The triclinic quadrature formula listed in the fifth volume of his book is the same as that given by Helen, an ancient Greek mathematician in A.D. 1 century. Volumes 7 and 8 also make the observation art in the book of island calculations carry forward and add luster. Qin said in the preface to the book of nine chapters that mathematics is "as big as knowing God and obeying fate; If you are small, you can run around the world, like everything else. " The so-called "understanding God" is to shuttle back and forth between unpredictable things and see the mystery; "Conforming to life" means conforming to the essence of things and their development laws. In Qin's view, mathematics is not only a tool to solve practical problems, but also a lofty realm of "understanding the gods and obeying life".
The book Shu Shu Jiu Zhang consists of nine chapters and nine categories, 18 volumes, and each category has 9 questions * * * 8 1 calculation questions. In addition, there are tributes under each category, which are concise and concise, and are used to describe the main contents of this kind of calculation problems, their relationship with the national economy and people's livelihood and their solutions.
This book is in the form of problem sets, without mathematical classification. Inscriptions not only talk about mathematics, but also involve natural phenomena and social life, which has become an important reference for understanding social politics and economic life at that time. Shu Shu Jiu Zhang has many innovations in mathematical content. China's counting method and its formula are completely preserved here; Natural numbers, fractions, decimals and negative numbers are all discussed in special articles, and decimals are used to represent the approximation of irrational roots for the first time. 1 Volume uses the greatest common divisor and the least common multiple flexibly, and creates the sequence equation to find the least common multiple of several numbers; On the basis of the problem of "things are unknown" in the mathematical classics of Sun Tzu's Art of War, it is summarized as a solution skill of great derivation, which makes the solution of a congruence group standardized and programmed, more than 500 years earlier than the similar method pioneered by Gauss in the west, and is recognized as "China's remainder theorem"; 17 records the calculus of the equation completely, and the book also follows Jia Xianzeng's multiplication method and then makes a positive and negative square root, so that the rational root or irrational root of any degree equation can be solved, which is more than 500 years earlier than the similar method of Horner in England in 19 th century.
In addition, Qin also proposed Qin algorithm. Until today, this algorithm is still a practical algorithm for polynomial evaluation. The algorithm seems simple, and the greatest significance lies in transforming the value of n-degree polynomial into the value of n-degree polynomial. In manual calculation, Qin's algorithm and its coefficient table can greatly simplify the operation.
Shu Shu Jiu Zhang is the inheritance and development of Jiu Zhang Arithmetic. It summarizes the main achievements of traditional mathematics in China in the Song and Yuan Dynasties and marks the peak of ancient mathematics in China. When it was a manuscript, it was successively included in Yongle Dadian and Sikuquanshu. 1842 has been widely circulated among the people since it was first printed. Qin's positive and negative cholesky decomposition and the great derivative method have long influenced the research direction of China's mathematics. The works of mathematicians, Li Rui, Zhang Dunren, Luo, Shi Yuechun and Huang Zongxian were all completed under the direct or indirect influence of the nine chapters. Qin's achievements also represent the mainstream and highest level of medieval mathematics development, and occupy a lofty position in the history of mathematics in the world. The polynomial f (x) = a [n] x n+a [n-1] x (n-1)+l+a [1] x+a [0] is rewritten as follows:
f(x)=a[n]x^n+a[n- 1]x^(n- 1))+l+a[ 1]x+a[0]
=(a[n]x^(n- 1)+a[n- 1]x^(n-2)+l+a[ 1])x+a[0]
=((a[n]x^(n-2)+a[n- 1]x^(n-3)+l+a[2])x+a[ 1])x+a[0]
=L
=(L((a[n]x+a[n- 1])x+a[n-2])x+L+a[ 1])x+a[0]。
To calculate the value of a polynomial, first calculate the value of the polynomial in the innermost bracket, that is
v[ 1]=a[n]x+a[n- 1]
Then the value of the polynomial is calculated layer by layer from the inside out, that is
v[2]=v[ 1]x+a[n-2]
v[3]=v[2]x+a[n-3]
......
v[n]=v[n- 1]x+a[0]
In this way, finding the value of n-degree polynomial f(x) is transformed into finding the value of n-degree polynomial.
(Note: Numbers in brackets indicate subscripts)
The above method is called Qin algorithm. Until today, this algorithm is still a practical algorithm for polynomial evaluation. There is a story in folklore-"Han Xin ordered soldiers".
At the end of Qin Dynasty, Chu and Han contended. Han Xin was once at war with Li Feng, the general of Chu State 1500 soldiers. After a bitter battle, the Chu army was defeated and retreated to the camp. The Han army also killed or injured four or five hundred people, so Han Xin reorganized his military forces and returned to the base camp. When we were on a hillside, a rear unit reported that Chu cavalry were chasing us. I saw the dust flying in the distance and the sound of ShaSheng was deafening. The Han army was very tired, and then the team was in an uproar. Han Xin's military forces reached the top of the slope and saw that the enemy was less than 500. They quickly ordered the troops to meet them. He ordered three soldiers in a row and ended up with two more; Then he ordered the soldiers to line up five times, and as a result, there were three more; He ordered a platoon of seven soldiers, only to get two more. Han Xin immediately announced to the soldiers: Our army 1073 Warriors, with less than 500 enemy troops. If we are condescending, we will surely defeat the enemy in numbers. The Han army believed in its commander-in-chief, and now believed that Han Xin was a "fairy" and a "SJ". So morale is greatly boosted. At that time, flags shook, drums roared, the Han army advanced step by step, and the Chu army was in a mess. Shortly after the battle, the Chu army was defeated and fled.
First, find the least common multiple of 3,5,7, 105 (note: because 3,5,7 are pairwise coprime integers, the least common multiple is the product of these numbers), multiply it by 10, and then add 23 to get 1073 (person).
In Sun Tzu's calculation more than a thousand years ago, there was such an arithmetic problem:
"I don't know the number of things today. 3322, 5522, 7722. What is the geometry of things? " According to today's words: divide a number by 3 and 2, by 5 and 3, by 7 and 2, and find this number.
This kind of problem is also called "Han Xin's point soldier", which forms a kind of problem, namely the solution congruence formula in elementary number theory. The conditional solution of this kind of problem is called "China's Remainder Theorem", which was first put forward by China.
① There is a number, divided by 3 and 2, divided by 4 and 1. What is this number divided by 12?
Answer: The number divided by 3 and 2 is:
2, 5, 8, 1 1, 14, 17, 20, 23….
The remainder after dividing by 12 is:
2,5,8, 1 1,2,5,8, 1 1,….
Divided by 4, the remaining 1 number is:
1, 5, 9, 13, 17, 2 1, 25, 29,….
The remainder after dividing by 12 is:
1, 5, 9, 1, 5, 9,….
The remainder of a number divided by 12 is unique. Only 5 in the upper two lines is the same * * *, so the remainder of this number divided by 12 is 5.
If you change the problem of ①, you will not find the remainder divided by 12, but this number. Obviously, there are many numbers that meet the conditions, which are 5+ 12× integers.
Integer can take 0, 1, 2, …, endless. In fact, after we find out 5, we notice that 12 is the least common multiple of 3 and 4, plus the integer multiple of 12, which are all numbers that meet the conditions. This is the division of "divide by 3 and 2, divide by 4 and 1".
② Divide a number by 3 and 2, by 5 and 3, and by 7 and 2 to find the smallest number that meets the conditions.
Solution: First list the numbers divided by 3 and 2:
2, 5, 8, 1 1, 14, 17, 20, 23, 26,…,
Then list the numbers divided by 5 and 3:
3, 8, 13, 18, 23, 28,….
In these two columns, the first common divisor is 8.3, and the least common multiple of 5 is 15. These two conditions are combined into an integer 8+ 15. List the numbers in this series as 8, 23, 38, …, and then list the numbers divided by 7+2, 2, 9, 16.
The minimum number to meet the requirements of the topic is 23.
In fact, we have combined the three conditions in the topic into one: divide by105,23.
Then Han Xindian's soldiers are between 1000- 1500, which should be105×/kloc-0+23 =1073.
There is a similar question in China's ancient mathematical work Sun Tzu's Art of War: "There are things today, I don't know their numbers, three or three numbers, two, five or five numbers, three or seven numbers, two, ask about the geometry of things? 」
Answer: "Twenty-three"
Technically, it says: "The number of three and three leaves two, take one hundred and forty, the number of five and five leaves three, take sixty-three, the number of seven and seven leaves two, take thirty, get two hundred and thirty-three, and then subtract two hundred and ten. Where the number of three is one, the number of seventy-five is one, the number of twenty-one is one, and the number of seventy-seven is one and fifteen, that's all. 」
The author of Sunzi Suanjing and the exact date of its completion cannot be verified, but according to the verification, its completion date will not be after the Jin Dynasty. According to this research, the solution of this problem was found earlier in China than in the west, so the generalization of this problem and its solution are called China's remainder theorem. China's remainder theorem plays a very important role in modern abstract algebra.