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What are the famous arithmetical works in ancient China?

Zhou pian ·suan Jing, Nine Chapters Arithmetic, Island Arithmetic Classic, Zhang Qiu Arithmetic Classic, Xiahou Yang Arithmetic Classic, Five Arithmetic Classics, Ancient Arithmetic Classics Compilation and Composition. It is "calculating ten books".

Zhouyi suan Jing

Among the ten books, The Book of Weekly Parallel Calculations is the earliest. I don't know who its author is. According to textual research, it was written no later than the end of the Western Han Dynasty (the first century BC). Zhou Kuai suan Jing is not only a mathematical work, but also an astronomical work about Gai Tian Shuo, a school of astronomical theory at that time. As far as mathematical content is concerned, the book records astronomical calculations with Pythagorean theorem, and there are more complicated fractional calculations. Of course, it cannot be said that these two algorithms were not mastered until the first century BC. It can only show that "Weekly Parallel Computing Book" is an earlier record in known data.

the nine chapters on the mathamatical art

Nine Chapters Arithmetic is the most important of ten arithmetic books, which comprehensively and completely introduces all aspects of ancient mathematics. Its influence on the later development of ancient mathematics in China is as profound as Euclid's Elements of Geometry's influence on western mathematics. In China, it has been directly used as a textbook for mathematics education for 1000 years. It also influenced foreign countries, and Korea and Japan used it as teaching materials.

I don't know the exact author of Nine Chapters Arithmetic, except that Zhang Cang (201-kloc-0/52), Geng Shouchang and others had added or deleted it in the early years of the Western Han Dynasty. There is no title of Nine Chapters Arithmetic in Hanshu Yiwenzhi, but there is a book Arithmetic by Xu Shang and Du Fu, so some people infer that it may also include the works of Xu and Du Fu. 1984, Shu Shu bamboo slips were unearthed from Zhangjiashan Tomb in Jiangling, Hubei Province in the early Western Han Dynasty. It is estimated that the book was written more than a century and a half before the nine chapters of arithmetic, and its content is very similar to the nine chapters of arithmetic, and some calculation problems are basically the same as the nine chapters of arithmetic.

It can be seen that there is a certain inheritance relationship between the two books. It can be said that Nine Chapters Arithmetic was gradually formed after many revisions in a long period, although some of its algorithms may have existed before the Western Han Dynasty. As the title shows, the book is divided into nine chapters, and a total of 246 math problems are collected. Together with the solutions to each problem, it is divided into nine categories, and each category is regarded as a chapter.

Judging from the mathematical achievements, the first thing to mention is that the book records the most advanced quartering operation and proportional algorithm in the world at that time. The book also records the algorithm of solving various area and volume problems and various measurement problems with Pythagorean theorem. The most important achievement of Nine Chapters Arithmetic is algebra. The method of square root and square root is recorded in the book, and on this basis, a numerical solution for solving the general quadratic equation with one variable (the first term coefficient is not negative) is obtained. There is also a whole chapter about solving simultaneous equations, which is essentially the same as the method in middle schools now. This is 1500 years earlier than similar algorithms in Europe. In the same chapter, the concept of negative number and the addition and subtraction algorithm of positive and negative numbers were recorded for the first time in the history of mathematics in the world.

Nine Chapters Arithmetic not only occupies an important position in the history of Chinese mathematics, but also has far-reaching influence abroad. In the Middle Ages in Europe, some algorithms in Nine Chapters Arithmetic, such as fractions and proportions, may have been introduced to India first, and then to Europe through Arabia. Another example is "surplus and deficiency" (which can also be regarded as a one-time interpolation method), which is called "China algorithm" in the early mathematical works of Arabia and Europe. Now, as a world-famous scientific work, Nine Chapters Arithmetic has been translated and published in many languages.

Sunzi suan Jing

This book was written in the fourth and fifth centuries, and the author's life and writing time are not clear. Sun Tzu's Art of War has three volumes. The paper describes the vertical and horizontal alternation system and the rules of multiplication and division, and illustrates the algorithm of calculating scores and the method of calculating Kaiping with examples.

China was the first country in the world to adopt decimal system for counting, which was widely used in the Spring and Autumn Period and the Warring States Period, that is, it strictly followed decimal system. Now the only information about counting methods is in the calculation of Sun Tzu's Art of War. Sunzi Suanjing is a three-volume book, which was written in the 4th century. The first volume of the book is a systematic introduction to the calculation rules, and the second volume has the famous "unknown things", also known as the "grandson problem". The title of the first volume, 3 1, can be described as the ancestor of the title of "chickens and rabbits in the same cage" in later generations, and later spread to Japan, becoming the "crane turtle count". The book describes it like this: "There are chickens and rabbits in the same cage today, with 35 heads on the top and 94 feet on the bottom. The geometry of chicken and rabbit? These four sentences mean: there are several chickens and rabbits in a cage, counting from the top, there are 35 heads; It's 94 feet from the bottom. How many chickens and rabbits are there in each cage?

It is of great significance to Volume 26: "There are unknown things today. Three and three numbers leave two, five and five numbers leave three, and seven and seven numbers leave two. What is the geometry of things? Answer: "Twenty-three". Sun Tzu's calculation not only provides the answer, but also gives the solution. Qin, a great mathematician in the Southern Song Dynasty, further initiated the research work of congruence theory and popularized the problem that "things are unknown". German mathematician Gauss (K.F.Gauss.A.D.1777-1855) clearly wrote the above theorem in "Arithmetic Inquiry" published in 180 1 year. In a.d. 1852, the British Christian priest William ((﹝alexander· Wiley 18 15- 1887) spread the solution to the problem of "unknown things" in Sun Tzu's Calculations to Europe. Marty was born in a.d./kloc.

Wucao suan Jing

Cao Wu suan Jing is an applied arithmetic book for local administrators (the author is unknown, but some people think it is Zhen Luan). The book is divided into five chapters: Cao Tian, Agropyron cristatum, Jicao, Cangcao and Caojin, so it is called "Cao Wu suan Jing". The solution of the problem is easy to understand, and the numerical calculation tries to avoid fractions. This book includes 67 questions altogether. Its author and age are not recorded. Ouyang Xiu's Book of the New Tang Dynasty (Volume 59) and Records of Arts and Literature (Volume 59) contain "Zhen Luan Wu Cao Shu Jing", and other books have similar records. Zhen Xuan was born around 535-566 AD.

This is the frontispiece of the Southern Song Dynasty publication "Wu Cao Shu Jing", which was engraved in Jiading five years in the Southern Song Dynasty (12 12). Cao Wu suan Jing is a mathematical work in ancient China. The author is Zhen Luan of the Northern Zhou Dynasty (Zun Shu, from Wuji, Hebei). He is familiar with astronomical calendars, and was once a doctor in Li Si and a satrap in Hanzhong. Li Tang, Feng Chun and others made notes on it.

Xiahou yangjing

Xiahou Yang is one of the ten books. The original book has been lost without inspection. Xiahou Yang Shujing was engraved in the ninth year of Yuanfeng in the Northern Song Dynasty (1084), which was a calculation book in the middle Tang Dynasty. Quoted the popular multiplication and division methods and agile methods at that time, answered the application questions in daily life, and saved a lot of mathematical historical materials.

Suan Qiujian suanjing

Zhang Qiujian's classic calculation book was written by Zhang Qiujian in the late 5th century. There are problems in the application of the greatest common divisor and the least common multiple, and there is no problem of Zhu difference sequence. The most famous is the indefinite equation group-Hundred Chicken Problem, but its solution is not specified. Xiahou Yang Shujing is probably the work of the Northern Wei Dynasty. Among them, the fast calculation rules and fractional rules of multiplication and division are briefly described, and the rules of division by law, division by steps, reduction and division, square root and cube are explained. In addition, the application of decimals is popularized, which is completely different from the current representation. The calculation result is odd zero, and the decimal part below the text is represented by the names of length units such as minutes, centimeters, millimeters and silk. This paper introduces that "Hundred Chickens Problem" is a famous mathematical problem in Zhang Qiu's Concurrent Calculations Classic, and gives the solution of an indefinite equation group consisting of two equations with three unknowns. The question of a hundred chickens is: "there is a chicken today, which is worth five;" One mother hen is worth three; Chicks, chicks are worth one. Buy 100 chickens for every 100 yuan, and ask the chicken geometry. According to the meaning of the question.

Since Zhang Qiujian, Chinese mathematicians have been deeply studying the hundred chickens problem, which has almost become synonymous with indefinite equations. From the Song Dynasty to the Qing Dynasty, the mathematical research on hundred chickens has achieved good results.

Classic of island computing

Calculations of Islands was written by Liu Hui (about 225-295) during the Three Kingdoms period. This book is all about using benchmarks to measure twice, three times, and the most complicated is four times to solve various mathematical problems of measurement. These surveying mathematics are the mathematical basis of the very advanced cartography in ancient China. In addition, Liu Hui is also famous for his annotation of Nine Chapters Arithmetic. Generally speaking, these notes can be regarded as mathematical proofs of several algorithms in Nine Chapters of Arithmetic. The secant in Liu Hui's annotation pioneered an important method for calculating pi in ancient China (see page 98 of this book), and he also applied the concept of limit to solving mathematical problems for the first time.

Ji suan suanjing

Wang Xiaotong wrote Ji Gu Shu Jing. In May of the eighth year of Tang Wude (625), Wang Xiaotong wrote the book "Jigu Shujing" in Chang 'an, which is the earliest existing book for solving cubic equations in China.

Wang Xiaotong's Ji Gu Shu Jing is the only work written by scholars in the Tang Dynasty. Wang Xiaomo lived mainly in the late 6th century and early 7th century. He came from a civilian background and began to devote himself to mathematics in his teens. He was an official in the Sui Dynasty on the basis of calendar calculation, and he was retained after entering the Tang Dynasty. He worked as a doctor of arithmetic in the early years of the Tang Dynasty, and was later promoted to Tong Zhilang and Tai Shicheng. I have been engaged in mathematics and astronomy all my life. In the sixth year of Tang Wude (623), the solar eclipse calculated by Fu's Wuyin Yuanli was inconsistent with the actual astronomical phenomena, so he was assigned to study the problems existing in Fu's calendar with Lang Zhongzu of the official department. In the ninth year of Tang Wude (626), he and Dali Qing made a letter to proofread Fu Li, and corrected more than 30 mistakes, and handed them over to a surname for execution. Wang Xiaotong's "Ancient Arithmetic" has always been regarded as a mathematical textbook of imperial academy Institute of Mathematics, and it is regarded as a mathematical classic, so it was later called "Ancient Arithmetic Classic". There are 20 questions in one volume (the new and old Tang books are called four volumes, but the number of questions in one volume is consistent with Wang Xiaotong's self-report, so there may be some differences in the classification of volumes). The first topic is to calculate the declination number of the moon, which belongs to the calculation of astronomical calendar. The second to fourteenth questions are the construction calculation problems of civil water conservancy projects such as building an observatory, building dams, digging ditches, building granaries and cellars. Questions 15 to 20 are Pythagoras problems. These problems reflected the actual needs of construction calculation of civil water conservancy projects such as canal digging, Great Wall construction and large-scale urban construction.

Five classics arithmetic

Written by Zhen Xuan in Northern Zhou Dynasty, in two volumes. In the book, Confucian classics such as the Book of Changes, The Book of Songs, Shangshu, Zhou Li, Yi Li, The Book of Rites, The Analects of Confucius, Zuo Zhuan, etc. It also makes detailed comments on the places related to numbers in its ancient notes, which may be helpful to those who study Confucian classics, but its value is limited in terms of mathematical content. The current biography is also copied from Yongle Dadian.

The legacy of numerology

Xu Yue? ——220) Notes on Mathematical Methods, which introduced the calculation method of 14 in the form of a question and answer with Liu Hong. "It is less than 100 words, but it is still the style of Tokyo." It is in this book that Xu Yue recorded the style of abacus calculation for the first time in China and named it for the first time, which wrote a glorious page in the history of abacus calculation in the world. Fourteen ancient algorithms were recorded. The first one is called "integral", which is the general calculation at that time. There are also Taiyi Calculations, Two Calculations, Three Calculations, Five Elements Calculations, Eight Diagrams Calculations, Jiugong Calculations, Operational Calculations, Knowledge Calculations, Number Calculations, Head Calculations, Turtle Calculations, Abacus Calculations and Number Calculations. A mental arithmetic method introduced in Notes on Mathematical Skills. The original text said:' Since you don't count, you should count. The note said,' People who use words to calculate numbers mean that they don't need to calculate funds, but should count according to their own intentions. This shows that the calculation is done only by mental arithmetic without tools such as beads, chips and needles. But judging from the examples cited in the note, the "calculation" here is different from the heart understood by modern people. Nowadays, mental arithmetic means that when calculating numbers, you don't need calculation tools, just use your brain. The scope of "calculation" is quite wide. In measurement, we not only don't need calculation tools, but also come up with an ingenious method, which can directly obtain the required digital results without digital operation. "

Composition technique

Seal script is the work of Zu Chongzhi, a famous mathematician in the Northern and Southern Dynasties. Unfortunately, this book was lost around the tenth century AD between the Tang and Song Dynasties. Song people used another arithmetic book found at that time to fill in the numbers when publishing the Ten Books of Arithmetic Classics. Zu Chongzhi's masterpiece Calculation of Pi (accurate to the seventh place after the decimal point) was included in Sui Shu Chronicle.