算更替,吴敬、王文素和程大位等学者进一步将定位法归纳为完整的珠算定位系统。欧洲笔算传入后,中算家结合东西方传统定位法,发展出中西合璧的笔算定位法。通过梳理乘除定位法的发展变迁史,从算理视角分析可知,不同的计算顺序和计算方式导致了定位法的不同表达,反映了不同历史时期人们对乘除运算及数本身的不同认知。同时,以定位法之变迁窥见乘除算法在不同运算工具体系下的演变过程,说明筹算向珠算的转变模式并非只是计算工具的简单改变,而是伴随着数学规则和范式的一种内在转变。

Determining the position(ding wei)is a method of dealing with the order of magnitude and unit of the product or quotient of the actual problem.The earliest record can be seen is Yang Hui's""SuanFaTongBianBenMo"of the Southern Song Dynasty.By the Ming Dynasty,the counting rods was gradually replaced by abacus,and the method of determining the position was developed in the books of Wu Jing,Wang Wensu,and Cheng Dawei,which was summarized as a complete abacus positioning system.With the introduction of European manual calculation,Chinese mathematicians combined it with traditional positioning methods to develop manual calculation positioning methods with the characteristics of Chinese and Western integration.By combing the history of the development and change of the method of determining the position,it is pointed out from the perspective of arithmetic that different calcu-lation orders and modes lead to different expressions of the positioning method,reflecting the different cognitions of the multiplication and division opera-tion and the number itself in different periods.At the same time,the evolution of the multiplication and division algorithm under dfferent arithmetic tool systems is glimpsed by the change of the positioning method,especially the transformation process from counting rods to abacus,which shows that this is not a simple change of calculation tools,but also accompanied by the internal transformation of arithmetic and rules.