What was the origin and development of the Nine Chapters of Arithmetic like?

The Nine Chapters of Arithmetic, an ancient Chinese mathematical monograph, is one of the most important of the ten books of arithmetic. It is the first book of arithmetic that has been handed down in China to summarize ancient mathematics in a systematic way, and his appearance marks the formation of the Chinese primary mathematical system. Its influence on Chinese mathematics is the same as that of Euclid's Geometrical Principles on Western mathematics. It was used as a textbook for more than 1,000 years after the Western Han Dynasty.

Social development from the Warring States period to the Qin and Han periods posed a number of computational and measurement problems for mathematics to solve, such as the government's taxation according to the number of acres of land, the proportional apportionment of labor, and large-scale earthworks, etc. The reform of the calendar also involved the computation of a variety of data on the number of days. All of this promoted the development of mathematics. Zhou Li" recorded the noble children of the six subjects, there are "nine numbers" one, which refers to mathematics is divided into nine subheadings, "nine chapters of arithmetic" may be in the "nine numbers" on the basis of absorbing the emergence of the "Hsu Shang arithmetic", "Du Zhong arithmetic" before the results of it The book was written on the basis of "Nine Numbers".

There are different views on the time of its publication: some people believe that it was written in the early years of the Eastern Han Dynasty; some people believe that it was written in the middle of the Western Han Dynasty; some people believe that it was written in the era of Wang Mang.

"Nine chapters of arithmetic" of the author has been difficult to prove, only that it is a gradual improvement of the results of many people, the upper bearing pre-Qin Mathematical development of the flow, into the Han after many scholars by the collation, deletion and revision of the early years of the Eastern Han Dynasty (the 1st century A.D.) into the book is the crystallization of several generations of people **** with the labor. Early Western Han mathematicians Zhang Cang and Geng Shouchang carried out additions. Because the Han Shu - Arts and Letters Zhi contains "Xu Shang arithmetic" 26 volumes, "Du Zhong arithmetic" 16 volumes, but not the "nine chapters of arithmetic". The Nine Chapters of Arithmetic may have been written after the works of Xu and Du, so it may contain the contents of the two books of Xu and Du.

Most of the ancient mathematicians of later times began their study and research of mathematics from the Nine Chapters of Arithmetic, and many of them had made annotations for it, the most famous of whom were Liu Hui (263), Li Chunfeng (656) and others. It was prescribed as a textbook by the state in the Tang and Song dynasties, and was published by the court of the then Northern Song Dynasty in 1084, making it the earliest printed math book in the world. The Nine Chapters of Arithmetic was introduced to Korea and Japan during the Sui and Tang dynasties.

The Nine Chapters of the Mathematical Art is a summary of the development and consolidation of mathematics in the Warring States period, the Qin and Han feudal societies, and is considered to be one of the world's most famous mathematical works in terms of its mathematical achievements. For example, the four operations of fractions, the art of present and future (known as the three-rate method in the West), open square and open cube (including the numerical solution of quadratic equations), the art of surplus and deficit (known as the double method in the West), a variety of area and volume formulas, the solution of systems of linear equations, the rules of addition and subtraction of positive and negative numbers, and the solution of the hook-share form (especially the hook-share theorem and the method of finding the number of hook-shares), etc. The level of the book is very high. Among them, the method of solving systems of equations and the rules for adding and subtracting positive and negative numbers are far ahead in the development of mathematics in the world. As far as its characteristics are concerned, it has formed an independent system centered on fund-raising and completely different from ancient Greek mathematics.

The contents of the Nine Chapters of Arithmetic are divided into nine chapters: Chapter 1, Square Field, with 38 questions, including the area of plane shapes and the algorithm of fractions;

Chapter 2, Cornucopia, with 46 questions, talking about various proportionality problems;

Chapter 3, Decay of Points, with proportionality problems of distributing materials according to rank and apportioning taxes according to rank;

Chapter 4, Lesser and Wider, with 24 questions, talking about the square and open square of area and volume. and volume problems of squaring and cubing;

Chapter V, Shang Gong, with 28 questions, on problems of calculating the area of three-dimensional shapes and calculating earth labor;

Chapter VI, Even Loss, with 28 questions, on problems of proportional distribution of compound and even proportions of taxes and of assignments of labor according to the law of equalization;

Chapter VII, Surplus and Deficit, with 20 questions, on the solution of the surplus and deficit problems of arithmetic, and also a few proportional problems.

Chapter VIII, Equations, 18 questions, on the application of multiple systems of equations, including 2-6 unknowns, and the current secondary school textbook "addition, subtraction and elimination method" is basically the same;

Chapter IX, Chart of 24 questions, on the use of the collinearity theorem and calculations. The use of the Pythagorean Theorem and the calculation of various "high, deep, wide, and distant" problems, which are directly related to measurement and drawing.

The Nine Chapters on Arithmetic covers most of the content of elementary mathematics, including arithmetic, algebra, and geometry. It is characterized by an emphasis on theory, but not detached from practice. It recorded the four operations of fractions and proportional operations, which were the most advanced in the world at that time. The book's solution to the profit and loss problem is a creation that occupies an important place in the history of world mathematics. India in the third and fourth centuries, there and our surplus and deficit algorithm exactly the same, in Arabia, Central Asia and medieval Europe is popular "double managed" (with two assumptions), may be affected by our surplus and deficit problem and the development of the influence. The book of the concept of positive and negative numbers and the addition and subtraction algorithms, but also the world's first, the first foreign recognition of negative numbers is the Indian mathematician Brahmanshastra in the 7th century, the 16th century in Europe to recognize the negative numbers. The hook and strand chapter of the Chinese hook and strand theorem to solve the "Phragmites central problem", and Indian mathematics in the famous "Lotus problem" is the same, only the data are different, the rest of the exact same, but India is more than 1,000 years later than China.

The nine chapters of arithmetic and the earliest Western mathematical monograph "Geometry Originally" compared, found that the "Geometry Originally" to the formal logic of the whole book throughout, while the "Nine Chapters of arithmetic" to the nature of the problem of the organization. Geometria was mainly geometric, with a little bit of arithmetic, whereas Nine Chapters on Arithmetic included a wide range of contents such as arithmetic, algebra, geometry, and so on. The Origin of Geometry pays more attention to theory and does not talk about practical applications, while The Nine Chapters of Arithmetic pays more attention to theory while touching on practical applications. The two books have their own strengths and weaknesses, forming a different style of mathematics between the East and the West.

After the birth of the Nine Chapters of Arithmetic, it has been used as a major textbook to teach mathematics, and all Chinese mathematical works before the 16th century were written in the style of the Nine Chapters of Arithmetic. It has been annotated by famous mathematicians throughout the ages, in which new mathematical concepts and algorithms have been introduced, promoting the development of ancient Chinese mathematics.

The Nine Chapters of Arithmetic spread to Korea and Japan, where it also became a textbook. It was transferred to Europe through India and Islamic countries. The problem of profit and loss was introduced to the Arab countries as the "Khitan algorithm" (i.e., Chinese algorithm). The book is now available in Japanese, English, Russian, German and other translations. By the world's attention.