Nine Chapters Arithmetic, written in the first century BC, is the most important mathematical classic in China. It reflected the highest level of development in the field of mathematics in China at that time. The whole book takes calculation as the center, and basically takes the form of algorithm command application questions. Many of its achievements are in the leading position in the world, which have had a far-reaching impact on the development and education of mathematics in China and laid the foundation for China's mathematics to be at the forefront of the world for more than 1000 years. After the book Nine Chapters Arithmetic was written, there were many creations. Liu Hui, a master of mathematics in Wei and Jin Dynasties, made the greatest contribution to the annotation of Nine Chapters Arithmetic, which had far-reaching influence. There are many important contents and viewpoints in the field of mathematics education in Nine Chapters and its annotators in previous dynasties, which are worth learning. Generally speaking, Nine Chapters was not a book of mathematics enlightenment education at that time, and its content far exceeded the teaching requirements of primary schools for six years today. However, with the rapid progress of society and the rapid development of mathematical science, the profound content in the early stage may become the basic content of popularization in the later stage. Compared with today's primary school mathematics syllabus, some arithmetic contents in Nine Chapters have become important contents in the teaching of senior primary schools. Some mathematical ideas and methods embodied in "Nine Chapters" also have important enlightenment and reference significance for primary school students. Now, this is summarized for teachers to know and understand in teaching: 1. Decimal position value system symbol. China is the first country in the world to produce and establish a perfect decimal numerical notation. Numbers began to be used as early as four or five thousand years ago, and the Shang Dynasty had mastered decimal numbers within 30 thousand and recorded them in the position system. This notation is much superior to the Babylonian 60-decimal system, the Mayan 20-decimal system, the Roman 5- 10 decimal system and the decimal system in ancient Egypt and Greece. China's decimal notation was praised by Marx as "one of the most wonderful inventions" in the process of human civilization. On this basis, Liu Hui created decimals, which did not appear in foreign countries until 14 and 15 centuries, and decimal points did not begin to be used until 17 centuries. 2. The invention of calculation tools-calculation is a unique calculation tool in ancient mathematics in China. "Arithmetic" means the use of computing technology, which appropriately summarizes the characteristics of using calculators and focusing on calculation in ancient mathematics in China. Nine Chapters is a math textbook with calculation tools. As the most dexterous computing tool in the world at that time, it was convenient and accurate to use, and it became a scientific tradition in the history of China that lasted for more than 1500 years. The abacus developed after Yuan Dynasty is the development, reform and continuation of the preparatory system. Teachers should understand the position and role of abacus and abacus in the history of world mathematics development and play their unique educational functions in teaching. China's calculation is mainly operation and counting, and then a complete formula is formed. This characteristic of calculation determines that the simplest and most direct method in the teaching process is "learning by doing", which is especially suitable for children's teaching methods based on demonstration and operation guidance, and conforms to the psychological characteristics of children's action thinking. In addition, China's mathematical formula has a long history, which is conducive to the use of "singing" and "swimming" two teaching methods. The popularization and continuous development of mathematical rhymes and the continuous improvement of algorithms and calculators not only promote the development of primary school mathematics education, but also directly affect the emergence and development of abacus. As "genuine" treasures in China's cultural treasure house, abacus and abacus are not only excellent calculation tools, but also good teaching aids and learning tools. Compared with foreign calculators, calculators and calculators, they can play a better intermediary role from concrete to abstract, help students to form clear representations of number order and number size, and thus improve their ability to recognize numbers. It is precisely because of its special value and function that abacus has been widely used in modern business circles with highly developed electronic technology. In China, Japan and other Southeast Asian countries, abacus is still popular. In addition, educators in the western world believe that abacus is of special significance in mathematics education. 3. The operation of four fractions and its application The knowledge of fractions in nine chapters (including reduction, general fractions, addition, subtraction, multiplication and division) was the most systematic and complete fraction theory in the world at that time. In the formula, there is a clear fractional algorithm, and there are many fractional application problems in other chapters. A) Fractional addition and subtraction Fractional addition are called division; Fractional subtraction is called subtraction. Its rule is: multiply the numerator and denominator of a fraction, the result of product addition and subtraction is regarded as "real", the denominator is multiplied as "method", and "real is like a method", which is expressed by today's symbols. For example, Fang's question 8. The general fraction is used here, not the lowest common denominator. Instead, it is complicated to add, subtract and drop points. Shao goes further, and its program can find smaller common multiples, and some even the smallest common multiples. B) Fractional multiplication and division Method Fractional multiplication and division method is called multiplication and division method, and its rule is: take the product of denominator as denominator, and take the product of numerator as numerator, just like today's method. Fractional division is called division, and its rule is to divide the real number and the normal number, and then divide the numerator. Later, Liu Hui added a simpler rule: multiply the denominator and numerator of the method by the real number in turn, which is the reverse multiplication in the current primary school mathematics textbook. C) First observe the law of fractions. If both numerator and denominator are even numbers, divide by 2 first. Otherwise, the numerator and denominator will be "reduced less and more unfavorable", and finally "equal number" will be obtained, which is the greatest common divisor of the original numerator and denominator. Reducing it by an equal number, that is, the simplified number, is consistent with Euclid's method of finding the greatest common divisor, from which the alternating division in modern arithmetic textbooks comes. It should be pointed out that the ancient people's calculation method is calculation rather than the above-mentioned modern pen calculation. For example, in the reduction of the sixth question of Fang, the "equal number" 7 is calculated by calculation, and then the numerator and denominator are divided by 7 to get the simplest number with a score or more, which is the earliest fractional arithmetic in the world. Fractional algorithm was not adopted in Europe until about15th century, and it was not until the 7th century that the Indian algorithm was the same as China's. It is very beneficial to draw lessons from ancient fractional theory and its achievements to enrich teachers' teaching, especially the principle of fractional multiplication and division, which still has important guiding significance for today's primary school teaching. Historically, the concept of fraction and its operation came into being before decimal. In the teaching order, decimals precede fractions, which is because decimal operation is close to integer and more convenient than fractions. When arranging teaching procedures, acceptability is given priority, and teachers should be aware of this. 4. Various proportional algorithms "Millet Nine Chapters" The current technology is a complete proportional algorithm: if all numbers, all rates and demand rates are known, then demand number = all numbers × demand rate ÷ all rates. This method spread to India and the west, known as the "three laws." In the "Nine Chapters", all the examples of the current technique are the problem of millet exchange. For example, it is known that the millet rate is 50 and the bran rate is 30. "Now there is a bucket of millet. How do you ask if you want to make bran rice? " Here 1 barrel is a number, and 50 and 30 are rates and demand rates respectively. According to the current technology, rice bran is 10 liter ×30÷50 = 6 liters. This problem is the proportion of primary school textbooks now. According to the current solution, if the required meter is x liters, there will be a ratio formula of 50: 10 = 30: X, so x = that is, x = 6. In addition, there are some complicated proportional problems in Chapter 9, such as complex proportional problems and chain proportional problems, which are not found in primary school mathematics textbooks. Regarding all kinds of proportions, Liu Hui's notes systematically expounded the principle of unity with the ratio as the key link. If these concepts are properly infiltrated into relevant textbooks, it will be beneficial to teaching. For example, Liu Hui put forward the meaning of "every phase number is called rate", and "phase" means "correlation". The number of rate relations increases or decreases by the same multiple at the same time, and the rate relations remain unchanged. If there is a relationship between a, b and c: a: b = a: b1; B: C = B2: C, commonly known as A, ask C geometry? "Nine Chapters" has been applied twice. Today's skill A is changed to B =, and B is changed to C =. This is called paying attention to today's skills. Liu Hui thinks that the rate of B in the two rate relationships can be changed to the same value b 1b2 first. In order to keep the rate relationship unchanged, the rate of A must be changed to ab2, and the rate of C must be changed to cb 1, which is called connecting with B, that is, A: B: C = AB2: B1B2: CB1. Liu Hui called this transformation the homogeneity principle. It comes from the general fractional division, which must make the denominator and bd the same, and then make the numerator and denominator the same, that is, ad and bc respectively, and the two fractions become. This is called "Qi Zi, follow his mother". 5. Preliminary knowledge of geometry a) The concept of rectangular area is vividly described in Fang and Liu Hui's Notes on Nine Chapters. "Tian Fang Shu said, multiply series widely to get the product step size." Square field refers to the rectangular field, width refers to the bottom of the rectangle, from (that is, vertical) refers to the height of the rectangle, and step is the unit of length, so the area of the rectangle is equal to the bottom multiplied by the height. Teachers can refer to the current textbooks, compare ancient and modern, and further understand why. B) Calculation of triangle area: "Keita is skillful, and half of the triangle is right-handed". The triangular field was called "Guitian" in ancient times, and "Zhengcong" refers to the height perpendicular to the bottom, so the area of the triangle is equal to half the height multiplied by the bottom. C) Calculation of trapezoidal area: The trapezoidal field is called "Yoshida", and its area is also given by multiplying the height by half of the sum of the upper and lower bottoms. The triangular area formula and trapezoidal area formula in Chapter Nine and Liu's notes are all derived from the traditional principle of supplement and supplement. The so-called "internal and external complementarity" is what Liu Hui said "to make up for the shortage with surplus". According to modern theory, the area of a plane figure remains unchanged before and after moving; A plane figure is cut into several blocks, and the sum of the areas of each block is equal to the area of the original figure (the same is true for solids). According to the rectangular area formula and the principle of channel complementarity, we can get the triangular and trapezoidal area formulas, such as triangular area = x bottom x high trapezoidal area =x (upper bottom+lower bottom) x height = width x high channel. The principle of complement is a traditional method to deal with the problem of area and volume in ancient China, which can be transformed into the problem of area and volume. It is widely used, intuitive and ingenious, which is equivalent to giving proof and adapting to the acceptance ability and psychological characteristics of primary school students.

The nine chapters are the first classics, covering the Six Classics of Confucianism, the difficulties of doctors and the Art of War of Sun Tzu. Scholars in the afterlife have to rely on their own doors and walls, follow in their footsteps, or get one or two things to form their own books. Nine Chapters Arithmetic is China's basic book of mathematics, which contains excellent mathematical methods. If compared with Greek mathematics, I am slightly inferior in geometry and number theory, but I am definitely superior to Greek mathematics in arithmetic and algebra. There are 246 questions in Nine Chapters Arithmetic, which are included in nine chapters, namely Tian Fang, Xiaomi, Decline, Shaoguang, Shanggong, Average Loss, Profit and Loss, Equation and Pythagorean. In essence, it is a classification manual of bureaucratic mathematical formulas. Historians believe that "Nine Chapters of Arithmetic" was written by Chang 'an government, based on the books of Qin and Han Dynasties, rather than the teachers' handouts. "This should be an appropriate judgment. As for his mathematical knowledge background, it can be traced back to the Zhou and Qin Dynasties and the Western Han Dynasty. On the one hand, its compilation process and style met the needs of the society at that time, on the other hand, it also reflected the purport of specific academic thoughts. According to historians' research, Confucius inherited the tradition of Zhou Dynasty city-state ministers. His "The Teaching of Six Handed Down Arts: Rite, Music, Shooting, Imperial, Calligraphy and Numbering" is probably the result of collecting historical experience and also depends on the needs at that time. Xi Li Le thought he was a relative, practiced shooting and running the army, and counted books to be a vassal. Therefore, learning mathematics is only one of the ways for them to learn from feudal officials to officials after the reunification of Qin and Han Dynasties. " Yan's Family Instructions put it well: "Arithmetic is also the key to the six arts. Since ancient times, Confucian scholars have learned all the laws of heaven and justice, but they are eclectic. Although there are few scholars who regard mathematics as a way to settle down, Nine Chapters Arithmetic is a summary of mathematical knowledge in Zhou, Qin and Western Han Dynasties, and has its own considerable achievements. Generally speaking. In the category of elementary mathematics, its method has modern significance, that is, its content can be immediately incorporated into a part of modern mathematics by changing its form. In arithmetic, Nine Chapters Arithmetic establishes four operations of fractions and points out the rules of reduction and general division. In addition, various proportional problems are dealt with, and the arithmetic solution of linear algebraic equations-"remainder" is correctly pointed out. In geometry, Nine Chapters Arithmetic lists many area formulas related to land measurement and volume formulas related to civil buildings. Except for a few inaccurate approximations, the rest are completely correct. In addition, it also includes the application problems (including measurement problems) solved by Pythagorean theorem. As for algebra, Kaiping method and Kaiping method have been clearly defined in Chapter 9 of Arithmetic, and naturally extended from Kaiping method are "taking a square with a square" (equivalent to the numerical solution of quadratic equations), the solution of multivariate linear equations ("equation skills") and the addition and subtraction of positive and negative numbers ("addition and subtraction skills"). As can be seen from the introduction, there is a strong tendency to analyze and summarize from the actual problem-solving in the writing process of Nine Chapters Arithmetic, but this does not highlight its background of occurrence and development. As we all know, the mathematical achievements of ancient Egypt and Babylon are no more than that. If you want to know how ancient scholars in China practiced their mathematical thoughts through the knowledge of Nine Chapters of Arithmetic, what questions did they first ask? Why did you bring it up? Why should they solve it? Finally, why do they view their math scores? Then we can't limit mathematics to itself. For example, if Euclid is cut off from Plato and Aristotle, the Elements of Geometry will really become the territory of a few mathematicians. Therefore, under a certain academic background, an in-depth study of the formation process of the knowledge of "Nine Chapters of Arithmetic" can not only help us to confirm the role of mathematics in the progress of human civilization. 2007-01-1721:54: 05 Supplement: Chapter 1, Square field: methods and algorithms for measuring the area of plane graphics, such as quadrature formulas of rectangular, triangular, circular, arc and circular fields, and fractional algorithms, including addition, subtraction, multiplication and division. The second chapter is "millet": the calculation between various grain exchanges, and the proportional algorithm is discussed. Chapter three, "Decline": the problem of proportional distribution. 2007-01-1721:54: 20 Supplement: Chapter IV, "Wider": square root of multiple digits, publisher's rule. Chapter five, "quotient work": the calculation of three-dimensional shape and volume. Chapter six, "lose equally": deal with the itinerary and reasonably solve the taxation problem, especially the time required for people to transport grain from the local area to Beijing to pay taxes, and some problems about taxation according to population, which are also mixed with attenuation, proportion and various miscellaneous problems. 2007-01-1721:54: 28 Supplement: Chapter VII "Surplus and deficiency": profit and loss problem's arithmetic is actually the current linear interpolation method, which has many names, such as trial and error method, double hypothesis method and so on. Chapter 8 "Equation": the content of linear equation, and finally the indefinite equation. By calculating, the coefficients and constant terms of the equation are put into the "equation set", which is the method to solve the equation more than once in "Nine Chapters Arithmetic", and the whole elimination process is equivalent to the linear transformation in algebra. In the chapter of equation, different representations of positive and negative numbers and the rules of addition and subtraction of positive and negative numbers are put forward. Chapter 9 "Pythagorean Theorem": The method of solving practical problems with Pythagorean Theorem is discussed.

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