Jia Xian's main contribution is to create the "Jia Xian Triangle" and the "multiply-multiply-divide method". The method of increasing, multiplying and opening is a positive root method for finding higher power. At present, the principle and procedure of comprehensive division in middle school mathematics are similar. Compared with the traditional method, multiplication-multiplication-division is concise and easy to program, especially in high power, which shows its superiority. The calculation program of multiplication and division is basically the same as that of European mathematician Horner (AD 18 19), but it is 770 years earlier than him.
Jia Xian first discovered the Jia Xian Triangle in the history of Chinese mathematics. Yang Hui made a chart of Jia Xian's prescription in the chapters "Detailed Explanation of Nine Chapters Algorithm" and "Original Prescription", and explained that "Jia Xian used this technique when interpreting books". The square painted by Jia Xian is Jia Xian Triangle. Yang Hui also explained in detail the unlocking and multiplication methods invented by Jia Xian. Although the information about Jia Xian has not been completely preserved, we can still find some unique mathematical ideas and methods from the Fine Grass recorded by Yang Hui, mainly including the following two points.
Abstract analysis method
In the process of studying the Nine Chapters, Jia Xian used abstract analysis, especially when solving the Pythagorean problem. He first proposed "Thirteen Pictures of the Birth and Change of Pythagoras". He said: "Pythagoras' strings are combined to be sum, simplification to be comparison to be change, multiplication to be segment, multiplication to be product to be power." Thirteen people refer to Gou (a), Gu (b), Xian (c), Gou Gubi (b-a), (c-a), Gu Xianbi (c-b), Gou (a+b), Gou Xianhe (a+c) and Gu Xianhe (b+c).
For example, when asked about "going out of the north and south gates to measure the city", the method in the Nine Chapters is: in the operation, it is true that the number of steps out of the north gate is multiplied by the number of steps out of the west, and the number of steps out of the south gate is the method, and the prescription is the city. Jia Xian's method is: Technically speaking, I am linked to the stock, the times are true, the second link, and the prescription is excluded. It was only by mastering this method that he was able to rewrite the techniques of "Nine Chapters" by pure mathematics, leaving a formulaic example of solving problems for future generations. He also widely used this method in Fine Grass in other chapters such as Equation Skills.
programming approach
Procedural method mainly refers to the thinking procedure, process and steps of exploring problems. It is suitable for solving similar problems under the same theoretical system. Jia Xian's "Multiplication by Multiplication" and "Law of Cheap by Multiplication" particularly embody this method. For example, Shao said, "Today, there are 1, 868,667 feet of products. Q: Is it solid geometry? " This is a cubic problem of 1860867. Jia Xian's method is as follows: Cao Yue: (1) Actually, the first place in business is 100. (2) The above-mentioned businessmen adopt the next method, and set the low price of 100 yuan, and take the low price as the square of 10000 yuan. (3) After more than 100 merchants, the next method will be used to enter Lian * * * 200 and the next method will be used to enter * * * 30000. (4) From the law to the cheap * * * three hundred. (5) The first, second and third are ten. (6) After the number of the first quotient, the number of the second quotient will be 20, so as to take the next method to enter Lian * * * 320, take Lian * * * 36400, cancel the business, and the balance is 132867. (7) After that, the sub-suppliers will enter the joint * * * 20 times, and the joint * * * 43,200 feet. (8) From the law to the cheap * * * three hundred and sixty. 9. The withdrawal of the first party, the second party and the third party is the same as before. (10) The third place in Shang Dynasty is three feet, and if you take the next method to enter Lian * * 363, and if you take Lian * * 44289, you will get three feet in addition to reality, and you will get the number of cubes.
The modern expression is as follows:
Cheap methods and practical business.
( 1) 1000000+00- 1860867 1
+ 1000000+ 1000000 1000000
(2) 1000000+ 1000000+ 1000000-860867
+ 1000000+2000000
(3) 1000000+2000000+3000000
+ 1000000
(4) 1000000+3000000+3000000-860867
1000+30000+300000-8608672
+2000+64000728000
(6) 1000+32000+364000- 132867
+2000+68000
(7) 1000+34000+432000
+2000
(8) 1000+36000+432000- 132867
1+360+43200- 1328673
+3+ 1089+ 132867
( 10) 1+363+442890
We noticed that this card issuing process has formed a fixed procedure. Contemporary scholars have found that the programmed mathematical thinking method is an important feature of China's ancient mathematics, and Jia Xian's work makes the prescription procedure systematic and standardized. Jia Xian's mathematical methodology had a far-reaching influence on mathematicians in Song and Yuan Dynasties. Throughout the "Song and Yuan Dynasties", we all learned the essence from it. We don't know whether Jia Xian is engaged in mathematics teaching, but as far as the activity and status of private schools are concerned, we can't rule out the possibility that he teaches mathematics knowledge. "The constitution is also wonderful, and the book is handed down from generation to generation" should be evidence. We know that one of the purposes of ancient scholars' writing books is to educate the world, so we have reason to discuss Jia Xian's thoughts on mathematics education. Studying Weeds carefully, we can find the bright spot of his mathematics education thought.
Attach importance to abstraction
In the two cases of "multiplication and division", it can be clearly seen that the algorithm is obtained by eliminating numbers. And this fine grass method runs through his works (in itself). The reason why Jia Xian did this should be deeply influenced by China's ancient educational thought that "it is better to teach people to fish than to teach them to fish". As far as we know, Nine Chapters of the Yellow Emperor was written around 1050. After the book was published, it was widely circulated in the society and gradually replaced Nine Chapters of Arithmetic to some extent. Bao Huanzhi in the Southern Song Dynasty said in 1200: "Since Guan Yi crossed the south, this subject (arithmetic) has been abolished, and fewer people are not good at it, and Nine Chapters of Arithmetic has almost disappeared. The modern folk foundation is called "Nine Chapters of the Yellow Emperor" ... "This is also the recognition of his mathematics education thought by the society at that time.
Focus on generalization
After Jia Xian gave the method of "setting locks to unlock", he also put forward the method of "increasing power to get cheaper" and gave the sixth-order Jia Xian Triangle to illustrate the relationship between powers. When discussing the Pythagorean problem, first discuss the Pythagorean birth and death diagram, and then talk about the solution of the problem, which gives people a clear sense of system. All his attempts reflect the importance attached to the outline of knowledge. Mr. Guo Shuchun believes that this is Jia Xian's "Introduction to Sequences" and "a development of deductive logic". It is also a good teaching method to pay attention to the generalization of knowledge outline in mathematics education.
Systematic thinking
According to the available data, Jia Xian has not set foot in the field of Liu Hui's fractional and differential calculus (that is, limit theory), and his other disciples Zhu Ji once criticized him for "abandoning the remainder and failing to complete his reasoning". In addition, the root problem discussed in Nine Chapters of the Yellow Emperor does not involve infinite situations, and he even changed the fractional solution in Nine Chapters into an integer solution. These indications show that his work is based on integer sets. On this basis, he summed up the Pythagorean and square root problems, and gave general solutions to many other problems, from which the traces of systematic methods can be faintly seen. With Jia Xian's level of mathematical knowledge, it is impossible for him not to be familiar with fractions, nor to understand Liu Hui's idea of seeking differentiation, but he did not deeply study the endless problems of prescribing. Therefore, he avoided fractions in his works, aiming at systematically spreading his mathematics knowledge to the world, which is valuable in the history of ancient mathematics education.
Distracted thinking
When discussing the Nine Chapters and other issues, Jia Xian did not stick to the previous ideas and algorithms, but found many new calculation methods. In the chapter of equal loss, he put forward the methods of "dividing points", "subtracting points" and using "equation technique" to find the difference rate; In Ying Xu Pian, he put forward the methods of "having skills now", "combining skills", "dividing skills", "equating skills" and "two virtual skills". On the issue of "harmony between the house and the valley", he put forward the "method of distinguishing between the valley and the valley" and so on It can be seen that Jia Xian not only pays attention to summarizing theoretical research methods, but also devotes himself to the training of divergent thinking, which is of great benefit to the innovation of knowledge. Jia Xian has a set of unique views on systematization, outline, generalization (abstraction) and diversification of mathematics education thinking, which can be used for reference in many aspects. Nine Chapters of Arithmetic is the most famous mathematical work in China before 1 1 century. In the process of its spread, many people made grass for it. There are three mathematicians who have made outstanding contributions to mathematical theory-Liu Hui (on the basis), Jia Xian (on the improvement of theoretical level) and Yang Hui (on the basic perfection of theory). Jia Xian plays the role of connecting the past with the future. On the other hand, the upsurge of mathematical research in Wei, Jin, Southern and Northern Dynasties was interrupted since the Tang Dynasty. Jia Xian's mathematical methodology stimulated the upsurge of mathematical research in Song and Yuan Dynasties, and he played a role in fueling the situation. It is embodied in the following two aspects.
Mathematical thought
Jia Xian made an abstract analysis of the problems raised in Nine Chapters of Arithmetic to reveal the essence of mathematics. The principle of this method is explained by programming. Outline and sort out the context of knowledge; Pay attention to the systematization of knowledge and avoid paradox. These thinking methods had a far-reaching influence on mathematicians in Song and Yuan Dynasties. Yang Hui's Detailed Explanation of Nine-chapter Algorithm draws lessons from Jia Xian's abstract and exploration achievements, and reclassifies the problems in the nine chapters. Ye Li's book Measuring the Round Sea Mirror inherits and develops these mathematical methods and establishes a logical deduction system. Zhu Shijie's Meet with Siyuan also used these thinking methods, which made me reach the peak in the history of ancient mathematics. Qin wrote "Introduction to Mathematical Methods" (nine chapters of mathematics), and imitated Jia Xian, showing the vitality of his methodology. Of course, these mathematical thinking methods are not original by Jia Xian, but also the result of mathematical writings, research and accumulation in past dynasties, which Jia Xian refined and passed on. First of all, Jia Xian's "method of increasing, multiplying and opening" initiated the research topic of higher power, and was improved by Qin's "positive and negative leveling method", which solved the problem of finding the equation of higher power. In addition,/kloc-finally established a complete set of equations theory at the beginning of the 4th century, from Ye Li's celestial technique (one-dimensional linear or higher-order equations) to Zhu Shijie's quaternary technique (four-dimensional linear or higher-order equations), which became the most successful discipline in the Song and Yuan Dynasties. Secondly, Jia Xian Triangle initiated the research direction of higher-order arithmetic progression summation. Zhu Shijie found the summation formula of higher-order arithmetic progression such as triangle and scattered star from each diagonal of triangle. Thirdly, the "multiply-multiply-open method" actually simplifies the calculation procedure, makes the programming more reasonable, and inspires the future calculation, rapid calculation and even the improvement of calculation tools. Fourthly, the writing form of Fine Grass created a mathematical research method, which was widely used for reference by later mathematicians. When preserving and sorting out mathematical works, Ganjia School once said to a batch of calculations or notes or sketches.