清华简《算表》的形制特征与运算方法

The Structure Characteristic and Operation Method of the Mathematical Table in the Tsinghua Collection of Bamboo Slip Manuscripts

清华大学入藏的战国竹简有一组简形制特殊,宽于其他简,正面画有朱色栏线,是一份表格形式的实用运算表。这组简自成一篇,原无标题,整理者据形制与内容定名为《算表》。《算表》未见于传世古籍及以往之出土文献,是研究中国古代数学的重要史料。对这组简的形制特征作了较为细致的介绍,并对《算表》的运算方法进行了初步的分析,指出《算表》核心是由"九"至"一"及其乘积"八十一"至"一"诸数构成的乘法表,其他部分则是其核心的扩展与延伸。此《算表》应用了十进制计数方法,并且用到了乘法的交换律、乘法对加法的分配律及分数等数学原理和概念,它不仅能直接用于两位数的乘法运算,而且可用于除法运算,并能对分数1/2或含有1/2的分数进行某些运算,可能还可以用于开平方运算。

In the Tsinghua Collection of Bamboo Slip Manuscripts, there is a form of practical arithmetical operations table. It contains 21 special slips, which are wider than other slips, and are painted with red lines on the obverse side. Because this set of slips has no title originally, the re- searchers named it as Mathematical Table according to its form and content. The Mathematical Table is an important historical material for the research of ancient Chinese mathematics, and has never been found in the ancient books and other excavated documents. This article describes the structure characteristic of these slips explicitly, and preliminarily analyses the arithmetical operation method, discusses the structure of the Tsinghua Mathematical Table, and indicates that its core is a multipli- cation table composed of the numbers from 9 to 1 and the products from 81 to 1, and the other parts are the expansion and extension of the core. The Mathematical Table has a decimal place value sys- tem, and uses the commutative law of multiplication, the distributive law of multiplication over addi- tion, and also other mathematical principles and concepts, such as fraction. It may be used to multi- ply a double-digit number by another double-digit number, and even to operate the multiplication be- tween two double-digit mixed numbers with a remainder of one-half. The table may also be used to carry out divisions and even the extraction of square roots of certain numbers.