Life is unknown. He used to be a local administrative officer in the Southern Song Dynasty, and his footprints were all over Suzhou and Hangzhou. He has made great contributions to summarizing agile algorithms such as folk multiplication, division, "superposition", vertical and horizontal diagrams, and mathematics education. He is the first mathematician in the world to draw a rich vertical and horizontal map and discuss its composition law. He also proved the sagittal formula, which was called "Shu Hui" at that time. Together with Qin, Zhu Shijie, they are called "the four great mathematicians of Song and Yuan Dynasties".
He has written five kinds of mathematical works, 2 1 volume, namely, Detailed Explanation of Algorithms in Chapter 9 12 (12 1), Everyday Algorithms 2 (1262) and Multiplication, Division and Change 3 (. The latter three are collectively called Yang Hui algorithm. Korea, Japan and other countries have published translations and spread them all over the world.
Basic introduction of real name: Yang Hui font size: Zi Qianguang age: Southern Song nationality: birthplace of Han people: Qiantang (now Hangzhou, Zhejiang). Main work: Nine chapters of algorithm, daily algorithm and Yang Hui algorithm are explained in detail. Main achievements: improvement of addition, vertical and horizontal diagram and superposition technology; Multiplication and division and agile algorithm of prime numbers: Yang Hui's Triangle and other great achievements, great writings, great research achievements, people's stories and great achievements Yang Hui left a lot of writings all his life. His mathematical masterpiece * * * has five kinds of 2 1 volumes, namely: nine chapters of detailed algorithm (12 (126 1 year) and daily algorithm. There are three volumes (1274, co-edited with others), two volumes (1275) and two volumes (1275, co-edited with others), of which the last three volumes are Yang. He attaches great importance to the popularization and development of mathematics education. Under the background of algorithm reform, Yang Hui's Learning Plan for Beginners is an important document in the history of Chinese mathematics education. The current version of "Detailed Explanation of Nine Chapters Algorithm" is not complete, and its arrangement is also chaotic. As can be seen from the Preface, the book gives a detailed explanation of 80 problems in Annotation, Annotation in Tang Dynasty and Jia Xian's Nine Chapters Arithmetic in the Northern Song Dynasty. On the basis of nine chapters and nine volumes of arithmetic, three volumes have been added, one is a graph, and the other is about multiplication and division algorithm, before the ninth chapter. One volume is a compilation of categories, which occupies the first number at the end of the book. The multiplication and division method of volume L, the decay and anti-decay problems of volume 2 square field, volume 3 millet, volume 4 decay point and volume 6 quotient work are all lost. Volume 4 is divided into the second half, volume 5 is the remnant of Yongle Dadian, and the rest is the Yi Family Hall Series. Judging from the style of the incomplete edition, the detailed explanation of Nine Chapters of Arithmetic in this book can be divided into: 1. Solve the problem. The content is to explain the terminology, the meaning of the topic, the arrangement of the text and the comments on the topic. Second, with Cao. In the arrangement, Yang Hui clearly distinguished Jia Xian's Fa and Cao from his own detailed explanation with big characters. Third, compare classes. Select the same or similar problems with the topic algorithm in "Nine Chapters Arithmetic" for comparative analysis. Fourth, continue to publish notes. On the basis of predecessors, this paper makes further comments on 80 questions in Nine Chapters of Arithmetic. Yang Hui's Code breaks through the classification mode of nine chapters of arithmetic, and is divided into nine categories according to the nature of solutions: multiplication, division, combination, interchange, decay, superposition, remainder, equation and pythagorean. Yang Hui's research Yang Hui also drew a triangle figure representing the coefficient of binomial expansion in the book "Detailed explanation of nine chapters of algorithm", which is called "the origin of root-opening exercise" and is now referred to as "Yang Hui triangle". Yang Hui triangle is a triangular numerical table arranged by numbers. The general form is as follows:11113314641151. The most basic feature of ..................................... Yanghui Triangle is that its two hypotenuses are both composed of the number 1, and the rest of the numbers are equal to the sum of the two numbers on its shoulders. The daily algorithm, the original is not handed down from generation to generation, only a few topics have been handed down from generation to generation. From the preface quoted by Yang Hui in Miscellanies of Algorithms, we can know the outline of the book: "Take multiplication, division and addition as the method, take the weighing field as the topic, compile thirteen poems and set sixty-six topics." Usage must carry the source, proposition must be responsible, divided into upper and lower volumes. "This book is undoubtedly a popular practical calculation book. The origin and end of multiplication and division all have their own topics, which have made great contributions to summarizing the improvement of folk equivalent multiplication and division. The first volume is called "The Origin of Algorithm Change". Firstly, the Outline of Learning Calculation is put forward, which is an important document in the history of mathematics education and also discusses the multiplication and division algorithm. The middle book is called "Multiplication and Division, Turning Waste into Treasure", and discusses the skills of addition, subtraction, multiplication and division, seeking one and nine; The second volume is called "Using Background through Calculation", which is a comment on the middle volume. The first volume of Comparative Method of Field Multiplication and Division is an extension of Fang's Detailed Explanation of Nine Chapters Algorithm, and the selected examples are very close to reality. The second volume mainly refers to Liu Yi's work. Yang Hui said in the preface of Tian Mu Multiplication and Division that "Mr. Wang wrote On Ancient Roots". ..... written as a hundred straight fields, I believe that the field is endless, and the method of quoting positive and negative profits and losses from prescriptions is unheard of. If you work too far, you don't study the source, but you can't know it unless you explore it. Hui Xuan can be a key questioner to learn more works and promote the significance of Liu Jun's training. The Agile Method of Multiplication, Division and Ratio of Field Mu cited 22 problems when discussing the ancient origin, mainly the solution of quadratic equation and quartic equation. The algorithm of extracting odds from ancient times, the first volume lists 20 vertical and horizontal diagrams, namely the Rubik's cube. The first one is a river map, the second one is Luo Shu, the second one is a Rubik's cube with four lines, five lines, six lines, seven lines and eight lines, and the other one is a Rubik's cube with nine lines and ten lines. Finally, there are pictures such as "Gathering Five", "Gathering Six", "Gathering Eight", "Saving Nine", "Eight Arrays" and "Serialization". Some pictures are described by words, but each picture has a construction method to make the natural numbers in the picture equal to each other. It is also of great scientific value to comment on Island. Yang Hui's research Yang Hui's works mostly focus on the application of arithmetic, which is easy to understand. His works also widely quoted mathematical classics and arithmetic books at that time, as well as some outstanding achievements of ancient mathematics in China, such as Liu Yi's Positive and Negative Recipes, Jia Xian's Source Map of Prescription Science, Multiplication and Incremental Method, etc. Thanks to Yang Hui's quotation, otherwise, it would no longer be known to us today. The main research achievement of Yang Hui's mathematical research and education work is to improve the calculation technology of multiplication and division and summarize various agile multiplication and division algorithms, which is determined by the social situation at that time. Since the mid-Tang dynasty, the social economy has been greatly developed, and handicrafts and commercial transactions have reached a considerable scale. Therefore, the opportunities for people to need mathematical calculations in their production and life have greatly increased. This situation urgently needs mathematicians to provide people with easy-to-master, fast and accurate calculation methods. In order to meet the social demand for mathematics, some practical arithmetic books appeared in the middle and late Tang Dynasty. However, all these books have been lost except Han Yan Arithmetic, which was mistaken for Xiahou Yang Arithmetic Classic by Song people. Han Yan algorithm was written around 770 AD, and many examples of agile algorithms of multiplication and division were introduced in the book. For example, a number multiplied by 42 can be converted into a number multiplied by 6 and then multiplied by 7; A number divided by 12 can become a number divided by 2 and then divided by 6. This can also be done for more complex problems. By decomposing the multiplier and divisor into one digit, the operation can be realized in one line, which simplifies the operation and improves the speed. Han Yan also introduced some other simple algorithms. For example, "body plus four" and "interval plus two". Shen Kuo, a scientist in the Northern Song Dynasty, also summarized agile algorithms such as addition and gravity. Yang Hui lived in Suzhou-Hangzhou area, where commerce was developed in the Southern Song Dynasty, and further developed multiplication, division and agility algorithms. He said: "The multiplication and division method is based on extensive and profound methods. The wizard algorithm uses' addition and subtraction',' nine returns' and' seeking one' to find shortcuts. Scholars are ignorant and should use them at the same time. "On the basis of predecessors, he put forward the' six multiplication': one is' single cause', that is, multiplying by a single-digit multiplier; The second is "double factor", that is, the multiplier can be decomposed into the product of two one-digit products; Third, it is called "antecedent", that is, two digits are multiplied by the last digit of a multiplier of one, such as 257× 21= 257× 201257. In fact, antecedent is accomplished by multiplication and addition of multiple digits into one digit through multiplication and distribution law. Fourth, multiplication, that is, the usual multiplication; Fifth, "multiplication" means that the multiplier can be decomposed into the product of two factors and multiplied twice; The six-word "loss multiplication" is a kind of subtraction multiplication. For example, when the multipliers are 9, 8 and 7, you can subtract the multiplicand of 1, 2 and 3 times from the multiplicand of1. Yang Hui further developed the algorithm of finding one, which was passed from Tang Dynasty to Song Dynasty, and summed up "five methods of addition and subtraction" and "four methods of addition and subtraction". Seeking one is actually to change the first digit of multiplication and division into one through multiplication, folding and factorization, so as to multiply and divide by addition, subtraction and multiplication. Yang Hui's five methods of multiplication, addition and addition, namely, adding one, adding two, adding again, adding spaces and adding together. If the multiplier is 1 1 to 19, add1; If the multiplier is 10L to 199, add two digits; Multiplier can be divided into the product of two factors, when one or two can be added, it is called heavy addition; When the multiplier is 10 1 to l09, one bit is added every other bit; The multipliers are 2 1 to 29, 20l to 299, with conjoined addition. For example, the calculation of 342×56 is written in modern symbols as follows: 342× 56 = 342×112 = (342001342× L2)12 = (342001342 × L2). The usage of "four-tree addition and subtraction" is similar to multiplication and addition, that is, "subtract one place", "subtract two places", "subtract again" and "subtract interval". Addition, a division that appeared in the early years of Northern Song Dynasty, was further improved by Yang Hui. The advantage of addition is that the trial quotient is avoided by using the method of double complement, but for the dividend with more digits, the operation is complicated, and later generations have improved it and summarized Jiugui Gukuo, which contains 44 formulas. Yang Hui quoted 32 phrases in "Nine Returns to New Enclosed" in his book "Multiplying and Dividing into Treasures", which were divided into three categories: "the return number is ten", "the return number is increased from the top" and "half is five". Objectively speaking, Yang Hui spared no effort to improve computing technology and greatly accelerated the pace of computing tool reform. With the popularization of calculation formulas, the operation speed is greatly accelerated, so that people feel that fiddling with calculation formulas can't keep up with them. In this context, abacus came into being, and by the end of Yuan Dynasty, it had been widely popular. Vertical and horizontal diagram, the so-called Rubik's cube. As early as Han Zheng Xuan's Annotations on Yi Wei and Shu Shu, the "Nine Palaces", that is, the third-order Rubik's Cube, has been covered with mysterious colors for thousands of years. Yang Hui created the name "vertical and horizontal map". In the book "Algorithm for Extracting Odds from Continuing Ancient Stories", various graphs are made. Graph ll is a fourth-order vertical and horizontal graph; Graph 12 is a hundred subgraph, that is, a ten-order vertical and horizontal graph. The sum of numbers in each row and column is 50-5 (the sum of diagonal numbers is not 505); Figure 13 is a schematic diagram of "Ju Ba". Yang Hui's Rubik's Cube of "Twenty-four for Thirty-two" has four circles, and the sum of the numbers of each circle is 100. Figure 14 is a picture of "saving nine", which is arranged with the first 33 natural numbers to achieve the effect of "one hundred and forty-seven circles are oblique and straight". Yang Hui not only gave the method of making these figures, but also got some knowledge of the general construction rules of figures, and solved the mystery of the Rubik's Cube. This is the earliest systematic research and record of the Rubik's Cube in the world. Since Yang Hui, mathematicians of Ming and Qing Dynasties in China have studied vertical and horizontal maps in succession. Another important achievement of Yang Hui is folding. This is Yang Hui's research on the summation of higher-order arithmetic progression after Shen Kuo's "gap product method". In Detailed Explanation of Algorithms in Chapter Nine and General Variations of Algorithms, several summation formulas of second-order arithmetic progression are described, among which three formulas are triangular stacks, quadrangular stacks and square stacks, which are equivalent to the following three formulas with today's symbols: the above three formulas can be derived from Shen Kuo's Chutong formula. Mathematical reclassification is also one of Yang Hui's important mathematical work. On the basis of explaining the nine chapters of arithmetic in detail, Yang Hui specially added a book "Classification", which reclassified the methods and 246 questions in the nine chapters into nine categories according to the nature of their methods: multiplication, division, combination, exchange, decline, intersection, profit, loss, equation and pythagorean. Yang Hui is not only an excellent mathematician, but also an outstanding mathematics educator. He devoted his life to the education and popularization of mathematics, and many of his works were written for the education and popularization of mathematics. The book "The Background of Algorithm Change" contains Yang Hui's "learning plan" specially formulated for beginners, which embodies Yang Hui's thoughts and methods of mathematics education. The story of Yang Hui's study of characters tells this achievement of Yang Hui, and it has to start from an accidental event. One day, Yang Hui, a local official in Taizhou, went out for a cruise. On the way, the gong cleared the way in front, and the official was behind the house. Carrying a big sedan chair in the middle, awesome. Charming spring generously exudes fragrance and brings joy and happiness to life. Rhododendrons are hidden in the branches of mango trees. Wake up people's hopes with its mellow and sweet voice. Crowds of thrush birds crouched on the branches like wedding banquets, making elegant calls. Neem, pear and chestnut trees all seem to be intoxicated by their own fragrance. Yang Hui lifted the curtain of the sedan chair and watched the peanut trees and birds pass through the forest. It's really a pleasant spring, calling the couple orioles a breeze. It's a good year and beautiful scenery. Walking, I saw the boring gong of escorting stopped, and there was a loud cry from the children in front, followed by a fierce reprimand from the chief. Yang Hui asked what was going on and sent someone to report: "The child won't let him go, saying that he won't let him go until he finishes the problem, or he will take a detour." When Yang Hui saw the interest, he stepped out of the sedan chair and came to the front. The village chief quickly said, "Did you coax the child away?" Yang Hui touched the child's head and said, "Why not let my officials pass by here?" The child replied, "it's not that I'm not allowed to pass." I am afraid that you will step on my formula, and I can't remember it again. " "What formula?" "Is to put the number of 1 to 9 in three rows, whether it is direct addition, horizontal addition or horizontal addition, the result is equal to 15. This afternoon, our husband asked us to do this problem well. I am counting the key points. " Yang Hui's cartoon image Yang Hui quickly squatted down and carefully looked at the child's formula. I feel that this number was mentioned in an article written in the book Da Dai Li compiled by Dade, a scholar of the Western Han Dynasty. Yang Hui and the children quickly got up together. It was not until after noon that they breathed a sigh of relief. The results came out. They checked it again, and the result was 15. They stood up. Let's put the formula out: (In the box on the left, the result is 15 regardless of horizontal addition, vertical addition or diagonal addition. The child looked at the amiable local official and said, "Thank you for your time. Come to my house for dinner! "Yang Hui listened and said," OK, OK, I'll see your husband this afternoon. "The child looked at Yang Hui with tears in his eyes. Yang Hui thought there must be something strange here and asked gently, "What's the matter?" The child explained the reason in detail: it turned out that the child didn't go to school, and the family was too poor to eat enough. Where did he get the money to study? The child drove the cattle to the landlord's house. Every time students leave school, he secretly hides under the students' windows to eavesdrop. Mr. Wang solved the problem this morning. The child taught himself and finally solved it. Yang Hui was deeply moved. It is not easy for a small child to have such painstaking efforts. He said to the child, "This is 10 silver. Take it home. Go to school in the afternoon and I will wait for you there. " In the afternoon, Yang Hui took the child to find Mr. Wang, told him about the child, and took out money to fill the quota for the child. The children's families are very grateful. Since then, the child has only one real husband. The teacher admired Yang Hui's incorruptibility, so they talked about mathematics. Yang Hui said: "The question I just did with my children seems to be from the book" Big Dai Li "?" The gentleman said with a smile: "Yes, although The Big Wear Ceremony is a collection of records of various etiquette systems, it also contains some mathematical knowledge. The topic you just mentioned is the math game problem I gave the children. " Seeing Yang Hui's puzzled expression, the teacher said: "Zhen Luan of the Northern and Southern Dynasties wrote in the book Numerology Legacy:" Nine palaces, two or four shoulders, six or eight feet, left three and right seven, wearing nine shoes, one and five living in the middle. " Yang Hui recited it again and found that what he said was exactly the same as the figures he and his children put out in the morning. He asked, "Do you know how this Nine palace map did it? "The teacher also don't know the source. Yang Hui returned home, pondered it over and over again, fiddled with these figures on the table whenever he had time, and finally found a rule. He summed up this rule in four sentences: the nine sons are obliquely arranged, the up and down are easy, the left and right are more harmonious, and the four dimensions are prominent. " That is to say, first, nine numbers are arranged in three rows obliquely from big to small, then 9 and 1 are interchanged, 7 on the left and 3 on the right are interchanged, and finally, 4, 2, 6 and 8 of the four corners move outward to form three rows, thus forming the Nine palace map. Let's demonstrate the following: (diagonal arrangement of nine children) (reciprocal up and down, more similar left and right) (four-dimensional projection) According to the similarity law, Yang Hui got the "flower 16 diagram", that is, the numbers from 1 to 16 were arranged into four rows and four columns of squares, so that each horizontal, vertical and diagonal line Readers, you might as well have a try. Later, Yang Hui sorted out this kind of problems scattered in predecessors' works and circulated among the people, and got many similar pictures, such as Five-Five, Six-Six, Evolution, Yi Tu, Nine-Nine and Hundred-Child. Yang Hui has always called these graphs vertical and horizontal graphs, and wrote them into his own mathematical book "Algorithm for Continuation of the Ancient" at 1275, and passed them on to future generations. The vertical and horizontal charts are also called magic squares, which require that the natural numbers from 1 to n2 be placed in n2 grids. But for a long time, people used to treat it as a pure mathematical game and didn't pay due attention to it. With the development of modern combinatorial mathematics, vertical and horizontal graphs show more and more vitality, and find a place in graph theory, combinatorial analysis, game theory, computer science and other fields. Yang Hui can be said to be the first mathematician in the world to give such a rich vertical and horizontal diagram and discuss its composition law. In addition to this achievement, Yang Hui also made a major contribution, that is, the "Yang Hui Triangle". On one occasion, Yang Hui got a book "Nine Chapters of Yellow Emperor's Arithmetic Fine Grass", which was written by several Jia Xian in the Northern Song Dynasty. There are many great achievements, for example, Jia Xian drew a picture called "Source Map of Prescription Science". The characters in the picture are arranged in a big triangle. The numbers on both waists are 1, and the rest is equal to the sum of the two numbers above it. Starting from the second line, each line of numbers in this big triangle corresponds to a set of binomial expansion coefficients. The following example shows that in the third line, 1, 3,3, 1, these four numbers just correspond to (x+ 1) 3 = x3+3x2+3x+65433. The nine-chapter algorithm corresponds to the fourth line (x+1) 4 = x4+4x3+6x2+4x+1. And so on. Yang Hui faithfully recorded this painting by Jia Xian and kept it in his book Nine Chapters of Arithmetic. Later, it was found that this big triangle not only can be used to solve equations, but also has a close relationship with mathematical knowledge such as combination, higher-order arithmetic progression and interpolation. In the west, it was not until16th century that someone drew a similar figure on the cover of a book. Pascal, a French mathematician, discussed the properties of this figure in detail in his paper 1654, so it is also called "Pascal triangle" in the west. In addition to the above achievements, Yang Hui also wrote books such as Daily Algorithm, Multiplication and Division, Changing Background, and Field-to-Field Method of Multiplication and Division, which provided extremely important information for future generations to understand the mathematical outlook at that time. Yang Hui's works have greatly enriched the treasure house of ancient mathematics in China and made outstanding contributions to the development of mathematical science. He deserves to be one of the "four great masters in Song and Yuan Dynasties". His famous math books have five kinds and 21 volumes. There are twelve volumes (126 1 year), two volumes (1262 year), three volumes (1274 year) and two volumes (field ratio multiplication and division algorithm). Yang Hui's mathematical research and education work focuses on computing technology. He summed up and developed agile algorithms for calculating multiplication and division, and some even made up songs, such as Nine Centralized Decisions. In his Algorithm for Extracting Odds from Ancient Times, he introduced various forms of "vertical and horizontal graphs" and related construction methods. "Overlap" was Yang Hui's research on higher-order arithmetic progression after Shen Kuo's "Gap Product". In Classification, Yang Hui reclassified 246 problems in Nine Chapters of Arithmetic into nine categories, such as multiplication and division, coincidence, interchange, quadratic decline and Pythagoras, according to the order of solving problems from shallow to deep. Chapter 9 Algorithm How to attach great importance to the popularization and development of mathematics education. Under the background of algorithm reform, Yang Hui's "Learning Calculation Outline for Beginners" is an important document in the history of Chinese mathematics education. Yang Hui has many mathematical works. He has compiled five kinds of 2 1 volume mathematics books, which contain many lost problems and algorithms in ancient mathematics works. The focus of Yang Hui's mathematical research and education work is computing technology. Yang Hui summarized and developed the method of calculating multiplication and division and the agile algorithm of creating the name of "vertical and horizontal graph". After Shen Kuo's Gap Product Method, he studied higher-order arithmetic progression.