The Book of Changes records that "there was a rule of tying ropes in ancient times, and later sages changed their books." There are many numerals in Oracle Bone Inscriptions unearthed in Yin Ruins. From one to ten, as well as hundreds, thousands and tens of thousands, are special notation, with 13 independent symbols. The notation is written in the combined document, including decimal notation, and the maximum number is thirty thousand.
Calculation is a calculation tool in ancient China, and this calculation method is called calculation. The age of calculation cannot be verified, but it is certain that calculation has been very common in the Spring and Autumn Period.
There are two ways to calculate numbers by counting chips, vertical and horizontal:
When representing multi-digits, the decimal numerical system is adopted, and the digits of each digit are arranged from left to right, criss-crossing (the rule is: one is vertical and ten is horizontal, one hundred stands upright, one thousand is relative to ten, and ten thousand is equal to one hundred), and zero is represented by spaces. Calculation and financing establish good conditions for addition, subtraction, multiplication and division.
Calculation was not gradually replaced by abacus until the end of Yuan Dynasty in15th century. It was on the basis of calculation that China ancient mathematics made brilliant achievements.
In geometry, Xiabenji in Historical Records says that Yu Xia has used drawing and measuring tools such as ruler, moment, scale and rope to control water, and has discovered a special case of Pythagorean Theorem (called Pythagorean Theorem in the West). During the Warring States Period, the Work Inspection Book written by Qi people summarized the technical specifications of handicraft industry at that time, including some measurement contents, involving some geometric knowledge, such as the concept of angle.
A hundred schools of thought contended during the Warring States period also promoted the development of mathematics, and some schools also summarized many abstract concepts related to mathematics. As we all know, Mo Jing's definitions and propositions of some geometric terms, such as "a circle, an equal length", "flat, the same height" and so on. Mohist school also gave the definitions of finite and infinite. Zhuangzi records the famous theories of Hui Shi and others, and the topics put forward by debaters such as Huan Tuan and Gong Sunlong, emphasizing abstract mathematical ideas, such as "the greatest is the greatest, the smallest is the smallest", "one foot pestle, half a day, inexhaustible" and so on. Many mathematical propositions such as these definitions and limit ideas of geometric concepts are quite valuable mathematical ideas, but this new idea of attaching importance to abstraction and logical rigor has not been well inherited and developed.
In addition, the Book of Changes, which tells the gossip of Yin and Yang and predicts good and bad luck, has sprouted from combinatorial mathematics, reflecting the idea of binary system.
Second, the formation and foundation of China's mathematical system.
This period includes the 400-year history of mathematical development from Qin and Han Dynasties, Wei and Jin Dynasties and Southern and Northern Dynasties. The Qin and Han Dynasties was the formation period of China's ancient mathematical system. In order to systematize and theorize the growing mathematical knowledge, specialized mathematical books have appeared one after another.
The earliest mathematical monograph in the history of China is Shu Shu, a Han bamboo slip unearthed in Zhangjiashan, Jiangling, Hubei Province in 1984, which was written in the early years of the Western Han Dynasty. At the same time, a Resume of the Han Dynasty was written in the second year of Lv Hou (BC 186), so the book was written in BC 186 at the latest (it should be before).
Zhou Parallel Computation Classic, compiled at the end of the Western Han Dynasty [the first century BC], is an astronomical work about Gaitian's cosmology, but it contains many mathematical contents. There are two main achievements in mathematics: (1) put forward the special case and universal form of Pythagorean theorem; (2) Chen Zi's method of measuring the height and distance of the sun is the pioneer of gravity difference's Pythagorean method. In addition, there are more complicated root-finding problems and fractional operations.
Nine Chapters Arithmetic is an ancient mathematical classic that has been compiled and revised by several generations. Written in the early years of the Eastern Han Dynasty [1st century BC]. This book is written in the form of problem sets, collecting 246 questions and their answers, which belong to nine chapters: Tian Fang, Su Mi, Decline, Shao Guang, Shanggong, Equal Loss, Profit and Loss, Equation and Pythagoras. The main contents include four fractional and proportional algorithms, the calculation of various areas and volumes, and the calculation of pythagorean measurement. In algebra, the concept of negative number and the law of addition and subtraction of positive and negative numbers introduced in the chapter of equation are the earliest records in the history of mathematics in the world. The solution of linear equations in the book is basically the same as that taught in middle schools now. As far as the characteristics of Nine Chapters Arithmetic are concerned, it pays attention to the application and integration of theory with practice, and forms a mathematical system centered on calculation, which has had a far-reaching impact on the ancient calculation in China. Some of its achievements, such as decimal numerical system, modern skills and surplus skills, have also spread to India and Arabia, and through these countries to Europe, which has promoted the development of world mathematics.
During the Wei and Jin Dynasties, China's mathematics developed greatly in theory. Among them, the work of Zhao Shuang (unknown year of birth and death) and Liu Hui (unknown year of birth and death) is considered as the beginning of China's ancient mathematical theory system. Zhao Shuang, a native of Wu in the Three Kingdoms, was one of the earliest mathematicians who proved mathematical theorems and formulas in ancient China. He made a detailed annotation on Zhou pian Shu Jing, and strictly proved the pythagorean theorem in pythagorean square diagram by geometric method. His method embodies the idea of cut-and-fill principle. Zhao Shuang also put forward a new method to solve quadratic equation by geometric method. In 263, Liu Hui, a Ren Wei of the Three Kingdoms, annotated Nine Chapters Arithmetic, which not only explained and deduced the methods, formulas and theorems of the original book as a whole, but also systematically expounded the theoretical system and mathematical principles of China's traditional mathematics, which was creative and created the secant method in Volume 1 Square Field (that is, the method of connecting regular polygons in a circle to infinitely approximate the area of a circle). It lays a theoretical foundation for the study of pi and provides a scientific algorithm. He obtained the approximate value of pi by the method of "secant circle" as 3927/ 1250 (i.e. 3. 14 16). "Shanggongpian" constructs the geometric model of "Mouhe Square Cover", which solves the problem of spherical volume formula and opens the way for Zuxuan to get the correct result. In order to establish the polyhedral volume theory, Yang successfully proved equestrian by the limit method. He also wrote Calculation of Islands, which developed the ancient Pythagorean measurement method-gravity difference technique.
The society in the Southern and Northern Dynasties was in a state of war and division for a long time, but the development of mathematics was still vigorous. There are some books on arithmetic, such as Sun Tzu's Art of War, Xiahou Yangbing Law and Zhang Qiu's Art of War. Written in the 4th-5th century A.D., Sunzi Suanjing gave the question "Things are unknown" and gave the answer, which led to the solution of a congruence group of China. The "Hundred Chickens Problem" in Zhang Qiujian suan Jing leads to three unknown indefinite equations.
In the 5th century A.D., the most representative works in this period were Zu Chongzhi and Zuxuan. On the basis of Liu Hui's annotation of Nine Chapters Arithmetic, they greatly promoted traditional mathematics and became a model of attaching importance to mathematical thinking and reasoning. They also made outstanding contributions to astronomy. Their book seal script has been lost. According to historical records, they have made three great achievements in mathematics: (1) Calculate pi to the sixth place after the decimal point and get 3. 14 15926.
He Chengtian, a contemporary astronomer, invented the method of adjusting the sun, used rational fractions to approximate real numbers, and developed ancient indefinite analysis and numerical approximation algorithms.
Thirdly, the establishment of Chinese mathematics education system.
Large-scale architecture in Sui Dynasty objectively promoted the development of mathematics. At the beginning of the Tang Dynasty, Wang Xiaotong compiled the Classic of Ancient Calculations, mainly through practical problems such as earthwork calculation, engineering division and acceptance, and silo calculation, and discussed how to establish cubic polynomial equations by geometric methods, and developed the square root theory in Nine Chapters of Arithmetic.
The Sui and Tang Dynasties was the period when the feudal bureaucratic system in China was established. With the establishment of imperial examination system and imperial academy system, mathematics education has made great progress. In 656, imperial academy established a Mathematics Museum with doctors and teaching assistants in mathematics. Taishi ordered Li and others to compile and annotate ten calculation books (including Zhou Pi Ai Shu, Jiu Zhang Shu Shu, Dao Shu, Sun Zi Shu, Zhang Qiu Shu, Xiahou Yang Shu, Ji Gu Shu and Sun Zi Shu). It has played an important role in preserving ancient mathematical classics.
With the implementation of some important astronomical discoveries in the Southern and Northern Dynasties in the calendar compilation at the turn of Sui and Tang Dynasties, some important mathematical achievements appeared in the calendar of Tang Dynasty. In 600 A.D., the Sui Dynasty put forward the world's first quadratic interpolation formula with equal spacing when solving Huang Shi, which was an outstanding creation in the history of mathematics. In the Tang Dynasty, monks and their followers developed it into a quadratic interpolation formula with unequal intervals in Yan Li.
In the late Tang Dynasty, the computing technology was further improved and popularized, and many practical arithmetic books appeared, trying to simplify the multiplication and division algorithm.
Fourthly, the peak of mathematics development in China.
After the demise of the Tang Dynasty, the Five Dynasties and Ten Kingdoms remained the continuation of the warlord melee. Until the Northern Song Dynasty unified China, agriculture, handicrafts and commerce flourished rapidly, and science and technology advanced by leaps and bounds. 1 1 century to14th century (Song and Yuan Dynasties), computational mathematics reached its peak, which was the heyday of unprecedented prosperity and fruitful achievements in ancient Chinese mathematics. During this period, a number of famous mathematicians and mathematical works appeared, which are listed as follows: Jia Xian's Nine Chapters of the Yellow Emperor (165438+mid-20th century), Gu Gen Lun (65438+mid-2nd century), Shu Jiu Chapters (1247) and. Yang Hui's nine-chapter algorithm [126 1], daily algorithm [1262] and Yang Hui's algorithm [1274- 1275], Zhu Shijie's arithmetic enlightenment [65438] achieved in many fields. The main tasks are:
Around 1050, Jia Xian of the Northern Song Dynasty (the year of birth and death is unknown) created the method of increasing, multiplying and opening any height in the Nine Chapters of the Yellow Emperor. It was not until A.D. 18 19 that the Englishman William George Horner came up with the same method. Jia Xian also listed the binomial theorem coefficient table, and the similar "Basga Triangle" did not appear in Europe until17th century. (The Nine Chapters of the Yellow Emperor Arithmetic Fine Grass has been lost)
During the period of 1088- 1095, Shen Kuo put forward the "gap product method" from the production practice problems such as the number of restaurants and the volume of terraces, and began to study the high-order arithmetic progression summation, and established the correct summation formula. Shen Kuo also put forward the theory of "meeting circle" and obtained the first approximate formula of arc length in the history of ancient mathematics in China. He also used the idea of logistics to analyze and study the relationship between logistics food supply and troop transportation.
In A.D. 1247, Qin in the Southern Song Dynasty popularized the multiplication and division method in Shu Shu Jiu Zhang, and described the numerical solution of higher order equations. He listed more than 20 solutions of higher-order equations from practice, the highest of which is the decagonal equation. It was not until16th century in Europe that the Italian Scipio Del Ferro proposed the solution of cubic equation. Qin also systematically studied the theory of primary congruence.
In A.D. 1248, Ye Li (Li Zhi, 1 192- 1279) wrote The Sea Mirror for Measuring the Circle, which was the first work to systematically discuss the "celestial technique" (one-dimensional higher-order equation) and was an outstanding achievement in the history of mathematics. In the round sea mirror? In the Preface, Ye Li criticized the fallacy of belittling scientific practice and treating mathematics as "poor skills" and "playing with things and getting tired of it".
In A.D. 126 1 year, Yang Hui (the year of birth and death is unknown) in the Southern Song Dynasty used "piling technique" to find the sum of several kinds of high-order arithmetic progression. In A.D. 1274, he also described the "Nine-fold Agile Method" in his book The Origin of Multiplication and Division Transformation, and introduced various calculation methods of multiplication and division. In A.D. 1280, Wang Xun and Guo Shoujing listed the interpolation formula of triple difference when compiling the calendar method of the Yuan Dynasty. Guo Shoujing also found two formulas equivalent to the present spherical triangle by geometric method.
In A.D. 1303, Zhu Shijie of Yuan Dynasty (date of birth and death unknown) wrote a jade mirror with four elements. He extended "Tianshi skill" to "four-element skill" (four-element simultaneous equation) and put forward a solution to eliminate elements. It was not until A.D. 1775 that etienne bezout, a Frenchman in Europe, proposed the same solution. Zhu Shijie also studied the summation of finite series, and on this basis, obtained the interpolation formula of higher-order difference. Until 65438 to 1678, the Englishman James Gregory and the Englishman Isaac Newton put forward the general formula of European interpolation.
In the 4th century A.D./KLOC-,people in China already used abacus. Before modern computers appeared, abacus was a simple and effective computing tool in the world.
5. The decline of mathematics in China and the development of daily mathematics.
This period refers to 1582 from the establishment of the Ming Dynasty in the middle of the 4th century to the demise of the Ming Dynasty. In addition to abacus, mathematics is in a weak state as a whole, involving the limitations of abacus, the deletion of mathematics content in the examination system of13rd century, the eight-stage examination system of Daxing in Ming Dynasty and other complex issues. Many Chinese and foreign historians of mathematics are still discussing the reasons involved.
The greatest achievement of the Ming Dynasty was the popularization of abacus, and many abacus readers appeared. It was not until the publication of Cheng Dawei's Command Arithmetic [1592] that the abacus theory became a system, marking the completion of the transition from compilation to abacus calculation. However, due to the popularity of abacus calculation, calculation almost disappeared, and ancient mathematics based on calculation gradually disappeared, and mathematics stagnated for a long time.
Six, the introduction of western elementary mathematics and the combination of Chinese and western.
/kloc-At the end of the 6th century, western missionaries began to emigrate to China. Because of the need of making astronomical calendars in Ming and Qing Dynasties, missionaries began to introduce western elementary mathematics knowledge related to astronomical calendars into China. Under the domination of China mathematician's thought of "Western learning spreading to the east", the integration of Chinese and western mathematics research appeared.
/kloc-At the end of the 6th century, western missionaries and China scholars jointly translated many western mathematical monographs. Among them, the first and most influential one is the first six volumes of Geometrical Elements translated by Italian missionaries Matteo Ricci and Xu Guangqi [1607], whose rigorous logical system and translation methods are highly praised by Xu Guangqi. Xu Guangqi's "Measuring Similarities and Differences" and "The Meaning of Pythagoras" applied the logical reasoning method of "Elements of Geometry" and demonstrated China's Pythagorean observation. In addition, most of the nouns in the textbook "Elements of Geometry" are the first ones and are still in use today. In the imported western mathematics, trigonometry is second only to geometry. Before that, trigonometry had only sporadic knowledge, and then it developed rapidly. The works that introduce western trigonometry include Deiss [2 volumes, 163 1], the secant circle and eight-line table compiled by Deng [6 volumes], and giacomo Rowe's "Measuring Meaning" [10 volumes, 163 1]. In Xu Guangqi's almanac of Chongzhen (volumes 137, 1629- 1633), the mathematical knowledge about the curve of circular vertebra was introduced.
After entering the Qing Dynasty, Mei Wending, an outstanding representative of Chinese and Western mathematics, firmly believed that China's traditional mathematics must be refined, made in-depth research on ancient classics, and treated western mathematics correctly, which made it take root in China and had a positive impact on the climax of mathematics research in the middle of Qing Dynasty. Contemporary mathematicians include Wang Xizhi and Xirao Nian. Emperor Kangxi of the Qing Dynasty loved scientific research, and his Essentials of Mathematics [53 volumes, 1723] was a comprehensive elementary mathematics work, which had a certain influence on mathematical research at that time.
Seven. Arrangement and revival of traditional mathematics
During the reign of Ganjia, the school of Ganjia, which was mainly based on textual research, compiled Sikuquanshu, in which the mathematical works included Ten Books of Calculating Classics and the works of Song and Yuan Dynasties, which made important contributions to the preservation of endangered mathematical classics.
In the research of traditional mathematics, many mathematicians have made inventions. For example, Jiao Xun, Wang Lai and Li Rui, who are called "three friends who talk about the sky", have done a lot of important work. Li obtained the summation formula of triangular self-riding crib in the Class of Stack Ratio [about 1859], which is now called "Lie identity". These works are a step forward than the mathematics in Song and Yuan Dynasties. Ruan Yuan, Li Rui and others compiled a biography of astronomers and mathematicians, The Biography of Universe, with 46 volumes [1795-1810], which is the first study in the history of mathematics.
Eight, western mathematics once again eastward.
After the crow war of 1840, the closed-door policy was forced to stop. The second climax of translation and introduction began with the addition of "Arithmetic" in Wentong Pavilion and the addition of a translation pavilion in Shanghai Jiangnan Manufacturing Bureau. The main translators and works are: the last nine volumes of The Elements of Geometry [1857] jointly translated by Li and the English missionary William, giving China a complete Chinese translation of The Elements of Geometry; Algebra (13); The generation of micro products, volume 18 [1859]. Li and the English missionary Ai He translated 3 volumes of the Theory of Conic Curves, and Hua and the English missionary John Flair translated 25 volumes of Algebra [1872], 8 volumes of Tracing the Source of Differential Products [1874] and Doubts 10 [/kloc-]. In these translations, many mathematical terms and terms were created, which are still used today. 1898, Shi Jing University Hall was established and Wentong Museum was merged. 1905, the imperial examination was abolished and western-style school education was established. The textbooks used were similar to those of other western countries.
Nine, the establishment of modern mathematics in China
This period is a period from the beginning of the 20th century to the present, which is often divided into two stages marked by the establishment of 1949 New China.
Modern mathematics in China started from studying abroad in the late Qing Dynasty and the early Republic of China. 1903 Feng Zuxun who studied mathematics earlier, 1908 Zheng who studied in America, 19 10 Hu Mingfu who studied in America,191/kloc-0. 19 13 Chen who studied in Japan and Xiong qinglai who studied in Belgium [19 15], Su et al. who studied in Japan 19 19. Most of them became famous mathematicians and mathematicians after returning to China, and made important contributions to the development of modern mathematics in China. Among them, Hu Mingfu received his doctorate from Harvard University in the United States on 19 17, becoming the first mathematician in China to receive his doctorate. With the return of foreign students, mathematics education in universities all over the world has improved. At first, there was only the Mathematics Department of Peking University1912; 1920, Jiang Lifu established the Department of Mathematics in Nankai University, Tianjin; 192 1 and 1926, Xiong Qinglai established mathematics departments in Southeast University [now Nanjing University] and Tsinghua University respectively, and soon established Wuhan University, cheeloo university University and Zhejiang University. 1930, Xiong Qinglai initiated the establishment of the Mathematics Research Department in Tsinghua University, and began to recruit graduate students. Chen Shengshen and Wu Daren became the earliest mathematics graduate students in China. In 1930s, [1927], [1934], Hua [1936] and [1936] went abroad to study mathematics, and they all became the backbone of modern mathematics development in China. At the same time, foreign mathematicians also came to China to give lectures, such as Russell in Britain [1920], boekhoff in the United States [1934], osgood [1934], Wiener [1935] and Adama in France [/kloc-]. 1935 the inaugural meeting of chinese mathematical society was held in Shanghai, attended by 33 delegates. The publication of 1936 annals of chinese mathematical society and Journal of Mathematics marks the further development of modern mathematics research in China. Before liberation, mathematical research focused on the field of pure mathematics, and more than 600 theories were published at home and abroad. In terms of analysis, Chen's trigonometric series theory and Xiong Qinglai's research on meromorphic functions and whole functions are representative works, as well as functional analysis, variational methods, differential equations and integral equations; In the field of number theory and algebra, Hua's analytical number theory, geometric number theory, algebraic number theory and modern algebra have achieved remarkable results; In geometry and topology, Su's differential geometry, his algebraic topology, his fiber bundle theory and indicator theory have all done pioneering work. In probability theory and mathematical statistics, Xu Baozhen obtained many basic theorems and strict proofs in univariate and multivariate analysis. In addition, Li Yan and Qian Baoyu initiated the study of the history of Chinese mathematics, and they did a lot of basic work in the annotation and textual research of ancient historical materials, which made our national cultural heritage shine again.
China Academy of Sciences was established in June 1949 1 1. March 195 1 reissue of China Mathematics Journal [1952 changed to Mathematics Journal], March 195 10 reissue of China Mathematics Journal [1953]. 1951August, the Chinese Mathematical Society held its first national congress after the founding of the People's Republic of China to discuss the development direction of mathematics and the reform of mathematics teaching in various schools.
Since the founding of People's Republic of China (PRC), great progress has been made in mathematical research. In the early 1950s, Hua's theory of heap primes [1953], Su's introduction to projective curves [1954], Chen's sum of series of rectangular functions [1954], and Li Yan's series on the history of middle arithmetic [65438] were published. In addition to continuing to make new achievements in number theory, algebra, geometry, topology, function theory, probability theory and mathematical statistics, history of mathematics and other disciplines, they have also made breakthroughs in differential equations, computing technology, operational research, mathematical logic, mathematical foundation and so on. Many of them have reached the world advanced level, and at the same time, they have trained and grown a large number of outstanding mathematicians.
At the end of 1960s, China's mathematics research basically stopped, education was paralyzed, personnel were drained, and foreign exchanges were interrupted. After many efforts, the situation has changed slightly. 1970, Mathematics Magazine was reissued, and Practice and Understanding of Mathematics was founded. 1973, Chen Jingrun published a paper "A big even number is expressed as the sum of the products of a prime number and no more than two prime numbers" in China Science, which made outstanding achievements in the research of Goldbach conjecture. In addition, mathematicians in China have some original opinions on function theory, Markov process, probability application, operational research and optimization methods.
1978 165438+ the third congress was held in the Chinese mathematical society on 10, which marked the revival of mathematics in China. 1978 National Mathematics Competition resumed, 1985 China began to participate in the International Mathematical Olympiad. 198 1 year, Chen Jingrun and other mathematicians won the National Natural Science Award. 1983, the state awarded the first batch of 18 young and middle-aged scholars with doctorates, among which mathematicians accounted for 2/3. 1986, China sent representatives to the international congress of mathematicians for the first time and joined the international mathematical union. Wu Wenjun was invited to give a 45-minute lecture on the history of ancient mathematics in China. In the past ten years, mathematical research has achieved fruitful results, and the number of published papers and monographs has doubled and the quality has been rising. At the annual meeting of 1985 to celebrate the 50th anniversary of the founding of chinese mathematical society, the long-term goal of mathematics development in China was determined. The delegates are determined to make unremitting efforts to make China a new mathematical power in the world at an early date.
X. Characteristics of China's Mathematics
(1) takes algorithm as the center and belongs to applied mathematics. China's mathematics is not divorced from the reality of social life and production, and its goal is to solve practical problems. Mathematical research centers on establishing algorithms and improving computing technology.
(2) Strong sociality. In China's traditional mathematical culture, mathematics is one of the six arts (ritual, music, shooting, imperial command, calligraphy and number) cultivated by Confucianism. Its function is to "connect with God, obey orders, manage the affairs of the world and divide all things", so China's traditional mathematics is always branded with China's philosophy and ancient academic thoughts, and is often intertwined with mathematics. At the same time, mathematics education and research were often controlled by the feudal government. Mathematics education and imperial examination system in Tang and Song Dynasties and mathematicians in past dynasties were often astronomical officials of the government, which fully reflected this nature.
(3) The theory is highly generalized. Because China's traditional mathematics pays attention to solving practical problems, and because of China's comprehensive inductive thinking, China's traditional mathematics doesn't care about the formalization of mathematical theory, but this doesn't mean that China's tradition only stays at the empirical level without theoretical achievements. China's mathematical algorithms actually contain the theoretical basis for establishing these algorithms. Mathematicians in China are used to building mathematical concepts and methods on a few self-evident and intuitive mathematical principles, such as the theory of rate in algebra, the principle of complementary entry and exit in plane geometry, Yang Equestrian in solid geometry, the principle of section in surface theory (or Liu Zu's principle, that is, principle) and so on.
XI。 The Influence of China's Mathematics on the World
Mathematical activities have two basic tasks-proof and calculation. The former is because of accepting the axiomatic (deductive) mathematical cultural tradition, while the latter is because of accepting the mechanized (algorithmic) mathematical cultural tradition. In the world mathematical culture tradition, Greek mathematics represented by Euclid's Elements of Geometry is undoubtedly the foundation of western deductive mathematics tradition, while China mathematics represented by Nine Chapters of Arithmetic is undoubtedly the foundation of oriental algorithmic mathematics tradition, which reflects things and promotes the development of world mathematical culture together.
China's mathematics spread to India and Arabia through the Silk Road, and later to the West through Arabs. Moreover, in the cultural circle of Chinese characters, it has been affecting the mathematical development of Asian countries such as Japan, Korean Peninsula and Vietnam.