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 the annotations of Hui and Tang Li and the eighty topics in 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. Volume 1 is a compilation of categories, which is the first figure at the end of the book. Volume 1 multiplication and division, volume 2 square field, volume 3 millet, volume 4 decay point, anti-decay problem, volume 6 quotient function has been 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
12 1
133 1
1464 1
15 10 105 1
16 1520 156 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.