Seventeen achievements
Throughout the history of mathematics development in China, the achievements in mathematics in ancient China are actually enough to open an exhibition hall. Here, I think the most remarkable achievements of 17 are listed as follows:
(1) Decimal notation and the adoption of zero.
Decimal notation was formed in China's primitive society, completed in the Shang Dynasty in the early slave society, and developed into a complete decimal system in the Shang Dynasty, with special large number names such as "ten", "hundred", "thousand" and "ten thousand". 1899 hieroglyphics unearthed in Anyang, Henan province show that China adopted decimal notation in 1600 BC, more than 1000 years earlier than the second inventor. 0 is an extremely important number, and the discovery of 0 is called one of the great discoveries of mankind.
The invention of the mathematical symbol "0" should be attributed to Indians in the 6th century. They first used a black dot () to represent zero, and then gradually became "0".
In ancient China, 0 was called Jinyuan number (meaning extremely precious number). Speaking of the appearance of zero, it should be pointed out that the word "zero" appeared very early and was widely used in ancient Chinese characters.
(2) The origin of binary system. The Eight Diagrams method, which originated in Zhouyi, was more than 2000 years earlier than the second inventor, the German mathematician Leibniz (A.D. 1646- 17 16).
Leibniz (1646-1716) is a famous philosopher and mathematician. He invented the binary system, which is of great significance to modern computer systems, but he thinks that before that, China had already mentioned the preliminary idea about the binary system in the Book of Changes. From the Book of Changes, we can see the origin of binary system, and the application of ancient binary system in China is the same as that of modern electronic computers. There was Zhouyi in the Fuxi era in ancient China. Zhouyi is a science that studies the changes of the sun and the moon. Through divination and with the help of binary means, it explains the great laws of life and things changing between heaven and earth and within the sun and moon system.
(3) that origin of geometric thought. The Mohist Classic, which originated in Mo Zhai during the Warring States Period, was earlier than the second inventor Euclid (330-275 BC) 100 years.
In the famous Mo Jing, some definitions and propositions of geometric terms are given, such as "circle, an equal length", "flat, an equal height" and so on. Mohist school also gave the definitions of finite and infinite.
There are eight articles on geometrical optics in the Book of Mohism, which explain the imaging of shadows, pinholes, flat mirrors, concave mirror and convex mirrors, and also explain the relationship between focal length and object imaging, which is more than 100 years earlier than the optical records recorded by Euclid in ancient Greece (about 330-275 BC). The theory of mechanics is also a masterpiece of ancient mechanics. The definition of force, lever, pulley, shaft, slope, fluctuation, balance and center of gravity of objects are all discussed. Most of these arguments come from practice. Mo Jing's eight optical articles reflect the great achievements of physics in China during the Spring and Autumn Period and the Warring States Period.
(4) Pythagorean theorem (quotient height theorem). The inventor Shang Gao (from the Western Zhou Dynasty) was more than 550 years earlier than the second inventor Pythagoras (580-500 BC).
Pythagorean theorem is a dazzling pearl in geometry, known as "the cornerstone of geometry", and is also widely used in higher mathematics and other disciplines. Because of this, several ancient civilizations in the world have been discovered and widely studied, so there are many names. The west is called Pythagorean Theorem or Pythagorean Theorem.
Theorem or Pythagoras.
Theorem is a basic geometric theorem, which was first proved by Pythagoras in ancient Greece. It is said that after Pythagoras proved this theorem, he beheaded a hundred cows to celebrate, so it is also called "Hundred Cows Theorem".
France and Belgium are called donkey bridge theorem, and Egypt is called Egyptian triangle.
China is one of the earliest countries to discover and study Pythagorean theorem. Ancient mathematicians in China called the right triangle pythagorean, the short side of the right angle is called hook, the long side of the right angle is called strand, and the hypotenuse is called chord, so the pythagorean theorem is also called pythagorean chord theorem. In China, more than 65,438+0,000 years ago, the formula and proof of Pythagorean theorem were recorded in Zhouyi Shu Jing. According to legend, it was discovered by Shang Dynasty's Shang Gao, so it is also called Shang Gao Theorem. During the Three Kingdoms period, Zhao Shuang made a detailed annotation on the Pythagorean Theorem in Zhou Bi suan Jing, and gave another proof. At present, the proof method of junior high school mathematics textbooks is Zhao Shuang's string diagram, and the proof uses Green-Zhu path diagram.
Zhao Shuang's Chord Diagram
Green-Zhu visits the map
Pythagorean theorem is a basic geometric theorem, it is one of the most important tools to solve geometric problems with algebraic ideas, and it is one of the ties of combining numbers with shapes.
(5) Rubik's cube. The earliest records of magic methods in China are The Analects of Confucius and The Classic in the Spring and Autumn Period, while abroad, the Rubik's Cube appeared in the 2nd century A.D., more than 600 years earlier than in China.
Rubik's Cube, also known as Rubik's Cube, Square Matrix or Hall Square, originated in China, and was called a vertical and horizontal diagram by Yang Hui, a mathematician in the Song Dynasty. The magic of the magic square is that no matter which route you take, the final sum or product is exactly the same, that is, in a square composed of several neatly arranged numbers, the sum or product of any number of rows, columns and diagonals in the diagram is equal, and a diagram with this property is called the magic square. China called it "River Map" and "Luoshu" in ancient times.
In China's numerology of Han Dynasty, it was called Jiugong Calculation, also known as Nine palace map. Also called "vertical and horizontal map".
China's classic Book of Changes records the legend of Luo Shu: In the 23rd century BC, when Dayu was harnessing water, a huge turtle appeared in Luoshui, a tributary of the Yellow River. There are nine mottled patterns on the tortoise shell, which represent the nine numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively, but three lines, three columns and two diagonal lines.
Yang Hui, a mathematician in the Southern Song Dynasty, named a figure similar to Nine palace map as a vertical and horizontal diagram in his book Algorithm for Extracting Odds from Ancient Times, and listed magic squares of order 3, 4, 5, 6, 7, 8, 9, 10. The construction method of the third-order magic square comprises the following steps:
"Nine sons are inclined, easy to go up and down, more similar to the left and right, four-dimensional prominent, wearing nine shoes, left three and right seven, shoulders two and four, full of six or eight", which is better than the French mathematician Claude Gaspar.
Barchet's method was more than 300 years earlier.
Huang Rong, the third-order magic square, also recited the formula of this third-order magic square in The Legend of the Condor Heroes.
The Rubik's Cube was recorded for the first time in China's "Da Dai Li" in the Spring and Autumn Period of 500 BC, which shows that our people knew the arrangement law of Rubik's Cube as early as 2,500 years ago. Abroad, it was not until 130 that the Greek Saiweng first mentioned the Rubik's Cube.
Our country not only has the right to invent the Rubik's Cube, but also is a country that conducts in-depth research on it. Yang Hui, a mathematician in the 3rd century A.D.10, compiled a magic square of order 3- 10, which was recorded in his book "Algorithm for Continuing Ancient Stories" written in 275. In Europe, it was not until 15 14 that the famous German painter Diu Lei drew a complete fourth-order magic square.
(6) fractional arithmetic and decimals. China's complete fractional arithmetic appeared in "Nine Chapters of Arithmetic", and its transcript appeared in 1 century at the latest. The same law appeared in India in the 7th century, and it is considered as the "originator" of this law. China is more than 500 years earlier than Indian.
China used the least common multiple of western countries 1200 years ago. As early as 1 100 years ago, decimals were used in the west.
(7) The discovery of negative numbers. This discovery was first seen in Nine Chapters Arithmetic, more than 600 years earlier than India and 1600 years earlier than the West.
According to historical records, as early as 2000 years ago, China had the concept of positive and negative numbers and mastered the arithmetic of positive and negative numbers. Liu Hui, a scholar in China during the Three Kingdoms period, made great contributions to the establishment of the concept of negative numbers. Liu Hui first gave the definitions of positive numbers and negative numbers. He said: "Today's gains and losses are the opposite, and positive and negative numbers should be named." In other words, in the process of calculation, positive numbers and negative numbers should be used to distinguish. Liu Hui gave the method of distinguishing positive and negative numbers for the first time. He said: "The front is red and the negative is black; Otherwise, evil will be different. "
In China's famous ancient mathematical monograph "Nine Chapters of Arithmetic" (written in the first century AD), the law of addition and subtraction of positive and negative numbers was put forward for the first time: "Positive and negative numbers say: the same name is divided, different names are beneficial, positive and negative; Its synonyms are divided, the same name is beneficial, and there is no positive or negative. "
In addition to the positive and negative operation methods defined in Nine Chapters Arithmetic, Liu Hong (AD 206) at the end of the Eastern Han Dynasty and Yang Hui (126 1) in the Song Dynasty also discussed the addition and subtraction principles of positive and negative numbers, all of which were completely consistent with those mentioned in Nine Chapters Arithmetic. In particular, in Yuan Dynasty, Zhu Shijie gave not only the rules of addition and subtraction of positive and negative numbers with the same sign but different signs, but also the rules of multiplication and division of positive and negative numbers. Negative numbers are recognized and recognized abroad, much later than at home. In India, the mathematician Brahmaputra didn't know about negative numbers until AD 628. It was not until the17th century that the Dutchman Jirar (1629) first realized and used negative numbers to solve geometric problems.
(8) surplus and deficiency. Also known as double false positioning method. It was first seen in the seventh chapter of Nine Chapters Arithmetic. In the world, it was not until the 3rd century A.D./kloc-0 that the same method appeared in Europe, which was more than 200 years later than that in China.
Surplus and deficiency technique is an arithmetic method to calculate profit and loss problem in ancient China. Borrowing the surplus and deficiency to find the implied number is one of Zhou Li's nine numbers. "Nine chapters of arithmetic, lack of profit": "Today, there are * * * shopping, people out of eight, profit three; Seven out of seven, less than four. Q: What is the quantity and price? Answer: Seven people, the price is 53. " . In11-1the works of some Arab mathematicians in the 3rd century, there also appeared the technique of surplus and deficiency, which was called librarian or qidan algorithm. At that time, the "Khitan" referred to by Arabs refers to China, which also shows that the surplus and deficiency of ancient China was in the forefront of the world.
(9) Equation technology. Different from today, linear equations were called equations in ancient times, and their solutions were called equations. It first appeared in Nine Chapters Arithmetic, in which the method of solving linear equations was more than 600 years earlier than India and 1500 years earlier than Europe. China is 1800 years earlier than other countries in the world in solving linear equations by matrix arrangement method.
(10) The most accurate pi "ancestral rate". When Liu Hui, a mathematician in China, annotated Nine Chapters Arithmetic (AD 263), he got the approximate value of π only by inscribed a regular polygon into a circle, and the value of π was accurate to two decimal places. His method was later called the secant circle method, which contained the idea of seeking the limit. Zu Chongzhi, a mathematician in the Northern and Southern Dynasties, further obtained the π value accurate to seven decimal places (AD 466) by using the method of circle cutting, and gave the insufficient approximation of 3. 14 15926 and the surplus approximation of 3. 14 15927, and also got two approximate fractional values with the density of 355.
Otto) and the Dutchman Antuoni (a.anthonisz) reached the same result; This record has been kept in the world for 1000 years. In order to commemorate Zu Chongzhi's contribution to the development of China's pi, this calculated value was named "Zu Chongzhi pi" after him, or "ancestral rate" for short. 17 At the beginning of the 5th century, the Arabic mathematician Cassie got the exact decimal value of pi17, which broke the record kept by Zu Chongzhi for nearly a thousand years. 1596, the German mathematician Curran calculated the π value to 20 decimal places, and then spent his whole life calculating it to 35 decimal places of 16 10. This value is named Rudolph number after him.
(1 1) equal product principle. Also known as the "ancestral declaration" principle. Keep the world record of 1 100.
The principle of equal product was first put forward by Zu Xuan (mathematician and astronomer), the son of Zu Chongzhi, an outstanding mathematician in the Northern and Southern Dynasties. Together with his father Zu Chongzhi, he successfully solved the problem of calculating the sphere area and got the correct volume formula. The well-known "the principle of forming ancestors" in the current textbooks is the outstanding contribution of Zuxuan to the world mathematics in the 5th century. Zu Xuan summed up Liu Hui's related work and put forward that "if the potential is 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 Xuan axiom (or Liu Zu's principle). Zu Xuan applied this principle to solve Liu Hui's unsolved spherical volume formula. This principle was not developed by the western Italian mathematician cavalieri ·﹝bonavent until the17th century.
UraCavalieri﹞ (it was discovered later than Zuxuan 1 100 years ago.
(12) quadratic interpolation method. Liu Zhuo, an astronomer in Sui Dynasty, first invented it, which was more than 1000 years earlier than Newton, the world runner-up (1642- 1727).
China invented the interpolation method very early in ancient times (interpolation method is an approximate calculation method to find other values of unknown functions by using the values of independent variables of a group of known unknown functions and their corresponding function values, and it is a numerical approximation method, and Bessel interpolation method is often used in astronomy and lunar calendar calculation. At that time, interpolation was called difference interpolation. For example, the "remainder" in Nine Chapters Arithmetic around 1 century BC is equivalent to a difference interpolation (linear interpolation). In 600 AD, Liu Zhuo of Sui Dynasty put forward the world's earliest equidistance quadratic interpolation formula (parabolic interpolation) when drawing up the emperor's calendar. This is an outstanding creation in the history of mathematics, which was developed into an unequal interval quadratic interpolation formula by monks and their followers in the Tang Dynasty in their Da Yan Li. Guo Shoujing, who wrote the calendar in Yuan Dynasty, further invented the cubic difference interpolation method. Liu Zhuo 1000, 400 years after Guo Shoujing, Newton of England put forward the general formula of interpolation.
(13) multiplication and division method. Multiplication and division is a general method to find the numerical solution of higher order equations in ancient mathematics in China, and it is also called Horner method in modern mathematics.
Jia Xian, a mathematician of the Song Dynasty in China, was first invented in1/century, which was about 800 years earlier than that proposed by Horner, a British mathematician in19th century. It was initiated by Jia Xian in 1 1 century, passed through12nd century, and finally completed in Qin Yu13rd century. 19 the steps of European Horner method and the principle of comprehensive division in modern mathematics are the same as it. This method originated from the formula of Nine Chapters Arithmetic popularized by Jia Xian, Yang Hui and others. By the13rd century, it had developed into a systematic method for finding numerical solutions of higher-order equations, which was recorded in the works of Qin, Zhu Shijie and others, among which Qin's Shu Jiu Zhang was the most detailed. The example of Horner's paper "Solving all sub-equations" published in 18 19, its algorithm program and digital processing are far less orderly than that in Qin dynasty more than 500 years ago; Qin algorithm is not only earlier than Horner in time, but also more mature. Kaiping multiplication was invented by Jia Xian, a mathematician in the Northern Song Dynasty, and was first collected from the book "The Classic of Calculations". Jia Xian's original work has been lost, but his important contribution to mathematics was quoted by Yang Hui, a mathematician in the Southern Song Dynasty, and was copied into Yongle Dadian (16,344). Fortunately, it has been preserved and is now in the library of Cambridge University in England.
(14) Yang Hui Triangle. Yang Hui Triangle, also known as Jia Xian Triangle and Pascal Triangle, is the geometric arrangement of binomial coefficients in the triangle, and is actually a binomial expansion coefficient table. Originally created by Jia Xian, it can be found in his book The Nine Chapters of Yellow Emperor's Fine Grass Algorithm. Later, this book was lost, and Yang Hui, a native of the Southern Song Dynasty, compiled this table in his "Detailed Explanation of Nine Chapters of Algorithms", hence the name "Yang Hui Triangle".
The most essential feature of Yang Hui Triangle is that its two hypotenuses are all composed of the number 1, and the other numbers are equal to the sum of the two numbers on its shoulders. The number arrangement law contained in Yang Hui's triangle makes us feel the beauty of mathematics, and at the same time appreciate its interest and practicality.
In the world, besides China's Jia Xian and Yang Hui, the second inventor is the French mathematician Pascal (A.D. 1623- 1662), whose invention time was 1653, which was nearly 600 years later than that of Jia Xian.
(15) China's remainder theorem. Also known as Sun Tzu's Theorem, it is a method to solve a linear congruence group in ancient China. China's remainder theorem is actually a method to solve simultaneous simultaneous linear congruences. This method was first seen in Sun Tzu's calculation. 180 1 year, German mathematician Gauss (A.D. 1777- 1855) put forward this solution in "Arithmetic Inquiry". Westerners believe that this method is the first in the world and call it "Gauss Theorem", but it was later discovered than China.
This is an important theorem in number theory.
(16) digital higher-order equation method, also known as "astrophysics". China's ancient methods of solving higher order equations. /kloc-in the third century, the numerical solution of higher-order equations is one of the mathematical problems.
Astrology is one of the algebraic methods in ancient China, and it is also the method of establishing higher-order equations in ancient China. From 65438 to 0248 A.D., Ye Li, a mathematician in Jin Dynasty, systematically introduced the establishment of quadratic equation by celestial technique in her works "Measuring the Round Sea Mirror" and "An Ancient Analysis", and skillfully expressed it in calculation. Wang Xun, a mathematician in Yuan Dynasty, widely used celestial sphere technique to solve higher-order equations. This method is more than 300 years earlier than other countries in the world, which lays a good foundation for solving multivariate higher-order equations in the future.
(17) calls for differences. Higher-order interpolation is a commonly used interpolation method in modern computational mathematics, and it is also a summation method of higher-order arithmetic progression. Since the Northern Song Dynasty, many mathematicians in China have studied this problem. In the Yuan Dynasty, Zhu Shijie first invented the technique of calling the difference, which solved this problem. In the world, Newton, who was nearly 400 years later than Zhu Shijie, got the same formula. Whether the difference of China ancient high-order arithmetic progression sum can be separated from the method of finding interpolation formula. Zhu Shijie's "Four Jade Juanjian" (1303) discusses the problems in "Ruxiang Unique Skill".
Among them, Zhu Shijie gave a four-time difference formula:
This is consistent with Newton's interpolation formula, but Newton put forward this formula more than 300 years later than Zhu Shijie.
The creation, development and application of induction are great achievements of world significance in the history of mathematics and astronomy in China.
Generally speaking, the development of ancient mathematics in China lacks an axiomatic system. This is precisely the bottleneck of the development from elementary mathematics to advanced mathematics. China's mathematics has no tendency of axiomatization from the beginning, and it is more about solving some specific problems or summing up some laws. The most important work that Euclid, the representative of western mathematicians, did was the axiomatization of geometry. The Elements of Geometry is an axiomatic system work based on several self-evident axioms. The so-called harmonious beauty and simple beauty of mathematics established in this way. This ancient Greek mathematician has a far-reaching influence on the whole European science. Newton's most important work, Mathematical Principles of Natural Philosophy, is the process of this axiomatic system. Describe the phenomenon, then organize this regular phenomenon into the most basic axioms and laws, and then use these laws to explain more complex phenomena. Its most fundamental is the law of universal gravitation and the three laws of motion. At that time, it was enough to "predict the movement of everything".
In addition, the backward level of mathematics in ancient China is also related to the backward level of science and technology as a whole, and the two are * * * into * *. The decline of ancient science and technology in China is another big problem.
References:
1. Exploring Pythagorean Theorem Tongji University Press
2. "Magic Vertical and Horizontal Map" Wang Qianwei
3. "Nine Chapters of Arithmetic" Zhang Cang Geng Shouchang
4. Yang Hui Triangle and Chessboard Street Walking has been published in China.