Mathematical attribute is the measurable attribute of anything, that is, mathematical attribute is the most basic attribute of things. The existence of measurable attributes has nothing to do with parameters, but its result depends on the choice of parameters. For example, time is measured in years, months, days or hours, minutes and seconds; Space, whether measured in meters, microns, inches or light years, always has their measurable properties, but the accuracy of the results is related to these reference coefficients.
Mathematics is a science that studies quantitative relations and spatial forms in the real world. In short, it is a science that studies numbers and shapes. Due to the needs of life and labor, even the most primitive people know simple counting, and it has developed from counting with fingers or objects to counting with numbers.
The knowledge and application of basic mathematics will always be an indispensable part of individual and group life. The refinement of its basic concepts can be seen in ancient mathematical classics of ancient Egypt, Mesopotamia and ancient India. Since then, its development has made small progress, until the Renaissance in16th century, and the mathematical innovation generated by the interaction with new scientific discoveries accelerated the knowledge until today.
Today, mathematics is used in different fields of the world, including science, engineering, medicine and economics. The application of mathematics in these fields is usually called applied mathematics, which sometimes leads to new mathematical discoveries and the development of new disciplines. Mathematicians also study pure mathematics with no practical application value, even though its application is often discovered later.
The French Bourbaki School, founded in 1930s, believes that mathematics, at least pure mathematics, is a theory to study abstract structures. Structure is a deductive system based on initial concepts and axioms. According to the Bourbaki school, there are three basic abstract structures: algebraic structure (group, ring, domain ...), ordered structure (partial order, total order ...) and topological structure (neighborhood, limit and topological structure.
Mathematics, called arithmetic in ancient times, is an important subject in ancient Chinese science. According to the characteristics of the development of ancient mathematics in China, it can be divided into five periods: the germination period; The formation of the system; Development; Prosperity and the integration of Chinese and western mathematics.
The Germination of Ancient Mathematics in China
At the end of primitive commune, after the emergence of private ownership and barter, the concepts of number and shape were further developed. The pottery unearthed during Yangshao culture period is 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 with 65,438+0 ~ 8 points, and 65,438+000 small squares divided into squares. The buildings in Banpo site are all round and square. In order to draw squares and determine straightness, people have also created drawing and measuring tools such as rulers, moments, rulers and ropes. According to the historian Xia Benji, Yu Xia has been in charge of 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 things were represented by eight diagrams composed of yin and yang symbols, which developed into sixty-four hexagrams, representing sixty-four things.
The method of using high, deep, wide and distant moments in the early Western Zhou Dynasty was mentioned in the Parallel Calculation Classics of the Zhou Dynasty in the first century BC, and an example was given that the moments of the hook 3, the strand 4, the chord 5 and the ring of the Pythagorean shape could be circles. It is mentioned in the Book of Rites that the children of nobles in the Western Zhou Dynasty should learn counting and counting methods from the age of nine, and they should also accept gifts, music, shooting, control and so on.
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. In this period, econometrics has been widely used in production, and mathematics has also been improved accordingly.
A hundred schools of thought contended during 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 scholars think that the abstract concepts of nouns are different from their original entities. They put forward that "rules should not be round", and define "freshman" (infinity) as "nothing outside the biggest" and "smallest" (infinitesimal) as "nothing inside the smallest".
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, 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 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 between famous scholars and Mohists about the definition and proposition of mathematics is of great significance to the development of China's ancient mathematical theory.
The Formation of Ancient Mathematics System in China
Qin and Han dynasties were the rising period of feudal society, with rapid economic and cultural development. It was during this period that the ancient mathematical system of China was formed, the main symbol of which was that arithmetic became a specialized subject, and mathematical works represented by Nine Chapters of Arithmetic appeared.
Nine Chapters Arithmetic is a summary of the development of mathematics during the establishment and consolidation of feudal society in the Warring States, Qin and Han Dynasties. As far as its mathematical achievements are concerned, it is a world-famous mathematical work. For example, the operation of quartering, the present technique (called the three-rate method in the west), the square root and square root (including the numerical solution of quadratic equation), the surplus and deficiency technique (called the double solution in the west), various formulas of area and volume, the solution of linear equations, the addition and subtraction rules of positive and negative operations, the pythagorean solution (especially the solution of pythagorean theorem and pythagorean number) and so on. , are all high level.
"Nine Chapters Arithmetic" has several remarkable characteristics: it adopts the form of mathematical problem sets divided into chapters according to categories; Formulas are all developed from counting method; Mainly arithmetic and algebra, rarely involving graphic properties; Attach importance to application, lack of theoretical explanation, etc.
These characteristics are closely related to the social conditions and academic thoughts at that time. In Qin and Han dynasties, all science and technology should serve the establishment and consolidation of feudal system and the development of social production at that time, emphasizing the application of mathematics. Finally, the book Nine Chapters Arithmetic, which was written in the early years of the Eastern Han Dynasty, ruled out the discussion on the definition and logic of nouns by famous scholars and Mohists in the Warring States period, and focused on mathematical problems and their solutions, which was completely in line with the development of society at that time.
During the Sui and Tang Dynasties, Nine Chapters Arithmetic spread to Korea and Japan, and became the mathematics textbooks of these countries at that time. Some of its achievements, such as decimal numerical system, skills of present use and skills of surplus and deficiency, have also spread to India and Arabia, and then to Europe through India and Arabia, which has promoted the development of world mathematics.
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 argues for victory, uses logical thinking and analyzes righteousness, which are all conducive to the theoretical improvement of mathematics. At the end of the Han Dynasty, Zhou Pian, Nine Chapters Arithmetic, Nine Chapters Arithmetic and Nine Chapters Multiplication Diagram all appeared in this period. Zhao Shuang and Liu Hui.
Zhao Shuang was one of the earliest mathematicians who proved and deduced mathematical theorems and formulas in ancient China. Pythagoras' Square Diagram and Annotations, as well as the Sunrise Square Diagram and Annotations added by him in Zhouyi Suan Jing, are very important mathematical documents. In Pythagorean Square Diagram and Notes, he put forward five formulas to prove Pythagorean theorem and solve 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 "dissected" in order to make mathematical works concise and rigorous, which is 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 makes great progress in the discussion. Liu Hui created secant technology, proved the area formula of a circle with 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 completely solve the volume of sphere.
After the Eastern Jin Dynasty, China was divided by the war between the North and the South 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. They greatly developed traditional mathematics on the basis of Liu Hui's Notes on Nine Chapters of Arithmetic. Their mathematical work mainly includes: calculating pi from 3.1415926 ~ 3.65438+. Put forward the principle of ancestor (constant sky); The solutions of quadratic and cubic equations are put forward.
Presumably, Zu Chongzhi calculated the inscribed area of the regular 6 144 polygon and the regular 12288 polygon 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.
Zu Chongzhi's son Zu (Riheng) summed up Liu Hui's related work and put forward that "if the potentials are the same, the products cannot be different", that is, if the horizontal cross-sectional areas of two solids at any height are equal, the volumes of the two solids are equal, which is the famous Zu (Riheng) axiom. Zu (Riheng) applied this axiom to solve Liu Hui's unsolved spherical volume formula.
Yang Di the Great's exultation and great construction objectively promoted the development of mathematics. In the early Tang Dynasty, Wang Xiaotong's "Jigu Shujing" 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 number without using mathematical symbols, which not only solved the needs of society at that time, but also laid the foundation for the establishment of celestial science later.
In the early Tang Dynasty, the feudal rulers inherited the Sui system and set up an arithmetic hall in imperial academy in 656, with 30 doctors and teaching assistants. Taishiling Li and others compiled Ten Books on Arithmetic as teaching materials for students in Arithmetic Hall, and the arithmetic examination in Ming Dynasty was also based on these books. Ten Books on Arithmetic compiled by Li and others is of great significance for preserving classic works of mathematics and providing literature materials for mathematical research. Their notes on Zhou Bi Shu Jing, Jiu Ben Shu Jing and Island Shu Jing are very 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 compilation is the main calculation tool in ancient China, which has the advantages of simplicity, image and concreteness, but it also has some shortcomings, such as large occupation area and easy to fiddle with misoperation when the operation speed is accelerated, so it has been reformed very early. Among them, Taiyi calculation, two-meter calculation, three-talent calculation and abacus calculation are all abacus calculations with beads, which are important technical reforms. In particular, "abacus calculation" inherits the calculation of five liters and ten decimals. It also overcomes the shortcomings of inconvenient vertical and horizontal counting and preparation, and has obvious advantages. However, the multiplication and division algorithm at that time could not be carried out continuously, and the bead counting was not convenient to carry, so it was 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.
The Prosperity of Ancient Mathematics in China
In 960, the establishment of the Northern Song Dynasty ended the separatist regime of the Five Dynasties and Ten Countries. Agriculture, handicrafts and commerce in the Northern Song Dynasty flourished unprecedentedly, and science and technology advanced by leaps and bounds. Gunpowder, compass and printing are widely used in this situation of rapid economic growth. In 2008+0084, Ten Books of Calculation were printed and published for the first time in the Secretariat Province, 1265438+.
During the 300 years from 1 1 to14th century, a number of famous mathematicians and mathematical works appeared, such as Jia Xian's Nine Chapters on Arithmetic and Fine Grass of the Yellow Emperor, The Theory of Ancient Roots, The Book of Nine Chapters, The Survey of the Sea Mirror Circle and Yi Gu Yan Duan.
The leap from square root, square root to square root for more than four times is realized by Jia Xian Yang Hui's multiplication Kaiping method and the nine chapters of algorithm compilation, Jia Xian's multiplication Kaiping method. Detailed Explanation of Algorithms in Chapter Nine contains Jia Xian's Root Flow, Seeking Base Grass by Multiplication and Addition, and examples of using multiplication and addition to open the fourth power. According to these records, it can be confirmed that Jia Xian found the binomial coefficient table and created the method of multiplication and division. These two achievements had a great influence on the whole mathematics of Song and Yuan Dynasties, among which Jia Xian Triangle was put forward more than 600 years earlier than Pascal Triangle in the west.
It was Liu Yi who extended the multiplication and division method to the solution of digital higher-order equations (including the case of negative coefficients). Yang Hui's algorithm volume introduces 22 quadratic equations and 1 quartic equations in the original book. The latter is the earliest example of solving higher-order equations by multiplication and division more than three times.
Qin is an expert in solving higher-order equations. He collected 2 1 problems (the highest number is 10) of solving higher-order equations by adding, multiplying and opening. In order to adapt to the calculation program of increase, multiplication and opening, Jiu Shao defined the constant term as negative, and divided the solutions of higher-order equations into various types. When the root of the equation is non-integer, Qin continues to look for the root. Or use the sum of the coefficients of the power of the reduced root transformation equation as the denominator and the constant as the numerator to represent the non-integer part of the root, which is the development of the irrational number processing method in "Nine Chapters Arithmetic" and Liu Hui's note. When seeking the second digit of the root, Qin also put forward the method of dividing the coefficient of the first term by the constant term to try to divide the second digit of the root, which was more than 500 years earlier than the earliest method of Horner in the west.
Astronomers Wang Xun and Guo Shoujing in Yuan Dynasty solved the interpolation problem of cubic function in time calendar. Qin mentioned the interpolation method in the title of "Composition Pushing Stars", Zhu Shijie mentioned the interpolation method in the title of "Four Swords" (they called it a trick), and Zhu Shijie got an interpolation formula of quartic function.
Using Tianyuan (equivalent to X) as the symbol of unknown number, a high-order equation was established, which was called Tianshu in ancient times. This is the first time in the history of Chinese mathematics to introduce symbols and use symbolic operations to solve the problem of establishing higher-order equations. The earliest extant heavenly work is Ye Li's Rounding Sea Mirror.
It is another outstanding creation of mathematicians in Song and Yuan Dynasties to extend celestial sphere to higher-order simultaneous equations of binary, ternary and quaternary. It is Zhu Shijie's "Meeting with Siyuan" that has been handed down to this day and systematically discussed this outstanding creation.
Zhu Shijie's representation of high-order four-variable simultaneous equations was developed on the basis of celestial technology. He put the constant in the center, the powers of the four variables in the four directions of up, down, left and right, and the other terms in the four quadrants. Zhu Shijie's greatest contribution is to put forward the four-variable elimination method. Firstly, one variable is selected as the unknown, and the polynomial composed of other variables is used as the coefficient of this unknown, and it is listed as several high-order equations of one variable. Then, the unknown number is gradually eliminated by using the method of mutual multiplication and elimination. By repeating this step, other unknowns can be eliminated, and finally the solution can be solved by the method of increasing, multiplying and opening. This is an important development of linear method group solution, which is more than 400 years earlier than similar methods in the west.
The Pythagorean solution had a new development in the Song and Yuan Dynasties. Zhu Shijie put forward the method of solving Pythagorean formula with known chord sum and chord sum, which supplemented the deficiency of Nine Chapters Arithmetic. Ye Li made a detailed study of Pythagorean compatible circle in Measuring the Circle Sea Mirror, and obtained nine formulas of compatible circle, which greatly enriched the contents of ancient geometry in China.
Given the included angle between the ecliptic and the equator and the back arc of the ecliptic from winter solstice to vernal equinox, finding the back arc and declination number of the right meridian is the problem of solving the spherical right triangle. The traditional calendar is calculated by interpolation. In Yuan Dynasty, Wang Xun and Guo Shoujing used the traditional Pythagorean method to solve this problem, while Shen Kuo used the method of meeting the circle and Tianyuan. However, they got an approximate formula, and the result was not accurate enough. But their whole calculation steps are correct.
The climax of China's ancient computing technology reform also appeared in the Song and Yuan Dynasties. The historical documents in the Song and Yuan Dynasties contain a large number of practical arithmetic bibliographies in this period, far more than those in the Tang Dynasty. The main content of the reform is still multiplication and division. At the same time as the algorithm reform, the abacus of piercing beads may have appeared in the Northern Song Dynasty. However, if the modern abacus is regarded as both a bead-piercing abacus and a set of perfect algorithms and formulas, it should be said that it was finally completed in the Yuan Dynasty.
The prosperity of mathematics in Song and Yuan Dynasties is the inevitable result of the development of social economy and science and technology, and it is also the inevitable result of the development of traditional mathematics. In addition, mathematicians' scientific thinking and mathematical thinking are also very important. Mathematicians in Song and Yuan Dynasties all opposed the mysticism of image number in Neo-Confucianism to varying degrees. Although Qin once advocated that mathematics and Taoist thought have the same origin, he later realized that there is no mathematics that can "connect the gods", only mathematics that can "cross the sky". In the preface to the encounter with Siyuan, Mo Ruo put forward the idea of "taking the virtual image as the truth and asking the truth with the virtual image", which represents a highly abstract thinking method; Yang Hui studied the structure of vertical and horizontal diagrams, revealed the essence of Luo Shu, and strongly criticized the mysticism of image numbers. These are undoubtedly important factors to promote the development of mathematics.