Mathematics, which originated from the early production activities of human beings, is one of the six great arts in ancient China, and is also regarded as the starting point of philosophy by ancient Greek scholars. Mathematical Greek μ α θ η μ α κ? Mathematickó s) means "the basis of learning" and comes from μαρθξμα(máthema) ("science, knowledge and learning").
Mathematical Manuscripts: Evolution of Mathematics
The evolution of mathematics can be regarded as the continuous development of abstraction and the extension of subject matter. The first abstract concept is probably number, and its cognition that two apples and two oranges have something in common is a great breakthrough in human thought. In addition to knowing how to calculate the number of actual substances, prehistoric people also knew how to calculate the number of abstract substances, such as time-date, season and year. Arithmetic (addition, subtraction, multiplication and division) will naturally occur. Ancient stone tablets also confirmed the knowledge of geometry at that time.
In addition, writing or other systems that can record numbers are needed, such as Mu Fu or chips used by the Inca Empire to store data. There are many different counting systems in history.
Since the historical era, the main principles in mathematics have been formed to do many calculations related to taxation and trade, understand the relationship between numbers, measure land and predict astronomical events. These needs can be simply summarized as the study of quantity, structure, space and time in mathematics.
Contents of Mathematical Manuscripts: The Germination of Ancient Mathematics in China
At the end of primitive commune, after the emergence of private ownership and commodity exchange, the concepts of number and shape developed further. The pottery unearthed during Yangshao culture period has been engraved with the symbol representing 1234. By the end of the primitive commune, written symbols had begun to replace knotted notes.
Pottery unearthed in Xi 'an Banpo has an equilateral triangle composed of 1 ~ 8 dots, and a pattern of 100 small squares divided into squares. The houses in Banpo site are all round and square. In order to draw circles and determine straightness, people have also created drawing and measuring tools such as rulers, moments, rulers and ropes. According to Records of Historical Records Xia Benji, Yu Xia used these tools in water conservancy.
In the middle of Shang Dynasty, a set of decimal numbers and notation had been produced in Oracle Bone Inscriptions, the largest of which was 30 thousand; At the same time, the Yin people recorded the date of 60 days with 60 names, including Jiazi, Yechou, Bingyin and Dingmao, which were composed of ten heavenly stems and twelve earthly branches. In the Zhou Dynasty, eight kinds of things were previously represented by eight diagrams composed of yin and yang symbols, which developed into sixty-four hexagrams, representing sixty-four kinds of things.
The book Parallel Computation in 1 century BC mentioned the methods of using moments of high, deep, wide and distance in the early Western Zhou Dynasty, and listed some examples, such as hook three, strand four, chord five and ring moments can be circles. It is mentioned in the Book of Rites that the aristocratic children of the Western Zhou Dynasty have to learn numbers and counting methods since they were nine years old, and they have to be trained in rites and music, shooting, controlling, writing and counting. As one of the "six arts", number has begun to become a special course.
During the Spring and Autumn Period and the Warring States Period, calculation has been widely used and decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world. During this period, econometrics was widely used in production, and mathematics was improved accordingly.
The contention of a hundred schools of thought in the Warring States period also promoted the development of mathematics, especially the dispute of rectifying the name and some propositions were directly related to mathematics. Famous experts believe that the abstract concepts of nouns are different from their original entities. They put forward that "if the moment is not square, the rules cannot be round", and defined "freshman" (infinity) as "nothing beyond the maximum" and "junior" (infinitesimal) as "nothing within the minimum". He also put forward the idea that "one foot is worth half a day, which is inexhaustible".
Mohism believes that names come from things, and names can reflect things from different sides and depths. Mohist school gave some mathematical definitions. Such as circle, square, flat, straight, sub (tangent), end (point) and so on.
Mohism disagreed with the proposition of "one foot" and put forward the proposition of "non-half" to refute: if a line segment is divided into two halves indefinitely, there will be a non-half, which is a point.
The famous scholar's proposition discusses that a finite length can be divided into an infinite sequence, while the Mohist proposition points out the changes and results of this infinite division. The discussion on the definition and proposition of mathematics by famous scholars and Mohists is of great significance to the development of China's ancient mathematical theory.
Mathematical Manuscripts: The Development of Ancient Mathematics in China
Metaphysics, which appeared in Wei and Jin dynasties, was not bound by Confucian classics in Han dynasty and was active in thought. It can argue and win, use logical thinking and analyze truth, all of which are conducive to improving mathematics theoretically. During this period, the Nine Chapters Heavy Difference Diagram appeared in Wu and Zhao's annotation of Zhou Huishu, Xu Yue's annotation of Nine Chapters Arithmetic in the late Han Dynasty and early Wei Dynasty, and Liu Hui's annotation of Nine Chapters Arithmetic in the Wei and Jin Dynasties. The work of Zhao Shuang and Liu Hui laid a theoretical foundation for the ancient mathematical system of China.
Zhao Shuang was one of the earliest mathematicians who proved and deduced mathematical theorems and formulas in ancient China. The Pythagorean Grid Diagram and Annotation and the Daily Height Diagram and Annotation, which he supplemented in Zhou Pian Shu Jing, are very important mathematical documents. In Pythagorean Square Diagram and Notes, he put forward five formulas to prove Pythagorean theorem and Pythagorean shape with chord diagram; In Sunrise Picture, he proved the weight difference formula widely used in Han Dynasty with the graphic area. Zhao Shuang's work was groundbreaking and played an important role in the development of ancient mathematics in China.
Liu Jicheng, who was contemporary with Zhao Shuang, developed the thoughts of famous scholars and Mohists in the Warring States Period and advocated strict definition of some mathematical terms, especially important mathematical concepts. He believes that mathematical knowledge must be "analyzed" in order to make mathematical works concise, compact and beneficial to readers. His Notes on Nine Chapters of Arithmetic not only explains and deduces the methods, formulas and theorems of nine chapters of arithmetic as a whole, but also gets great development during the discussion. Liu Hui created secant, proved the formula of circle area by using the idea of limit, and calculated the pi as 157/50 and 3927/ 1250 by theoretical method for the first time.
Liu Hui proved by infinite division that the volume ratio of right-angled square cone to right-angled tetrahedron is always 2: 1, which solved the key problem of general solid volume. When proving the volume of square cone, cylinder, cone and frustum, Liu Hui put forward the correct method to solve the volume of sphere completely.
After the Eastern Jin Dynasty, China was in a state of war and north-south division for a long time. The work of Zu Chongzhi and his son is the representative work of the development of mathematics in South China after the economic and cultural shift to the south. On the basis of Liu Hui's Notes on Nine Chapters of Arithmetic, they greatly promoted traditional mathematics. Their mathematical work mainly includes: calculating pi between 3.1415926 ~ 3.1415927; Put forward the principle of ancestral pestle; The solutions of quadratic and cubic equations are put forward.
Presumably, Zu Chongzhi calculated the area of the circle inscribed by the regular polygon 6 144 and the regular polygon 12288 on the basis of Liu Hui secant method, and thus obtained this result. He also obtained two fractional values of pi by a new method, namely the approximate ratio of 22/7 and the density ratio of 355/ 1 13. Zu Chongzhi's work made China lead the west in the calculation of pi for about one thousand years.
Zu Chongzhi Zi Zuxuan summed up Liu Hui's related work, and put forward that "the potential is the same, but the product is not different", that is, two solids with the same height, if the horizontal cross-sectional area of any height is equal, the volumes of the two solids are equal, which is the famous axiom of Zuxuan. Zu Xuan applied this axiom to solve Liu Hui's unsolved spherical volume formula.
Emperor Yang Di was overjoyed and made great achievements, which objectively promoted the development of mathematics. In the early Tang Dynasty, Wang Xiaoyu's "Jigu Shujing" mainly discussed earthwork calculation, division of labor, acceptance and calculation of warehouses and cellars in civil engineering, which reflected the mathematical situation in this period. Wang Xiaotong established the cubic equation of number without using mathematical symbols, which not only solved the needs of the society at that time, but also laid the foundation for the establishment of the art of heaven. In addition, for the traditional Pythagorean solution, Wang Xiaotong also used the digital cubic equation to solve it.
In the early Tang Dynasty, the feudal rulers inherited the Sui system, and in 656, they set up the Arithmetic Museum in imperial academy, with 30 students, including arithmetic doctors and teaching assistants. Ten arithmetic classics edited and annotated by Taishiling Li are used as teaching materials for students in the Arithmetic Museum and as the basis for verifying arithmetic. Ten Books of Calculating Classics compiled by Li and others is of great significance in preserving classical works of mathematics and providing literature for mathematical research. Their notes on Zhoupian suan Jing, Nine Chapters Arithmetic and Island Suan Jing are helpful to readers. During the Sui and Tang Dynasties, due to the need of calendar, celestial mathematicians created quadratic function interpolation method, which enriched the content of ancient mathematics in China.
Calculation and compilation was one of the main calculation tools in ancient China. It has many advantages, such as simplicity, image and concreteness, but it also has some disadvantages, such as large compiling area and easy to make mistakes when the operation speed is accelerated. So the reform was carried out very early. Among them, Taiyi, Ermi, Sancai and Abacus are all abacus with beads, which is an important technical reform. In particular, "abacus calculation" inherits the advantages of calculating five liters and decimal places, overcomes the shortcomings of inconvenient calculation and preparation of vertical and horizontal numbers, and its advantages are very obvious. But at that time, the multiplication and division algorithm could not be performed continuously. The abacus beads have not been worn and are not easy to carry, so they are still not widely used.
After the middle Tang Dynasty, the prosperity of commerce and the increase of digital calculation urgently required the reform of calculation methods. It can be seen from the list of books left by New Tang Shu and other documents that this algorithm reform is mainly to simplify the multiplication and division algorithm. The algorithm reform in the Tang Dynasty enabled multiplication and division to be operated in parallel, which was suitable for both calculation and abacus calculation.