Chapter 27 Chapter 7 "Nine Chapters on Arithmetic"

"Nine Chapters on Arithmetic" is the oldest surviving monograph on mathematics in my country.It is not the work of one person at a time, but is completed through continuous revision and supplementation of several generations.It was written in the early Eastern Han Dynasty at the latest, and its basic content was basically finalized in the late Western Han Dynasty (the middle of the first century BC).The so-called "nine chapters" are divided into nine categories of mathematical problems. There are 246 mathematical problems and their solutions in the whole book, and their style is similar to the current math problem sets.

The first chapter, Fangtian, is about the calculation of the area of ​​various shapes of fields, including the calculation of the surface area of ​​squares, rectangles, triangles, trapezoids, circles, rings, bows, and truncated spheres.Among them, there are also the earliest descriptions in the world of arithmetic rules such as fraction reduction, general fraction, four arithmetic operations, and finding the greatest common divisor. The second chapter, Corn, deals with the calculation of various proportions, especially the exchange of various grains in proportion. The third chapter, decay, talks about the calculation of the proportion of materials distributed according to grades or taxes allocated according to grades.

The fourth chapter, Shaoguang, talks about the problem of finding the length of one side from a known area or volume, which involves the methods of square extraction and cubic extraction. The fifth chapter, Shanggong, talks about various engineering mathematics problems related to earthwork and labor volume, mainly describing the calculation methods of various engineering volumes. The sixth chapter, All Loss, talks about the distribution of taxes and the proportion of migrant workers based on conditions such as population size, distance, and price levels. It involves complex proportion calculations such as compound proportions and continuous proportions.

The seventh chapter of surplus and insufficiency talks about the algorithm that uses two assumptions to solve some difficult problems, most of which are for "how many people buy things together, if each person buys eight, it will be three more, and if each person buys seven, it will be four less. Ask the number of people How much is it and the price?” Answers to such questions.Because the solution requires two assumptions, it is now commonly called "double method". Chapter 8, Equations, deals with the solution of simultaneous linear equations.The solution is basically the same as the "addition, subtraction and elimination method" taught in the current middle school algebra class.This chapter also introduces negative numbers, and gives the operation rules of addition and subtraction of positive and negative numbers.These contents appear for the first time in the history of world mathematics.

The ninth chapter, Pythagorean, talks about using the Pythagorean theorem to measure and calculate height, depth, width, and distance, etc., and the content involves the solution of right-angled similar triangles and quadratic equations. The publication of "Nine Chapters of Arithmetic" marks the formation of traditional Chinese mathematics, and it has a profound influence on the development of mathematics in later generations.Mathematics works of later generations basically inherited its writing style, and inherited and developed the tradition it advocated to start from practical problems and provide mathematical solutions.Many outstanding mathematicians in later generations commented on "Nine Chapters on Arithmetic", introducing new mathematical concepts and methods in the notes.The famous ones include Liu Hui's annotations in the Three Kingdoms period, and Li Chunfeng's annotations in the Tang Dynasty.