Mathematics, which originated from the early production activities of human beings, is one of the six ancient arts in China, and is also regarded as the starting point of philosophy by ancient Greek scholars.
Mathematical Greek μ α θ η μ α τ κ? Mathematikós) means "the foundation of learning", which comes from ματθημα(máthema) ("science, knowledge, learning").
Mathematical Manuscript: the Evolution of Mathematics
The evolution of mathematics can be regarded as the continuous development of abstraction or the extension of subject matter.
The first abstracted concept is probably number, and its cognition that there is something in common between two apples and two oranges is a great breakthrough in human thought.
In addition to knowing how to count the number of actual substances, prehistoric humans also knew how to count the number of abstract substances, such as time-day, season and year.
Arithmetic (addition, subtraction, multiplication and division) also came into being naturally.
Ancient stone tablets also confirmed the knowledge of geometry at that time.
Further, writing or other systems that can record numbers are needed, such as Fumu or Chip, which was used to store data in the Inca Empire.
There have been many different counting systems in history.
From the beginning of the historical era, the main principles in mathematics were formed to do many calculations related to taxation and trade, to understand the relationship between numbers, to measure land, and to predict astronomical events.
These needs can be simply summarized as the study of quantity, structure, space and time in mathematics.
Content of Mathematical Manuscript: The Germination of Ancient Mathematics in China
At the end of the primitive commune, after the emergence of private ownership and exchange of goods, the concepts of number and shape were further developed. The pottery unearthed during Yangshao culture period was engraved with the symbol 1234.
By the end of the primitive commune, writing symbols had begun to replace knotting notes.
Pottery unearthed in Banpo, Xi 'an has an equilateral triangle consisting of1~ 8 dots and a pattern of 100 small squares divided into squares. The houses of Banpo site are all round and square.
In order to draw a circle and determine straightness, people have also created drawing and measuring tools such as rules, moments, standards and ropes.
According to Records of Historical Records Xia Benji, Xia Yu used these tools in water control.
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 used ten heavenly stems and twelve earthly branches to form 60 names such as Jiazi, Yechou, Bingyin and Dingmao to record the date of 60 days. In the Zhou Dynasty, eight kinds of things were represented by eight diagrams composed of Yin and Yang symbols before, and they were developed into sixty-four hexagrams, representing 64 kinds of things.
The Book of Parallel Calculation in the 1st century BC mentioned the method of measuring the height, depth, width and distance by moments in the early Western Zhou Dynasty, and cited some examples, such as the hook three, the strand four, the chord five and the ring moment can be a circle.
"The Book of Rites" mentioned that the aristocratic children in the Western Zhou Dynasty had to learn numbers and counting methods from the age of nine, and they had to be trained in etiquette, 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 the decimal notation has been used, which is of epoch-making significance to the development of mathematics in the world.
During this period, measurement mathematics was widely used in production, and mathematics was also improved accordingly.
The contention of a hundred schools of thought in the Warring States period also promoted the development of mathematics, especially the debate on correcting the name and some propositions was directly related to mathematics.
Famous experts think that the abstract concept of nouns is different from their original entities. They put forward that "the rules can't be round if the moment is not square", and define "big one" (infinity) as "nothing beyond the largest" and "small one" (infinitesimal) as "nothing within the smallest".
He also put forward some propositions, such as "a foot's worth, taking half of it every day and inexhaustible".
Mohism, on the other hand, believes that names come from things, and names can reflect things from different aspects and depths.
Mohism gives some mathematical definitions.
Such as circle, square, flat, straight, secondary (tangent), end (point) and so on.
Mohist school disagrees with the proposition of "one foot" and puts forward a proposition of "non-half" to refute it: if a line segment is divided in half and half indefinitely, there will be a "non-half" that can no longer be divided, and this "non-half" is a point.
The proposition of famous scholars discusses that finite length can be divided into an infinite sequence, while the proposition of Mohism points out the change and result of this infinite division.
The discussion of mathematical definitions and propositions of 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, but was active in thought. It can argue and win, and it can use logical thinking and analyze righteousness, which are beneficial to improve mathematics theoretically.
During this period, Zhao Shuang of the State of Wu annotated The Classic of Zhou Kuai Shu, Xu Yue of Wei Chu at the end of Han Dynasty wrote the annotation of Nine Chapters Arithmetic, Liu Hui wrote the annotation of Nine Chapters Arithmetic at the end of Wei Dynasty and at the beginning of Jin Dynasty, and the diagram of Nine Chapters Heavy Difference all appeared.
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.
His supplementary "Pythagorean Square Diagram and Notes" and "Daily Altitude Diagram and Notes" in the Book 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 by chord diagram; In the "Sunrise Map and Notes", he used the graphic area to prove the weight difference formula widely used in the Han Dynasty. Zhao Shuang's work was groundbreaking and played an important role in the development of ancient mathematics in China.
At the same time as Zhao Shuang, Liu Huiyue inherited and developed the thoughts of famous artists and Mohists in the Warring States Period, and advocated strict definitions of some mathematical terms, especially important mathematical concepts. He believed that mathematical knowledge must be "analyzed" in order to make mathematical works concise and close and beneficial to readers.
His annotation of Nine Chapters Arithmetic not only explains and deduces the methods, formulas and theorems of Nine Chapters Arithmetic in general, but also develops greatly in the process of discussion.
Liu Hui created the secant technique, proved the area formula of a circle 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 that the volume ratio of right-angled square cone to right-angled tetrahedron is always 2: 1 by infinite division method, 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 way to solve the volume of the ball 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 annotation of Nine Chapters of Arithmetic, they have greatly advanced the 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 regular 6 144 polygon and the regular 12288 polygon inscribed in the circle on the basis of Liu Hui's secant technique, 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 Xuan, the son of Zu Chongzhi, summed up Liu Hui's relevant work and put forward that "the power potentials are the same, but the products are not different", that is, two solids with the same height, and if the horizontal cross-sectional areas at any height are equal, the volumes of the two solids are equal, which is the famous Zu Xuan axiom.
Zu Xuan applied this axiom to solve Liu Hui's unsolved spherical volume formula.
Emperor Yangdi was overjoyed and made great achievements, which objectively promoted the development of mathematics.
At the beginning of the Tang Dynasty, Wang Xiaotong's "Ji Gu Shu Jing" mainly discussed the calculation of earthwork, 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 numbers 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 celestial art later.
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 doctors and teaching assistants in arithmetic and 30 students.
Ten Books of Arithmetic Classics compiled and annotated by Taishiling Li Chunfeng are used as textbooks for students in the Arithmetic Museum, and these books are also used as the basis for the examination of Ming Arithmetic.
The Ten Books of Calculating Classics compiled by Li Chunfeng and others is of great significance in preserving classic works of mathematics and providing literature materials for mathematical research.
Their annotations to Zhou Pian 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 the interpolation method of quadratic function, which enriched the content of ancient mathematics in China.
Calculation and preparation is one of the main calculation tools in ancient China. It has many advantages, such as simplicity, image and concreteness, but it also has some shortcomings, such as the large area occupied by the preparation and the mistakes caused by improper handling when the operation speed is accelerated. Therefore, the reform has been carried out very early.
Among them, Taiyi calculation, two-meter calculation, three-talent calculation and abacus calculation are all trough abacus with beads, which are important reforms in technology.
In particular, "abacus calculation" inherits the advantages of calculating five liters and decimal places, and overcomes the disadvantages of calculating vertical and horizontal numbers and inconvenient preparation, and its advantages are very obvious.
But at that time, the multiplication and division algorithm could not be carried out in a row.
The abacus beads have not been worn yet, so it is inconvenient to carry, so it is still not widely used.
After the mid-Tang Dynasty, the prosperity of business and the increase of digital calculation urgently require the reform of calculation methods. From the book list left by the New Tang Book and other documents, we can see that this algorithm reform is mainly to simplify the multiplication and division algorithm. The algorithm reform in the Tang Dynasty enables the multiplication and division method to operate in a row, which is suitable for both calculation and abacus calculation.