Does anyone know anything about the history of mathematics?

China is one of the ancient civilizations of the world, located in the eastern part of Asia and on the west coast of the Pacific Ocean. Mathematics has a long history in China, and its achievements have been brilliant. The following is an account of the historical development of mathematics in China.

1. The pre-Qin budding period

The Yellow River basin and Yangtze River basin is the cradle of Chinese culture, about 2000 BC, in the middle and lower reaches of the Yellow River produced the first slave state - Xia Dynasty. It was followed by the Shang and Yin Dynasties [about 1500 B.C -1027 B.C], and the Zhou Dynasty [1027 B.C -221 B.C]. The period from the eighth century B.C. to the establishment of the Qin Dynasty [221 B.C] is also known as the Spring and Autumn Period and the Warring States Period.

According to Yi. According to "The Rhetoric", "the ancient world ruled by knotted ropes, and the sages of later times changed the rules to books and deeds". In the oracle bone inscriptions unearthed in Yinxu, there are a lot of words for counting. From one to ten, and hundreds, thousands, ten thousand is the special notation text, *** there are 13 independent symbols, notation with the book written in the book, which has a decimal system of notation, the largest number appeared for thirty thousand.

Calculation chip is an ancient Chinese calculation tool, and this calculation method is called chips. The date of the creation of counting chips is not known, but it is certain that counting chips in the Spring and Autumn period has been very common.

With counting chips, there are vertical and horizontal two ways:

1 2 3 4 5 6 7 8 9

Signifying a multi-digit number, the decimal value system, the number of values arranged from left to right, vertical and horizontal [the law is: a vertical and ten horizontal, a hundred and a thousand stiffs, a thousand, ten look at each other, ten thousand, a hundred equivalent], and the empty space to indicate zero. Chips for addition, subtraction, multiplication, division and other operations to establish a good condition.

Calculation until the fifteenth century at the end of the Yuan Dynasty was gradually replaced by the bead calculation, ancient Chinese mathematics is based on the calculation of its brilliant achievements.

In terms of geometry, the Historical Records of China (史记...... Xia Benji", said Xia Yu, when he was ruling the water, has used a gauge, rectangle, quasi, rope and other mapping and measuring tools, and has long been found that "hook three, four string five" this hook and strand theorem [the West called the hook and strand theorem] of the special case. During the Warring States period, the "Kao Gong Ji," a book written by the Qi state, summarized the specifications of craft technology at the time, included some measurements, and touched on some geometric knowledge, such as the concept of angle.

The Hundred Schools of Thought during the Warring States period also contributed to the development of mathematics, and some schools of thought summarized and generalized many abstract concepts related to mathematics. Famous ones are the definitions and propositions of certain geometrical terms in the Mojing, such as: "Circle, the same length in one center", "Ping, the same height", etc. The Mojing also gives the concepts of exhaustion and infinity. The Mozi also gives the definitions of the infinite and the poor. The Zhuangzi records the famous doctrines of Huishi and others, as well as the theses put forward by debaters such as Huan Tuan and Gongsun Long, which emphasize abstract mathematical ideas, such as: "To the great and the great beyond is called the great one, and to the small and the small without the inside is called the small one," and "The flogging of a ruler is not exhausted by a million generations if half of it is taken in a single day," and so on. These definitions of many geometric concepts, the idea of limits and other mathematical propositions are quite valuable mathematical ideas, but this new idea that emphasizes abstraction and logical rigor has not been well inherited and developed.

In addition, the I Ching, which tells of the yin and yang gossip and predicts good and bad fortune, already has the germ of combinatorial mathematics and reflects the idea of binary.

2. The Han and Tang dynasties

This period includes the development of mathematics from the Qin and Han dynasties to the Sui and Tang dynasties for more than 1,000 years, and the dynasties experienced are Qin, Han, Wei, Jin, North and South Dynasties, Sui and Tang. Qin and Han was the period of formation of the ancient Chinese mathematical system. In order to systematize and theorize the ever-expanding mathematical knowledge, special books on mathematics appeared one after another.

The astronomical work Zhou Thigh Calculation Classic compiled in the late Western Han Dynasty [the first century B.C.] had two main achievements in mathematics: (1) the special case and universal form of the collinearity theorem; (2) the Chen Zi method of measuring the sun's altitude and distance, which was the forerunner of the later re-difference technique. In addition, there are more complex problems of squaring and operations with fractions, etc.

The Nine Chapters of the Mathematical Art is an ancient mathematical classic that has been compiled, deleted and revised by several generations, and was written in the early years of the Eastern Han Dynasty [the first century B.C.]. The book is written in the form of a problem set, **** collection of 246 problems and their solutions, belonging to the square field, corn, decay points, less wide, commercial work, average loss, surplus and deficit, equations and hook and strand nine chapters. The main topics include the four rules of fractions and the proportional algorithm, various calculations of area and volume, and calculations about hook-and-square measurements. In terms of algebra, the concept of negative numbers and the law of addition and subtraction of positive and negative numbers introduced in the chapter of Equations are the earliest records in the history of mathematics in the world; the method of solving systems of linear equations in the book is basically the same as that taught in secondary schools nowadays. In terms of the characteristics of the Nine Chapters of the Art of Arithmetic, it focuses on application and the connection of theory to practice, and forms a mathematical system centered on the preparation of calculations, which has a profound influence on the ancient calculations of China. Some of its achievements, such as the decimal value system, the present art, surplus and deficit art, etc. also spread to India and Arabia, and through these countries to Europe, promoting the development of world mathematics. During the Wei and Jin Dynasties, Chinese mathematics had a greater theoretical development. Among them, the work of Zhao Shuang and Liu Hui is considered to be the beginning of the theoretical system of ancient Chinese mathematics. Zhao Shuang was one of the earliest mathematicians in ancient China to prove mathematical theorems and formulas, and made a detailed commentary on the Zhou Thigh Arithmetic Classic. Liu Hui annotated the Nine Chapters of the Mathematical Art, not only explained and derived the methods, formulas and theorems of the original book in general, but also made many innovations in the process of exposition, and even wrote the "Sea Island Mathematical Scriptures", applying the re-difference technique to solve the problems related to measurements. One of Liu Hui's important works was the creation of the Circle Cutting Technique, which laid the theoretical foundation and provided a scientific algorithm for the study of pi.

Society in the North and South Dynasties was in a state of war and division for a long time, but the development of mathematics was still vigorous. Sun Tzu's Arithmetic Scriptures, Xiahou Yang's Arithmetic Scriptures, and Zhang Qiu Jian's Arithmetic Scriptures are works from this period. The Sun Tzu Arithmetic Classic gives the "Objects do not know the number" problem, which leads to solving the primary congruence group problem; the "Hundred Chickens Problem" of the Zhang Qiu Jian Arithmetic Classic leads to the indeterminate system of equations with three unknowns problem.

The work of Zu Chongzhi and Zu Rihuan, the father and son, is most representative of this period. Based on Liu Hui's commentary on the Nine Chapters of the Mathematical Art of Arithmetic, they took traditional mathematics one step further, becoming an example of the importance of mathematical thinking and mathematical reasoning. They also made outstanding contributions to astronomy. Their work "Splice Art" has been lost. According to historical records, they had three main achievements in mathematics: (1) calculating pi to the sixth decimal place, obtaining 3.1415926 <π< 3.1415927 and finding the approximate rate of π to be 22/7, and the dense rate to be 355/113; (2) obtaining the theorem of Zu Ri Huan [if the power potentials are the same, then the product can not be different] and obtaining the formula for the volume of the sphere; (3) obtaining the formula for the volume of the sphere; (4) obtaining the formula for the volume of the ball; and (5) obtaining the formula for the volume of the ball; and (6) obtaining the theorem of the power potentials of the sphere. spherical volume formula; (3) developed the solution of quadratic and cubic equations.

Sui-Dynasty construction, objectively promote the development of mathematics. At the beginning of the Tang Dynasty, Wang Xiaotong wrote the "Ancient Calculation Scriptures", which mainly discussed the calculation of earthworks in civil engineering, the division and acceptance of works, and the calculation of warehouses and cellars.

The Tang Dynasty made great strides in mathematical education, and in 656 the National Academy of Mathematics set up the Arithmetic Hall, with a doctor of arithmetic and an assistant professor, and the Ten Books of Arithmetic [including Zhou Thighs Arithmetic, Nine Chapters of the Mathematical Art, Island Arithmetic, Sun Tzu Arithmetic, Zhang Qiujian Arithmetic, Xiahou Yang Arithmetic, Jiegu Arithmetic, Wucao Arithmetic, Wujing Arithmetic, and Suffixing Arithmetic]. Arithmetic," and "Suffixed Art"], which were used as textbooks for the students of the Arithmetic Hall. It played an important role in preserving the ancient mathematical classics.

In addition, the Sui and Tang dynasties created the second interpolation method due to the calendar needs, which laid the foundation for the higher interpolation method in the Song and Yuan dynasties. And the late Tang Dynasty saw further improvement and popularization of computational techniques, with the emergence of many kinds of practical arithmetic books, for which the multiplication and division algorithms strove for simplicity.

3. The heyday of the Song and Yuan Dynasties

After the death of the Tang Dynasty, the Five Dynasties and Ten Kingdoms was a continuation of warlordism until the Northern Song Dynasty united China, with rapid prosperity in agriculture, handicrafts, and commerce, and rapid advances in science and technology. From the eleventh century to the fourteenth century [Song and Yuan dynasties], the mathematical calculations to reach the heights of the ancient Chinese math unprecedented prosperity, fruitful heyday. This period appeared a number of famous mathematicians and mathematical works, listed as follows: Jia Xian's "Huang Di nine chapters of the algorithm of fine grass" [11th century], Liu Yi's "discuss the ancient roots of" [12th century], Qin Jiushao's "book of nine chapters of the number of" [1247], Li Ye's "measurement of the round sea mirror" [1248] and the "benefit of the ancient evolution of the paragraph" [1259], Yang Hui's "nine chapters of the algorithm of the detailed explanation" [1261], Algorithm for Daily Use [1262] and Yang Hui's Algorithm [1274-1275], Zhu Shijie's Enlightenment of Arithmetic [1299] and Siyuan Yujian [1303], and so on. Song-Yuan mathematics reached the pinnacle of ancient Chinese mathematics, and even of world mathematics at that time, in many areas. Among the major works are:

1. numerical solution of higher equations;

2. tianyuan jiao and siyuan jiao, the legislation and solution of higher equations, which was the first time in the history of Chinese mathematics that symbols were introduced and symbolic arithmetic was used to solve the problem of establishing higher equations;

3. danyan qiyuyi jiao, the method of solving the group of simultaneous covariances at one time, which is now known as the Chinese Remainder Theorem;

4. the art of strokes and stacks, i.e., higher-order interpolation and higher-order sums of equal differential series. In addition, other accomplishments included new developments in the solution of hook-and-square forms, the study of solving spherical right triangles, the study of vertical and horizontal diagrams [phantom squares], the concrete application of decimals [decimal fractions], and the emergence of bead counting, among others. There was also some development of folk mathematical education during this period, as well as the exchange of mathematical knowledge between China and Islamic countries.

4. Western input period

This period from the mid-fourteenth century, the establishment of the Ming dynasty to the end of the twentieth century Qing dynasty **** more than 500 years. Mathematics in addition to the bead counting the situation of overall decline, which involves the limitations of the counting, the thirteenth century examination system has been deleted from the mathematical content, the Ming Dynasty, the rise of eight examination system and other complex issues, many Chinese and foreign mathematical historians are still exploring the reasons involved. At the end of the sixteenth century, Western elementary mathematics began to be imported into China, which led to the emergence of a fusion of Chinese and Western mathematical research. After the Opium War, modern higher mathematics began to be introduced into China, and Chinese mathematics was transformed into a period in which the study of Western mathematics was the main focus. It was not until the end of the nineteenth century that modern mathematical research in China really began.

The greatest achievement of the Ming Dynasty was the popularization of bead calculations, many bead readers appeared, and Cheng Dawit's "direct reference to the algorithm of the Unification of the Zong" [1592] came out, the theory of bead calculations has become a system, marking the completion of the transition from the chip calculations to bead calculations. However, due to the popularity of bead counting, almost extinct, built on the basis of the ancient mathematics of the counting is also gradually lost, math stagnation for a long time.

Sui and early Tang Dynasty, Indian mathematics and astronomy knowledge had been imported into China, but the impact was relatively small. By the end of the sixteenth century, Western missionaries began to China, and Chinese scholars translated many Western mathematical monographs. One of the first and influential is the Italian missionary Matteo Ricci and Xu Guangqi's translation of the first six volumes of the original geometry [1607], whose rigorous logical system and method of translation was highly respected by Xu Guangqi. Xu Guangqi himself wrote Measurement and Measurement of Similarities and Differences and Gouxue Yi, which applied the logical reasoning method of Geometry Originally to argue for Chinese Gouxue Measurement. In addition, most of the terms used in the textbooks of the Principia Geometrica were pioneered and are still in use today. Second only to geometry in the imported Western mathematics was trigonometry. Trigonometry was only sporadically known before this time, but it developed rapidly thereafter. The works that introduced western trigonometry include Deng Yuhang's compilation of The Great Measurement [2 vols., 1631], The Table of Eight Lines for Cutting Circles [6 vols.], and Luo Yagu's Measurement of the Whole Meaning [10 vols., 1631]. In Xu Guangqi presided over the compilation of the Chongzhen Calendar [137 volumes, 1629-1633], introduced the mathematical knowledge about the circular vertebral curve.

After the Qing Dynasty, the outstanding representative of the Chinese and Western mathematics is Mei Wending, who firmly believed that traditional Chinese mathematics "must have the essence of reason," the ancient masterpieces of in-depth study, but also correctly treat the Western mathematics, so that it is rooted in China, the high tide of mathematical research in the mid-Qing Dynasty has a positive impact. His contemporaries also included Wang Xizhu and Nian Xiyao. The Kangxi Emperor loved scientific research, and his "Royal Decree" of "The Essence of Mathematics" [53 vols., 1723], a relatively comprehensive book of elementary mathematics, had a certain influence on the mathematical research of the time.

During the Ganjia period, the Ganjia School, which was mainly based on the study of evidence, compiled the Siku Quanshu, in which the mathematical works of the Ten Books of Mathematical Schematics and those of the Song and Yuan dynasties made an important contribution to the preservation of mathematical texts that were on the verge of being annihilated.

While studying traditional mathematics, many mathematicians also made inventions, such as Jiao Chuan, Wang Lai, and Li Rui, who were known as the "Three Friends of Heaven and Earth", who did a lot of important work. Li Shanlan obtained a formula for the summation of triangular self-multiplying stacks in his "Stacks and Stacks of Comparison and Classification" [ca. 1859], which is now known as "Li Shanlan's Constant Formula". These works were a step forward from the mathematics of the Song and Yuan dynasties. Ruan Yuan, Li Rui and others wrote a biography of astronomers and mathematicians, "Chou Ren Biography," 46 volumes [1795-1810], which was the first study of the history of mathematics.

After the Crow War in 1840, the closed-door policy was suspended. In 1840, after the Crow War, the closed-door policy was suspended, and "arithmetic" was added to the Tongwenkan, and the Translation Hall was added to the Jiangnan Manufacturing Bureau in Shanghai, which began the second climax of the introduction of translations. The main translators and writings included: Li Shanlan and the British missionary Weili Yali translated the last 9 volumes of Geometry Originally [1857], which gave China a complete Chinese translation of Geometry Originally; 13 volumes of Algebra [1859]; and 18 volumes of Substitute Microproducts [1859]. Li Shanlan and the British missionary Joseph Ai translated "conic curve" 3 volumes, Hua Hengfang and the British missionary Fu Lanya translated "Algebra" 25 volumes [1872], "micro-products traceability" 8 volumes [1874], "to solve the problem of mathematics" 10 volumes [1880] and so on. In these translations, many mathematical terms and terminology were created and are still used today. In 1898, the Peking University Hall was established, and the Tongwenkan was incorporated into it. 1905 saw the abolition of the imperial examinations and the establishment of Western-style schooling, using textbooks similar to those used in other Western countries.

5. Modern mathematical development period

This period is a period of time from the beginning of the twentieth century to the present, often marked by the founding of the new China in 1949 is divided into two stages.

China's modern mathematics began in the late Qing Dynasty and early Republic of China to study abroad. Early to go abroad to study mathematics are Feng Zuxun in 1903 to stay in Japan, Zheng Zhifan in 1908 to stay in the United States, 1910 to stay in the United States, Hu Mingfu and Zhao Yuanren, in 1911 to stay in the United States, Jiang Lifu, in 1912 to stay in France, He Lu, in 1913 to stay in Japan, Chen Jiankong and stay in Belgium, Xiong Qinglai [1915 to stay in France], and in 1919 to stay in Japan, such as Su Bucheng. Most of them returned to China and became famous mathematicians and mathematics educators, making important contributions to the development of modern mathematics in China. Among them, Hu Mingfu obtained a doctoral degree from Harvard University in 1917, becoming the first Chinese mathematician to obtain a doctoral degree. With the return of overseas students to China, mathematics education in universities around the world took off. Initially, only Peking University was established in 1912 when the establishment of the Department of Mathematics, in 1920 Jiang Lifu in Tianjin Nankai University to create the Department of Mathematics in 1921 and 1926 Xiong Qinglai in Southeast University [now Nanjing University] and Tsinghua University to establish the Department of Mathematics, Wuhan University, Qilu University, Zhejiang University, Sun Yat-sen University, one after another, the Department of Mathematics, in 1932, all over the world has been set up by the University 32 departments of mathematics or mathematics and science. In 1930, Xiong Qinglai established the first mathematics research department in Tsinghua University and began to recruit graduate students, and Chen Shenshen and Wu Daren became the earliest graduate students of mathematics in China. In the 1930s, some mathematicians went abroad to study mathematics, such as Jiang Zeihan [1927], Chen Shengshen [1934], Hua Luogeng [1936], and Xu Baolong [1936], who became the backbone of China's modern mathematical development. At the same time, foreign mathematicians also came to China to give lectures, for example, Russell [1920] from Britain, Birkhoff [1934], Osgood [1934], Wiener [1935] from the United States, and Adama [1936] from France, etc. The founding meeting of the Chinese Mathematical Society was held in Shanghai in 1935, and 33 delegates attended the meeting. In 1936, the Journal of the Chinese Mathematical Society and the Journal of Mathematics appeared one after another, marking the further development of modern mathematical research in China. Mathematical research before liberation was concentrated in the field of pure mathematics, and more than 600 treatises were published at home and abroad***. In the field of analysis, Chen Jianguong's trigonometric number theory, Xiong Qinglai's study of subpure functions and integral function theory are masterpieces, in addition to the results of generalized analysis, calculus of variations, differential equations and integral equations; in the field of number theory and algebra, Hua Luogeng's analytic number theory, geometric number theory, and algebraic number theory, as well as the recent algebraic studies have achieved results that have attracted attention from all over the world; in the field of geometry and topology, Su Bucing's differential geometry, Jiang Zeihan's algebraic topology, and Su Bucing's differential geometry, have made the world's attention. In geometry and topology, Su Buqing's Differential Geometry, Jiang Zeihan's Algebraic Topology, and Chen Shengshen's Fibre Plexus Theory and Schematic Class Theory have done pioneering work. In probability theory and mathematical statistics, Paul Hsu has obtained a number of fundamental theorems and rigorous proofs in the area of univariate and multivariate analysis. In addition, Li Yan and Qian Baozheng pioneered the study of the history of Chinese mathematics, and they did a lot of groundbreaking work in the annotation and organization of ancient arithmetic historical materials as well as in the analysis of the evidence, which brought back the luster of China's national cultural heritage.

The Chinese Academy of Sciences was established in November 1949, and in March 1951 the Chinese Journal of Mathematics was reissued [changed to Mathematical Journal in 1952], and in October 1951 the Chinese Mathematical Journal was reissued [changed to Mathematical Bulletin in 1953]. In August 1951 the Chinese Mathematical Society held its first congress after the founding of the country, discussing the direction of mathematical development and the reform of mathematics teaching in various types of schools. The Chinese Mathematical Society held its first congress after the founding of China in August 1951, discussing the direction of mathematical development and the reform of mathematics teaching in various types of schools.

Mathematical research after the founding of the country has made great progress. the early 50's published Hua Luogeng's "stack prime number theory" [1953], Su Buqing's "Introduction to projective curves" [1954], Chen Jiankong's "right-angle function of the sum of grades" [1954], and Li Yanyan's "in the history of arithmetic series" 5 [1954-1955] and other monographs, to 1966, **** published By 1966, *** had published about 20,000 papers on various mathematics. In addition to the number theory, algebra, geometry, topology, function theory, probability theory and mathematical statistics, history of mathematics and other disciplines continue to achieve new results, but also in the differential equations, computational techniques, operations research, mathematical logic and mathematical foundations of the branches of breakthroughs, there are many treatises to reach the world's advanced level, and at the same time training and growth of a large number of outstanding mathematicians.

In the late 1960s, China's mathematical research basically stopped, education was paralyzed, the loss of personnel, and the interruption of foreign exchanges, and then the situation changed slightly through the efforts of many parties; in 1970, the Journal of Mathematics resumed publication, and the first issue of the Practice and Understanding of Mathematics was created; in 1973, Chen Jingrun published a paper entitled "The Large Even Numbers Represented by the Sum of the Products of a Prime Number and a Number Not More Than Two Primes" on Science China, which was the first paper on Goldbach's theory of mathematical mathematics in China. In 1973, Chen Jingrun published the paper "Representation of Large Even Numbers as a Sum of a Prime Number and a Product of Not More Than Two Prime Numbers" in Science of China, and made outstanding achievements in the study of Goldbach's Conjecture. In addition, Chinese mathematicians have made some innovations in function theory, Markov processes, probability applications, operations research, and preferential methods.

The third congress of the Chinese Mathematical Society was held in November 1978, marking the revival of mathematics in China. 1978 saw the resumption of the National Mathematics Competition, and 1985 saw China's participation in the International Mathematical Olympiad. 1981 saw Chen Jingrun and other mathematicians awarded the National Natural Science Prize, and 1983 saw the first batch of doctoral degrees conferred by the state on 18 young and middle-aged scholars, two-thirds of whom were mathematical scientists. 1986 saw the first batch of doctoral degrees awarded to Chinese mathematicians. In 1986, China sent representatives to the International Congress of Mathematicians for the first time and joined the International Mathematical Union, and Wu Wenjun was invited to give a 45-minute lecture on the history of ancient Chinese mathematics. In the last decade or so, mathematical research has been fruitful, the number of published papers and monographs has increased exponentially, and the quality has continued to rise. at the annual meeting celebrating the 50th anniversary of the founding of the Chinese Mathematical Society in 1985, the long-term goals for the development of mathematics in China had been set. The delegates aspired to work tirelessly to make China a new mathematical power in the world at an early date.

Ancient Egyptian Mathematics (Ancient Egyptian Mathematics)

The Nile Valley in northeastern Africa gave birth to Egyptian culture. A unified empire was established here between 3500 and 3000 BC.

Currently, our knowledge of ancient Egyptian mathematics stems mainly from two papyri written in the monastic language, one of which is the Moscow papyrus, written around 1850 BC, and the other is the Rhind papyrus, also known as the Ahmes papyrus, written around 1650 BC. The Ahmes papyrus is quite informative, describing Egyptian multiplication and division, the use of unit fractions, the method of trial places, the solution of problems of finding the area of a circle, and the application of mathematics to many practical problems.

The ancient Egyptians used hieroglyphics, their numbers were expressed in decimal, but not in place value, and there was a specialized notation for fractions. The arithmetic established by the Egyptian number system had an additive character, and its multiplication and division were accomplished using only successive doubling. The ancient Egyptians reduced all fractions to unit fractions (the sum of fractions whose numerator is 1), and in the Ames papyrus, there is a large table of fractions that represents 2/(2n+1)-like fractions as a sum of unit fractions, e.g., 2/5=1/3+1/15, 2/7=1/4+1/28, ..., 2/97=1/56+1/679+

1/776, and so on.

The ancient Egyptians were already able to solve some problems belonging to the primary and simplest quadratic equations, and there was also some rudimentary knowledge of the equivariant and isoperimetric series.

If the Babylonians developed excellence in arithmetic and algebra, on the other hand, it is generally believed that the Egyptians were superior to the Babylonians in geometry. One view is that the waters of the Nile flooded regularly once a year, inundating the valleys on both sides of the river. After the floods, the pharaohs had to redistribute the land, and the knowledge of land measurement accumulated over time evolved into geometry.

The Egyptians were able to calculate the area of a simple plane figure, calculating the circumference of a circle to be 3.16049; they also knew how to calculate the volume of prismatic vertebrae, circular vertebrae, cylinders, and hemispheres. One of the most amazing achievements was the calculation of the volume of a square prismatic vertebra with a flat truncated head, and they gave a procedure for calculating it that matches modern formulas.

As for the construction of the pyramids and temples, the fact that a great deal of mathematical knowledge was used shows that the Egyptians had accumulated a lot of practical knowledge, which had yet to be elevated to a systematic theory.

Indian mathematics (Hindu mathematics)

India is one of the earliest culturally developed regions of the world, the origin of Indian mathematics and the origin of mathematics of other ancient peoples, as in the production of practical needs on the basis of the generation. However, the development of Indian mathematics also has a special factor, that is, its mathematics and calendar, as in the Brahmin rituals under the influence of the full development. Coupled with Buddhist exchanges and trade, Indian mathematics and the Near East, especially Chinese mathematics, advanced in the process of mutual integration and mutual promotion. In addition, the development of Indian mathematics has always had a close relationship with astronomy, most of the mathematical works published in certain chapters of astronomical works.

The Rope Sutra, an ancient Brahmanical classic probably written in the 6th century B.C., is a religious work of significance in the history of mathematics, in which the geometrical laws embodied in pulling a rope to design an altar are taught and the Pythagorean Theorem is widely applied.

For about 1,000 years thereafter, little is known about the development of mathematics because of the lack of reliable historical sources.

The period from the 5th to the 12th centuries A.D. was a period of rapid development of mathematics in India, and its achievements occupy an important place in the history of world mathematics. Some famous scholars appeared in this period, such as Aryabhata (the first) (ryabhata) in the 6th century, who wrote the Aryabhata Book of Calendars; and Brahmagupta (Brahmagupta) in the 7th century, who wrote the Brahma-sphuta-sidd'h nta (Revised System of Brahmagupta), an astronomical work that includes the " Lectures on Arithmetic" and "Lectures on Indeterminate Equations" and other mathematical chapters; Mah vira in the 9th century; and Bh skara in the 12th century, with the Siddh nta iromani (The Ultimate of the Astronomical System), with important sections on mathematics such as the Lil vati (The Source of Algorithms) and V jaganita) and so on.

In India, the decimal value system of notation for integers arose before the 6th century, when any number could be written using nine digits and a small circle for zero, with the help of the place value system. They thus established arithmetic operations, including the four rules for integers and fractions; the rules for squaring and cubing. They did not only consider 'zero' as 'nothing' or empty space, but also treated it as a number for the purpose of arithmetic, which is a major contribution of Indian arithmetic.

The Indian numerical and place value notation was introduced to the Islamic world in the 8th century, and was adopted and improved by the Arabs, and then spread to Europe in the early 13th century through Fibonacci's Book of Arithmetic, where it gradually evolved into today's widely utilized numbers 1, 2, 3, 4, ..., etc., called the Indo-Arabic numerals.

India has made significant contributions to algebra. They used symbols for algebraic operations and abbreviated characters to represent unknown numbers. They recognized negative and irrational numbers, had a specific description of the quadratic rule for negative numbers, and realized that quadratic equations with real solutions have roots in two forms. Indians showed excellence in indeterminate analysis, and instead of being satisfied with finding only any one rational solution to an indeterminate equation, they devoted themselves to finding all possible integer solutions. Indians have also calculated the sums of arithmetic and geometrical series, and solved commercial problems such as simple and compound interest, discounts, and combinations of shares.

Indian geometry was empirical; they did not pursue logically rigorous proofs, but focused on developing practical methods, generally associated with measurement, and on calculating areas and volumes. Its contribution was far less than their contribution in arithmetic and algebra. In trigonometry, the Indians replaced the Greek's full sine with the half-sine (i.e., sine), made the table of sines, and proved some simple trigonometric constants, and so on. Their work in trigonometry was very important.

Arabic mathematics

From the ninth century onward, the center of mathematical development shifted to Arabia and Central Asia.

Since the creation of Islam at the beginning of the seventh century A.D., it quickly developed a powerful force that expanded rapidly over a vast area beyond the Arabian Peninsula, spanning Europe, Asia, and Africa. Within this vast area, Arabic was the official script in common use, and the Arabic mathematics described here refers to the mathematics studied in Arabic.

From the eighth century onwards for about one to one and a half centuries was the period of translation of Arabic mathematics, and Baghdad became a center of scholarship, with a palace of science, an observatory, a library, and an academy. Scholars from all over the world translated classical Greek, Indian and Persian works into Arabic in great numbers. In the process of translation, many texts were recalibrated, verified and added to, and a great deal of the ancient mathematical heritage was given a new lease of life. Arab civilization and culture developed rapidly on the basis of the acceptance of foreign cultures and remained vibrant until the 15th century.

Al-khowarizmi [Al-khowarizmi], the foremost mathematician of the early Arab period, produced the first work in Arabic to introduce Indian numbers and notation in the Islamic world. After the twelfth century A.D., Indian numerals and decimal notation were introduced to Europe, and after centuries of reforms, these numbers became the Indo-Arabic numerals we use today. Another famous book by Khorezm, ilm al-jabr wa'lmugabalah [Algebra], systematically discusses the solution of quadratic equations, and the formula for finding the roots of such equations appears for the first time in this book. The modern word "algebra" derives from the word "al jabr" which appears in the book's title.

Trigonometry occupies an important place in Arabic mathematics, and its emergence and development are closely related to astronomy. The Arabs developed trigonometry on the basis of the work of the Indians and Greeks. They introduced several new trigonometric quantities, revealed their properties and relations, and established some important trigonometric constants. Full solutions of spherical and plane triangles were given, and many more sophisticated tables of trigonometric functions were produced. Among the famous mathematicians are: Al-Battani [Al-Battani], Al-Battani [Al-Battani], Al-Battani [Al-Battani], Al-Battani [Al-Battani], and Al-Battani [Al-Battani]. Al-Battani, Abu'l-Wifa, and Al-Maqtada. Wefa [Abu'l-Wefa], Al-Beruni [Al-Beruni], and Al-Beruni [Al-Beruni]. Al-Beruni [Al-Beruni] and so on. Systematic and complete discussion of trigonometry work is completed by the thirteenth-century scholar Nasir ed-din [Nasir ed-din], the work of trigonometry from astronomy and become an independent branch of mathematics, the development of trigonometry in Europe has a great impact.

In terms of approximate calculations, the fifteenth century, Al-Kashi [Al-kashi]. Kashi [Al-kashi] in his "Theory of Circumference", described the calculation of pi π, and get the pi accurate to 16 decimal places, thus breaking the record maintained by Zu Chongzhi for a thousand years. In addition, Al. Al-Qassi did important work on decimals and was the first Arab scholar we know of to treat the binomial theorem in the form of "Pascal's triangle".

Arabic geometry was less accomplished than algebra and trigonometry. The rigorous logical arguments of Greek geometry were not accepted by the Arabs.

On the whole, Arab mathematics was less creative, but at a time when most of the world was scientifically barren, its achievements appear relatively large, and it is to their credit that they acted as the preservers of a great deal of the world's spiritual wealth, which came back to Europe only after the Dark Ages had passed. It was chiefly through their translations that Europeans learned of the achievements of ancient Greek and Indian and Chinese mathematics.