The history of multivariate linear equations, such as Inventor and Nine Chapters Arithmetic, Nine Chapters Arithmetic Examples.

Nine Chapters Arithmetic is a mathematical monograph in ancient China, and it is the most important one among the "Ten Arithmetic Classics" (ten ancient arithmetic books that appeared between Han and Tang Dynasties). Liu Hui commented on "Nine Chapters Arithmetic" in Wei and Jin Dynasties, saying: "The Duke of Zhou made a ceremony, and there were nine numbers, while nine chapters were a ceremony." He also said, "Zhang Cang, Hou Peiping in Han Dynasty, and Geng Shouchang, a senior farmer, all made good use of fortune telling. Because of the remnants of old texts, Cang and others each called it deletion and supplement, so the purpose of the school is different from ancient times or today, and the theory is closer. " According to textual research, Zhang Cang and Geng Shouchang in the Western Han Dynasty have made supplements. The final edition was written at the latest in the early years of the Eastern Han Dynasty, but its basic content was basically finalized in the late years of the Eastern Han Dynasty. There are only two kinds of mathematics books recorded in Han Shu Literature and Art Annals (written by Ban Gu according to Liu Xin's Seven Views): Xu Shang Arithmetic and Du Zhong Arithmetic, but there is no Nine Chapters Arithmetic, so it can be seen that Nine Chapters Arithmetic appeared later than Seven Views. The Biography of Ma Yuan in the Later Han Dynasty records that his grandnephew Ma Xu was "well-read and good at arithmetic in nine chapters" and was born in the last two or three decades of 1 century. According to the official names and place names in Nine Chapters Arithmetic, it can be inferred that the modern version of Nine Chapters Arithmetic was written in the second half of 1 century. Nine Chapters of Arithmetic divides all the mathematical problems in the book into nine categories, edited by Chen Yu 1984, and unearthed in Hubei. According to textual research, it is more than a century and a half earlier than Nine Chapters of Arithmetic. Some contents in the book are very similar to Nine Chapters of Arithmetic, and some contents are basically the same. Some people speculate that the two books have a certain inheritance relationship, but there are also different views that Nine Chapters of Arithmetic has not been directly influenced by Arithmetic Book. Most mathematicians in later generations began to study and study mathematics from the Nine Chapters of Arithmetic, and many people commented on it. The most famous ones are Liu Hui (263) and Li (656). The notes of Liu, Li and others were circulated together with Nine Chapters of Arithmetic. During the Tang and Song Dynasties, Nine Chapters Arithmetic was clearly defined as a textbook by the state. In the Northern Song Dynasty, the government also published Nine Chapters of Arithmetic (1084), which was the earliest printed mathematics book in the world. Among the modern editions of Nine Chapters Arithmetic, the earliest edition is the Southern Song Dynasty reprint of the Northern Song Dynasty (12 13), which is now in Shanghai Library (only the first five volumes are left). In Qing Dynasty, Dai Zhen copied and collated Nine Chapters of Arithmetic from Yongle Dadian. Since then, most of the editions of Sikuquanshu, Wuyingsi Rare Book and Ten Calculations inscribed by Kong (1773) are based on Dai Pai. As a world-famous mathematical work, Nine Chapters Arithmetic was introduced to Korea and Japan during the Sui and Tang Dynasties. It has been translated into Japanese, Russian, German and French.

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The nine chapters on arithmetic are very rich in content. In the form of problem sets, the book contains 246 applied questions related to production and life practice, in which each question has questions (questions), answers (answers) and skills (steps to solve problems, but there is no proof), some have one skill for one question, and some have many skills for many questions. According to their properties and solutions, these problems belong to,, decline (Cui) points, Shao Guang, Shang Hong, continuous loss, profit, loss, equation and Pythagorean respectively. * * * Nine chapters are as follows. The original edition has illustrations, and this edition only leaves the text. Nine Chapters Arithmetic contains 246 math problems, which are divided into nine chapters. Their main contents are as follows: Chapter 1 "Square field": It mainly describes the calculation method of plane geometric figure area. Include rectangle, isosceles triangle, right-angled trapezoid, isosceles trapezoid, circle, sector, bow and ring. In addition, the four algorithms of fraction and the method of finding the greatest common divisor of numerator and denominator are systematically expounded. Chapter two "millet": conversion of grain proportion; Propose a proportional algorithm, which is called the prior art; The law of proportional distribution was put forward in the chapter of decay, which is called decay class; Chapter 3 "Decline": the problem of proportional distribution; This paper introduces the method of square root and square root, and its steps are basically the same as today. This is the earliest multi-digit and fractional root rule in the world. It laid a foundation for China to lead the world in numerical solution of higher order equations for a long time. The fourth chapter is "less but wider": knowing the area and volume, the length of one side and the length of the diameter are calculated backwards; Chapter 5 "commercial engineering": geotechnical engineering and volume calculation; In addition to various solid volume formulas, there are also engineering allocation methods; Chapter VI "lose-lose": reasonable allocation of tax revenue; Solving the problem of reasonable burden of tax service by decreasing method. The existing technology, diminishing technology and its application methods constitute a whole set of proportional theory, including today's positive and negative proportion, proportional distribution, compound proportion and chain proportion. It was not until the end of 15 that a similar method was formed in the west. Chapter seven "surplus and deficiency": the problem of dual management; This paper puts forward three types of profit and loss problem: insufficient surplus, sufficient and insufficient surplus, two surpluses and two shortages, and the solutions to some general problems that can be transformed into insufficient surplus through two assumptions. This is also the world's leading achievement, which has a great influence after it spread to the west. Chapter 8 "Equation": the problem of linear equations; Linear equations are expressed by separation coefficient method, which is equivalent to current matrix; The direct division used to solve linear equations is consistent with the elementary transformation of matrices. This is the earliest solution of completely linear equations in the world. In the west, it was not until17th century that Leibniz put forward a complete law for solving linear equations. This chapter also introduces and uses negative numbers, and puts forward the addition and subtraction rules of positive and negative numbers, which are exactly the same as those in modern algebra. When solving linear equations, the multiplication and division of positive and negative numbers are actually performed. This is a great achievement in the history of world mathematics, which broke through the range of positive numbers for the first time and expanded the number system. Foreign countries did not realize negative numbers until the Yarlung Zangbo River in India in the 7th century. Chapter 9 "Pythagorean Theorem": Various problems solved by Pythagorean Theorem. Most of them are closely related to the social life at that time. The general solution formula of Pythagorean number problem is put forward: if A, B and C are Pythagorean, strand and chord respectively, then m >；; In the west, Pythagoras, Euclid, etc. Only a few special cases of this formula were obtained, and similar results were not obtained until Diophantine in the 3rd century, which was about three worlds later than Nine Chapters Arithmetic.