What are Liu Hui's outstanding achievements in mathematics?

Liu Hui (born around 250 AD), wei ren, an outstanding mathematician in ancient China, was one of the founders of China's classical mathematical theory. History books rarely record his birth, death and life story. According to limited historical data, he was from Zouping, Shandong Province in Wei and Jin Dynasties.

Liu Hui's main works are: Nine Chapters of Arithmetic Notes (volume 10); In the Tang Dynasty, 1 was renamed as "island calculation"; There are 1 volumes in Nine Chapters, but the last two volumes were lost in the Song Dynasty.

Liu Hui's mathematical achievements are roughly in two aspects:

First, clarify the ancient mathematical system of China and lay its theoretical foundation. This aspect is embodied in Nine Chapters of Arithmetic Notes. It has actually formed a relatively complete theoretical system:

In the theory of number system, the arithmetic rules of general division, reduction, four operations and simplification of complex fraction are expounded by using the same number and different number. In the annotation of prescription, he discussed the existence of irrational roots from the infinite meaning of prescription, introduced new numbers, and created a method of infinitely approaching irrational roots with decimals.

In calculus theory, the ratio is clearly defined, and based on three basic operations, such as multiplication and division, the unified theoretical basis of number and formula operation is established. He also used "rate" to define the "equation" in China's ancient mathematics, that is, the augmented matrix of linear equations in modern mathematics.

In pythagorean theory, pythagorean theorem and the calculation principle of pythagorean solution are demonstrated one by one, and a theory similar to pythagorean form is established, and pythagorean measurement is developed. Through the analysis of typical figures such as "crossing in the hook" and "straight in the stock", a similar theory with China characteristics was formed.

In the theory of area and volume, the Liu Hui principle is put forward by using the complementary access principle, complementary deficiency and the limit method of "cyclotomy", which solves the problem of calculating the area and volume of various geometric shapes and geometries. The theoretical value of these aspects is still shining.

Second, on the basis of inheritance, put forward your own ideas. This aspect is mainly reflected in the following representative innovations:

Circumcision and Pi: Liu Hui in Nine Chapters of Arithmetic? In the annotation of roundness field, the exact formula of circle area is proved by secant technique, and the scientific method of calculating pi is given. He first cut the circle from the hexagon inscribed in the circle, and doubled the number of sides to calculate the area of 192 polygon, π= 157/50=3? 14, and calculate the area of 3072 polygon, and get π=3927/ 1250=3? 14 16 is called "emblem rate".

Liu Hui's Principle: Chapter 9 Arithmetic? Yang Equestrian Notes, when he solved the volume of cone by infinite division, he put forward Liu Hui's principle of calculating the volume of polyhedron.

"Harmonious Housing Reform" says: Nine chapters of arithmetic? He pointed out the inaccuracy of the formula V=9D3/ 16(D is the diameter of the ball) and introduced the famous geometric model "Mouhe Square Cover". "Mouhe Square Cover" refers to the intersection of inscribed cylinders with two perpendicular axes.

New technology of equation: Chapter 9 Arithmetic? Equation ",he put forward a new method to understand linear equations, using the idea of ratio algorithm.

Weighted Difference Method: In the White Paper of Calculation Classics of Island, he put forward the weighted difference method, and adopted the methods of height measurement and distance measurement, such as weight table, cable connection and accumulated torque. He also developed gravity difference technology from two observations to three observations and four observations by analogy. In the 7th century, India and Europe only began to study the problem of two observations in15 ~16th century.

Liu Hui's Nine Chapters Arithmetic is one of the earliest mathematical monographs in China, which was written in the Western Han Dynasty. The completion of this book has gone through a historical process. Some of the mathematical problems collected in the book were handed down in the pre-Qin period, and were edited by many people for a long time, and finally sorted out by mathematicians in the Western Han Dynasty. The content of the final version circulated today was formed before the Eastern Han Dynasty. Nine Chapters Arithmetic is China's most important classic mathematical work. Its completion laid the foundation for the development of ancient mathematics in China and played an extremely important role in the history of Chinese mathematics. In this issue, 246 application problems and solutions to various problems are collected in nine chapters, namely, Tian Fang, Xiaomi, Descent, Shaoguang, Working, Average Loss, Insufficient Income, Equation and Pythagorean.

The appearance of nine chapters arithmetic is the result of social development and long-term accumulation of mathematical knowledge, which brings together the labor achievements of mathematicians in different periods. Liu Hui thinks: "The Duke of Zhou has nine ceremonies. If there are nine ceremonies, nine chapters are enough. Zhang Cang, Hou Peiping of Han Dynasty, Cheng Gengshouchang and the old farmer are all good fortune tellers. Cang et al. are called deletion and supplement because of the remnants of old texts. Therefore, the purpose of the school is different from the ancient or different, and the theory is closer. " According to Liu Hui's research results, Nine Chapters Arithmetic originated from Nine Numbers in Duke Zhou. The Nine Chapters Arithmetic he saw was edited by Zhang Cang and Geng Shouchang in the Western Han Dynasty on the basis of inheriting the legacy of the pre-Qin Dynasty, which contained a lot of supplementary contents in the Western Han Dynasty. According to historical documents and unearthed cultural relics, what Liu Hui said is credible.

All kinds of algorithms contained in Nine Chapters Arithmetic were supplemented and revised to meet the needs of the time on the basis of mathematics handed down by mathematicians in the pre-Qin and Han Dynasties. According to Liu Hui's textual research, Zhang Cang and Geng Shouchang are both major mathematicians involved in the revision. Historical records? According to Biography of Prime Minister Zhang, Zhang Cang (about 250 BC-0/52 BC) experienced two dynasties: the Qin Dynasty and the Han Dynasty. In the sixth year of Emperor Gaudi (2065438 BC+0 BC), he was named Hou of Beiping for his meritorious service in attacking Tibetan tea. "Since the history of Qin Wei, the book tomorrow. And make good use of the arithmetic calendar. " He also "wrote 18 books to explain the law of yin and yang." Geng Shouchang's date of birth is unknown. When Emperor Gaozu Xuan Di became an official, he became a senior farmer, and he was favored by the emperor for "taking goodness as a calculation and doing business with utility". He advocated the theory of "Huntian" in astronomy, and in the second year of Ganlu (the first 52 years), he played "to spend the moon with a round instrument and test the shape of the sky." Zhang Cang and Geng Shouchang are both famous mathematicians and hold high positions. Naturally, they will preside over the revision of arithmetic handed down from the pre-Qin period. According to Liu Hui's records, his annotation Nine Chapters Arithmetic was finally edited by Geng Shouchang. We believe that the time when Geng Shouchang edited Nine Chapters of Arithmetic can be set as the time when this book was completed.

Nine Chapters Arithmetic is an official mathematics textbook compiled by the state, which has a great influence on the development of mathematics in Han Dynasty. Guang Yun consists of four chapters, namely, Nine Chapters and The History of the Later Han Dynasty, which were composed by Xu Shang, Wu and Wang Shen? The Biography of Ma Yuan is written by Ma Xu (about 70 ~ 14 1) in nine chapters. He is knowledgeable and good at arithmetic. In addition, Zheng Xuan (127 ~ 200), Liu Hong and others recorded the Nine Chapters of Arithmetic. It can be seen that this book was an important teaching material for learning mathematics at that time. The inscription on a bronze plate in the second year of Guanghe in the Eastern Han Dynasty (179) stipulates: "Big Sinon takes five seals (138? ) letters, ... multi-state copper bucket, oblique name. According to Huang Zhong's calendar, "Nine Chapters Arithmetic" is equal in length, weight and size, which is consistent throughout the country. " This shows that the book was not only widely circulated in the Eastern Han Dynasty, but also the mathematical problems involved in the development of weights and measures should be based on the algorithm in the book. Xu Shang and Du Zhi may be the first mathematicians to study the Nine Chapters Classic after it was written. Both Xu Shang and Du Zhi were mathematicians in the late Western Han Dynasty. Hanshu? Yiwenzhi recorded 26 volumes of Xu Shang Arithmetic and Du Zhi Arithmetic 16. These two books were written by Yin Xian before correcting his mathematical works in the third year (the first 26 years) of Emperor Han Cheng. The completion date of Xu Shang's and Du Zhi's works is not far from the time when Geng Shouchang deleted and supplemented Nine Chapters of Arithmetic. Their mathematical works should be completed on the basis of studying nine chapters of arithmetic.

Nine Chapters Arithmetic not only occupies an important position in the history of Chinese mathematics, but also makes an important contribution to the development of world mathematics. Fraction theory and its complete algorithm, proportion and proportion distribution algorithm, area and volume algorithm, and solutions to various application problems are described in detail in the chapters of the book, such as square domain, millet, decay, quotient work and even loss. The opening method, profit and loss (double hypothesis method), the concept of positive and negative numbers, the solution of linear simultaneous equations and the general formula of integer pythagorean string in Shaoguang, Profit and Loss, Equation and Pythagorean are all outstanding achievements in the history of mathematics in the world.

Liu Hui's Notes on Nine Chapters not only made important achievements in sorting out the ancient mathematical system and perfecting the ancient calculation theory, but also put forward rich and colorful ideas and inventions. He used the ratio theory to establish the unified theoretical basis of the number sum formula, applied the principle of complementarity in and out and the limit method to solve many problems of area and volume, and established a unique theory of area and volume. He used nine chapters to strictly prove many conclusions, and some of his methods have great inspiration for later generations and even today's mathematics.

Liu Hui's work not only had a far-reaching impact on the development of ancient mathematics in China, but also laid a lofty historical position in the history of mathematics in the world. In view of Liu Hui's great contribution, many books call him "Newton in the history of Chinese mathematics".