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.
He introduced various forms of vertical and horizontal diagrams and related construction methods in "Arithmetic of Odd Stories in Ancient Times". Stacking is Yang Hui's research on higher-order arithmetic progression after Shen Kuo's gap product. In the aspect of classification, Yang Hui reclassified 246 problems in Nine Chapters of Arithmetic into nine categories according to the order of solving problems from shallow to deep, such as multiplication, division, integration, interchange, secondary descent and pythagorean.
He attaches great importance to the popularization and development of mathematics education. In the book Origin of Algorithm Change, Yang Hui's Learning Plan for Beginners is an important document in the history of Chinese mathematics education. (1) major works
Yang Hui left a large number of works in his life, namely: Detailed Explanation of Nine Chapters' Algorithms (12), Daily Algorithms (II) (1262) and Multiplication and Division to Change the Background (1274).
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 Classification breaks through the nine-chapter arithmetic classification mode, and is divided into nine categories according to the nature of the solution: multiplication, division, combination, intersection, decline, overlap, surplus, loss, equality and pythagorean.
Yang Hui also drew a triangle figure representing the coefficient of binomial expansion in the book "Detailed Explanation of Nine Chapters Algorithm", which is called "the origin of roots" and is now referred to as "Yang Hui Triangle".
Yang Hui Triangle is a triangular numerical table arranged by numbers, and its general form is as follows:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
.....................................
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 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 further understand the writings and publicize the significance of Liu Jun's training. " The Agile Method of Multiplication, Division and Ratio of Fields cited 22 problems when discussing the origin of ancient times, 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.
Most of Yang Hui's works focus on applied arithmetic and are 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.
(II) Main research results
The focus 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 Xiahouyang Arithmetic Classic by Song people and printed and circulated to this day. Han Yan Arithmetic was written around 770 AD, and many examples 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 "six methods of multiplication": one is "single cause", that is, multiplying by a one-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 10 1 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 109, one bit is added for every other bit; The multipliers are 2 1 to 29, 20 1 to 299, with conjoined addition. For example, the calculation of 342×56 is written in modern symbols: 342× 56 = 342×112 ÷ 2 = (34200+342×12) ÷ 2 = (34200 ten 3428.
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 Variation of Algorithms", several summation formulas of second-order arithmetic progression are described, among which there are three kinds, namely triangular crib, quadrangular crib and square crib, which are equivalent to the following three kinds with today's symbols:
The above three formulas can be derived from Shen Kuo's 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.