100 must-read books for college students

Chapter 21 Mathematical Scriptures in History
Chapter 20 The first arithmetic in history: "Nine Chapters of Arithmetic"

In the history of Chinese mathematics, there is a book comparable to Euclid's "Elements of Geometry".

"Nine Chapters on Arithmetic".It is the earliest existing monograph on mathematics in my country, and its biography includes the original text of "Nine Chapters on Arithmetic", Cao Wei and Liu Hui

Notes, Li Chunfeng and other notes in the early Tang Dynasty are divided into three parts.

"Nine Chapters of Arithmetic" is a collection of the culmination of Chinese mathematics knowledge from the pre-Qin to the Western Han Dynasty, and it was included in the "Ten Books" that all students of the Guozijian School of Mathematical Sciences must read in the Tang Dynasty

One of the "Computing Classics". The book collects 246 questions, which are divided into 246 chapters according to the nature and solution of the questions, and systematically summarized
Many achievements in mathematics from the pre-Qin period to the Western Han Dynasty were in the leading position in the world at that time.Its completion marks the Chinese
The determination of the traditional mathematical system.

"Nine Chapters of Arithmetic" Annotated by Liu Hui

The author of "Nine Chapters of Arithmetic" is still a mystery, but from 1983 to 1984 in the Han Tomb No. 247 Zhangjiashan, Jiangling County, Hubei Province

The several bamboo slips found in the book have clarified some background information.Among them is a book called "Book of Mathematics". For its style and content, see

It seems to be closely related to the style and content of "Nine Chapters of Arithmetic".However, this work was seriously damaged, so far only published
A brief and preliminary account of its content.Xu Yue (active in 220 AD) may be the earliest commentator on "Nine Chapters of Mathematical Sciences"

Left and right), it is said that he is also the author of the surviving book "Shu Shu Ji Yi".His annotations were known until the Sui and Tang dynasties

, but it was lost in the Song Dynasty.The earliest surviving commentary was written by Liu Hui. According to "Sui Shu" Volume 404, page [-],

Liu Hui's annotations were made in the Cao Wei era, and its date is 263 years.However, in a note on the area of ​​Yuantian, Liu Hui mentioned preserving

The bronze dendrobium in the arsenal of the Jin Dynasty (built in 265), it can be seen that he has been studying the "Nine Chapters of Arithmetic" while serving the new dynasty.

"As a note.

Liu Hui's annotation is one of the great achievements of ancient mathematics, but it is a pity that it is difficult to verify his life experience today.

It is said that he is from Zouping, Shandong.Liu Hui

Is a great mathematician, especially known for his research on pi.Liu Hui proved the value of pi

The final value corresponds to π=314.

In addition, it should be pointed out that after Liu Hui finished his notes on "Nine Chapters of Arithmetic", he added a new chapter on "Heavy Differences", discussing the use of

A method of looking with a sundial.During the Sui and Tang Dynasties, this part was still attached as Chapter 10 after "Nine Chapters of Arithmetic", but in the Tang Dynasty

In the [-]th century, it began to spread in a single volume, and was named "Sea Island Mathematical Classics" according to the name of its first question.
It still exists today.

In 656, Li Chunfeng was ordered to compile the mathematics textbook used by Taixue ("Old Tang Book" Volume 79

, p. 2719), for this purpose, he made a re-note of "Nine Chapters of Mathematical Sciences" based on the book with Liu Hui's annotations.

Book.Some interesting early data are preserved in Li's Notes.It is because of his work that the current edition is fixed.

In addition, Li Ji, a member of the Imperial Library of the Northern Song Dynasty, also provided annotation materials, and before the book was printed in 1084, he added

An appendix titled "Nine Chapters of Arithmetic, Sound and Meaning".

The coherence and orderliness of "Nine Chapters of Arithmetic" is enough to show that it was probably written or compiled by a single author.it belongs to han

It has never been disputed that it is an ancient work, but much of the mathematical knowledge it contains dates back at least to the Warring States period.

The authenticity of the popular edition of "Nine Chapters of Arithmetic" is beyond doubt, and all surviving editions of this work can be traced back to the Northern Song Dynasty
A series of nine mathematical works published by the Royal Library in 1084 (including "Zhou Bi Suan Jing")
. "Nine Chapters of Arithmetic" was published together with Liu Hui's and Li Chunfeng's annotations and Li Ji's appendix. This printed copy has not been preserved.

It has been preserved to the present, and what we have is the copy that Bao Huanzhi kept after it was reprinted in 1213.According to his postscript,

After the fall of the Northern Song Dynasty in 1126, the research on "Nine Chapters of Arithmetic" almost completely stagnated.
This question is interesting, and the only available copy lacks the necessary annotations such as Liu's and Li's notes, and it is said to be written by the Yellow Emperor

.

Fortunately, in the summer of 1200, a book originally printed in the Royal Library in 1084 fell into the hands of a Hangzhou friend of Bao Huan.
So he copied it and printed it.In the Ming Dynasty, Bao Ben was copied into the "Yongle Dadian".A remnant of Bob's (
Only chapters 1 to 5) are now in the Shanghai Library.

The head of arithmetic leads the world

"Nine Chapters of Arithmetic" collected 246 questions by category, and gave each

answer to this question, it is clear that the book aims to give a comprehensive account of the mathematical knowledge of the time.and

After each question is cited, it is followed by a numerical answer, and then a detailed process of solving the problem.Each question starts with a special number

Word examples to express.In the absence of algebraic representation, problem-solving methods are expressed rhetorically, that is, describing the
A language instruction that performs a series of arithmetic operations.It never explicitly attempts to demonstrate the effectiveness of the methods used.246 in the book
This application problem belongs to Fangtian, Corn, Decline, Shaoguang, Shanggong, Junshu, Insufficient Profit, Equation, and Pythagorean Nine Chapters.

Fang Tianzhang gave the method to obtain the area of ​​fields in various shapes such as rectangle, triangle, trapezoid, circle, bow, fan, ring, etc.
Of the law.

Set π=3 in calculating the area of ​​the circle;

However, the given rules for bow and sector are approximate.

The rules given are exact as far as figures formed by straight lines are concerned.

In addition, this chapter also clarifies the rules of fractional operations.score pass

Complete rules for division, reduction, and the four arithmetic operations of addition, subtraction, multiplication, and division.The latter predates Europe by more than 1400 years.Mi Zhang proposed a proportional algorithm,

It is called Jinyoushu.Decline chapter puts forward the law of proportional distribution, which is called decay technique.In addition to the various three-dimensional volume formulas given by Shang Gongzhang

, as well as the project assignment method.Jun Shuzhang used the technique of decay to solve the problem of reasonable burden of taxes and servitude.

Jinyoushu, Decaying Shushu and their application
The method constitutes a whole set of proportional theories including today's positive and negative proportions, proportional distribution, compound proportions, and chain proportions.West
Fang did not form a similar set of methods until after the end of the 15th century.

The Chapter Profit and Deficiency puts forward three types of profit and loss problems: surplus and deficiency, surplus and deficiency, double surplus and double deficiency, and some

A solution to a general problem that can be transformed into a surplus-deficit problem through two assumptions.

The equations chapter uses the method of separating coefficients to represent linear equations, which is equivalent to the current matrix.This is the world's first complete

Solutions of linear equations.

The chapter also introduces and uses negative numbers and proposes
Learned the plus and minus technique - the addition and subtraction rules of positive and negative numbers are exactly the same as those in modern algebra; when solving linear equations, they are actually implemented
Multiplication and division of positive and negative numbers.This is a major achievement in the history of world mathematics, breaking through the range of positive numbers for the first time, expanding the

number system.In foreign countries, it was not until the Brahma and Duo of India in the 7th century that negative numbers were recognized.

The Pythagorean chapter puts forward the general solution formula of the Pythagorean number problem: if a, b, and c are respectively the hook, strand, and string of the Pythagorean shape, then c: b: a=12 (m12+n[-]): mn: [-] (m[-] +n[-]), m>n.exist
In the West, Pythagoras, Euclid, etc. only obtained a few special cases of this formula, and it was not until Diophantus in the 3rd century that

A similar result was obtained, which is about 3 centuries later than "Nine Chapters of Arithmetic".There are still some content in the Pythagorean Chapter, but in the West it is still
Modern things.For example, the last question gives such a set of formulas: a=2(ca)(cb) +(cb
), b=2(ca)(cb) +(ca), c=2(ca)(cb) +(ca)+(cb).this achievement
In other words, it was not obtained by the American number theorist Dixon until the end of the 19th century abroad.

From the above, it can be seen that "Nine Chapters of Arithmetic" is centered on calculation, closely related to
It is a practical style aimed at solving mathematical problems in people's production and life.Its influence is so deep that in the future, most of our country
Academic works generally take two forms: either annotate it, or write a book imitating its style; even after Western calculations were introduced to China, people wrote
When writing books, mathematical knowledge including Western calculations is often incorporated into the framework of "Nine Chapters".

The algorithms in "Nine Chapters of Arithmetic" are abstract, and the interrelationships are not obvious, making them seem messy.Liu Hui greatly developed and deepened the

The concept of rate and the principle of equivalence have been used for a long time, and they are regarded as the discipline of computing.Many questions, just find out where the various rates
The relationship, through "multiplying to disperse, coordinating to gather, and aligning to communicate", can all be attributed to Jinyoushu's solution.

Decompose a plane (or three-dimensional) figure
It can be divided into several parts, and the sum of the areas (or volumes) of each part is equal to the area (or volume) of the original figure.A plane (or three-dimensional) graphic classic

Through translation or rotation, its area (or volume) remains unchanged.Based on these two self-evident

The principle of complementarity between in and out of the premise is the most commonly used principle in ancient Chinese mathematics for geometric deduction and proof.Liu Hui developed

The principle of phase complementation has successfully proved many area, volume, and Pythagorean and square root formulas that can be transformed into area and volume problems.

The correctness of formulas and algorithms.

The most outstanding contribution of Liu Hui is the successful use of infinitesimal division and limit thinking in mathematical proofs.

"Nine Chapters of Arithmetic" puts forward the circle area formula as S=l2π·r (S is the circle area, l is the circumference, r is half
径)
.Based on this formula, Liu Hui calculated the two approximate values ​​of π, 15750 and 3927
In 1250, the scientific method of calculating pi was established for the first time in China, which laid the foundation for the study of pi in my country in the world.

The foundation of long-term leadership.

Liu Huizhu’s research on the volume problem has touched the core issue of the modern volume theory, and believes that the solution of the volume of the tetrahedron is
It is the key to the volume theory of polyhedrons, but its volume cannot be solved by finite partition and check method.To solve this problem, he proposed
An important principle has come out: "Evil and blockage, one is the yang horse, and the other is the turtle. The yang is the second, and the turtle is the first, and the rate is not easy." This is the so-called

Liu Hui

principle.It has been more than 100 years since the modern mathematics masters Gauss and Hilbert discussed this issue.

Liu Hui elaborated on the principle of the ancestors in many ways, and the

This proves the error of the sphere volume formula in "Nine Chapters of Arithmetic".He designed the Mouhe square cover, pointing out the volume of the ball and the Mouhe square cover

The ratio is π:4, and the volume of the sphere can be obtained by only asking for the volume of the latter.Although he failed to find out the volume of Mouhe Fanggai

, but sincerely said "to be able to speak", showing Liu Zheng's open mind.This question was later asked by Zu Chongzhi

Father and son solved it thoroughly. Li Chunfeng described Zu's method in detail when annotating "Nine Chapters of Arithmetic".

There are other brilliant achievements in Liu Hui's annotations, such as the inexhaustibility of the formulas, and he proposed to continue to formulate the formulas and find their "micro numbers".

The number approached the irrational root, which was the first of its kind for decimal numbers; he also realized that indefinite equations have infinitely many solutions, and so on, Liu Huizhu formed
He developed a mathematical system, and he said, "Things are deduced from each other, and each has its own purpose, so although the branches are divided, they have the same basic knowledge, and they only come from one end."

He believes that mathematics is "rules" and "measures", that is, empty
The unity of the relationship between form and quantity.Based on these profound understandings, his proofs are clear except for some mistakes.

The evidence is sufficient, well-organized, and rigorous in reasoning; and most of them use deductive reasoning, not circular arguments, but strict mathematical proofs
.With Liu Hui's proof, the formula solution of "Nine Chapters of Arithmetic" can be recognized by future generations.

"Nine Chapters of Arithmetic" and its annotations by Liu Hui, with its outstanding mathematical achievements and unique mathematical system, have a great influence on the development of mathematics in the whole world.

It has had a profound impact and occupies an extremely important position in the history of science.it appears, sign
From the 1st century BC, China replaced ancient Greece as the center of world mathematics, laying the foundation for China to lead the world in mathematics for more than 1500 years.Today, with the emergence and development of computers, the algorithms and programming ideas contained in them are still unknown.

Mathematicians take enlightenment.Mr. Wu Wenjun pointed out that "the ideological influence contained in "Nine Chapters" will become more and more significant, and in the next century
It is not only possible, but also almost a foregone conclusion, to surpass the ideological system of "Original", and I think it is by no means too much
Speculative words. "

(End of this chapter)