The narrative mode of Nine Chapters Arithmetic is mainly inductive. Firstly, some examples are given, and then the general methods to solve this kind of problems are listed. This is obviously different from the deductive narrative mode of Euclid's Elements of Geometry (about 330 ~ 275 BC), the representative work of ancient Greek mathematics. It has always had a great influence on the development of mathematics in China. It has been stipulated as a teaching material for mathematics education in past dynasties, and it is one of the so-called "ten classic books on calculation".
There are 246 math problems in Nine Chapters Arithmetic, which are divided into nine categories, namely "Nine Chapters". The first chapter "Square field" mainly talks about the algorithms of various fields. The second chapter, "Millet", mainly talks about various algorithms of proportional exchange of grains; The third chapter, "Decreasing scores", mainly talks about the algorithm of distribution according to grade or proportion; The fourth chapter "less width" mainly focuses on the algorithm of one side with known area and volume; The fifth chapter "commercial workers" mainly talks about the algorithm of earthwork and labor consumption in various projects: the sixth chapter "equal loss" mainly talks about the algorithm of sharing taxes and distributing labor (corvee) according to population, distance and other conditions; The seventh chapter, "Remainder and Shortage", mainly talks about the algorithm of two hypotheses to solve some difficult problems; The eighth chapter "Equation" mainly talks about the solution of linear equations and the law of addition and subtraction of positive and negative numbers; Chapter 9 "Pythagorean Theorem" mainly talks about the application of Pythagorean Theorem, right-angle similar triangles, and the solution of a quadratic equation.
The contents of the problem of "five * * * wells" are as follows: five companies share a well, and if two ropes are used in one company, one company 1 rope will be extended; Put it down from the wellhead and just reach the water surface; In addition, use 3 ropes of Party B and/kloc-0 ropes of Party C; Or use article 4 of family C and article L of family D; Or we used the family's 5 pieces, the family's 65,438+0 pieces, or the family's 6 pieces and the family's 65,438+0 pieces, all just arrived. What is the well depth and rope length of each family?
Because the original problem contains more than two unknowns, and the scope of the answer and other specific conditions are not given, there are infinite groups of solutions after the equation is discharged. Such an equation is called "indefinite equation". If the length of the problem is in inches, then its minimum positive integer solution is as follows:
Well depth is 72 1 inch, Jia Jia rope length is 265 inches, armour 19 1 inch, armour 148 inch, armour 129 inch and armour 76 inch.
The earliest person who studied indefinite equations in the west was Diophantine in Alexandria, ancient Greece, about the 4th century AD. He is more than 300 years later than Nine Chapters of Arithmetic. /kloc-in the 3rd century, Qin, a mathematician in the Song Dynasty in China, put forward the idea of "pan-analysis and seeking technique" in Nine Chapters (1247). In fact, this is the general method to solve an indefinite equation. In Europe, it was not until18th century that the Swiss mathematician Euler created a general solution to the indefinite equation.
Qin's theory of "Great Circumcision" was not only invented much earlier than in Europe, but also had a lofty position in history. Moreover, its method is more concise and concrete than the European method, and it is easy to carry out numerical calculation. Today, compared with the "one-time congruence" method in modern number theory, it still has its advantages. Therefore, this algorithm has been praised by European and American scholars and is called "Chinese remainder theorem".