Jia Xian

Jia Xian

Jia Xian, a native of the Northern Song Dynasty, completed the nine chapters of the Yellow Emperor Suan Jing Xi Cao about 1050. The original book was lost, but its main content was copied by Yang Hui (about the middle of the 13th century), so it was handed down to the world. Yang Hui's detailed explanation of the nine chapter algorithm (1261) contains the diagram of "the origin of prescription method", which indicates that "Jia Xian uses this technique". This is the famous "Jia Xian triangle", or "Yang Hui triangle". At the same time, Jia Xian's "method of increasing multiplication and opening" is recorded in the detailed explanation of nine chapter algorithm.

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Jia Xian, a mathematician of Northern Song Dynasty in the first half of the 11th century. Jia Xian was an outstanding mathematician in the first half of the 11th century (Northern Song Dynasty) in China. He once wrote nine chapters of the Yellow Emperor, algorithmic fine grass (Volume 9) and algorithmic ancient collection (Volume 2), both of which have been lost. According to the records of the history of Song Dynasty, Jia Xian studied astronomy and calendar calculation under the guidance of Chu Yan, a mathematician, and wrote nine chapters of the Yellow Emperor, the algorithm of the fine grass, the explanation of the lock calculation book and other books. Jia Xian's works have been lost, but his important contribution to mathematics has been cited by Yang Hui, a mathematician in the Southern Song Dynasty, and has been preserved.

Jia Xian's main contribution is to create the "Jia Xian triangle" and "the method of increasing multiplication". The open method of multiplication is the positive root method of higher power. At present, the principle and program of comprehensive division in middle school mathematics are similar to it. Compared with the traditional method, the method of increasing multiplication and opening is more orderly, simple and more procedural, so it shows its advantages especially in opening higher power. The calculation procedure of the method of multiplication is roughly the same as that of Horner (1819), but 770 years earlier.

Jia Xian first discovered Jia Xian triangle in the history of Chinese mathematics. Yang Hui made a diagram of Jia Xian's method of prescription in the chapter of "detailed explanation of nine chapter algorithm" and "the essence of prescription method", and explained that "Jia Xian used this method to explain and lock the calculation book". Jia Xian's method chart is Jia Xian's triangle. Yang Hui also explained in detail the lock release method, lock release method, multiplication method and multiplication method invented by Jia xianhuan.

Mathematics achievement

Chu Yan, Jia Xian's teacher, was a famous astronomer and mathematician in the early Northern Song Dynasty. At that time, Wang Zhu (997-1057) recorded: "Shi Si Tian Suan, Chu, the leader. He is old and faint, and his sons Jia Xian and Zhu Ji are famous. The constitution is now the Zuoban palace, and it is the supreme history of Jili. The operation of the constitution is also wonderful. Some books are handed down to the world. " According to the records in the history of the Song Dynasty, Yi Wen Zhi, Jia Xian wrote nine volumes of the nine chapters of the Yellow Emperor's Suan Jing Xi Cao. According to the records in the history of the state of Ming Jiao Hong, he wrote two volumes of algorithmic ancient collection and Shi Suo. Unfortunately, both of them have been lost. Yang Hui quoted Jia Xian's "origin of square root method" graph (i.e. binomial expansion coefficient table with positive index, now called "Yang Hui triangle") and "multiplication and opening method" (positive root method for solving higher power) in his book detailed explanation of nine chapter algorithm (1261). The former is 600 years earlier than Pascal Blaise's triangle (1623-1662), and the latter is 770 years earlier than William George Horner's method (1819). In addition, the presentation of "Li Cheng Shi lock open method", the perfection of "Pythagorean thirteen Diagrams" and the establishment of "law of increasing power and seeking honesty" all show that Jia Xian has made important contributions to the abstraction, programming and mechanization of the algorithm.

Mathematical methods

Although the information about Jia Xian has not been preserved completely, we can still find some of his original mathematical ideas and methods from Yang Hui's detailed records, mainly including the following two points.

Abstract analysis

In the process of studying the nine chapters, Jia Xian used the abstract analysis method, especially in solving the Pythagorean problem. He first put forward the "Thirteen diagrams of Pythagorean birth and change". He said: "Pythagorean strings merge into sum, reduce into comparison, and so on into change, multiplication, segment, self multiplication, product, power." Thirteen names refer to gou (a), Gu (b), Xian (c), Gou (B-A), Gou (C-A), Gu (C-B), Gou (a + b), Gou (a + C), Gu (B + C), Xian (c + (B-A)), Xian (c + (a + b) - C), Xian (C - (B-A)). He has completed all the relations of Pythagorean string and its sum and difference, and said that these relations are "useful for taking, useless for not taking, and drawing up a plan for testing", which shows that he has put aside the calculation itself in the ninth chapter and made an abstract analysis of the Pythagorean problem.

For example, the method of "going out of the south gate and north gate to test the city square" is: Shu says: multiply the number of steps out of the north gate by the number of steps out of the west gate, double it, and take the number of steps out of the South Gate as the subordinate method, and divide the square into the city square. Jia Xian's method is: Shu said: more hook by shares, double for real, and two more hook for from, prescription except not. It was only by mastering this method that he was able to use the pure mathematical method to rewrite the nine chapters, leaving a formula example for later generations. He also widely used this method in other chapters such as equations.

Procedure method

Procedural method mainly refers to the thinking procedure, process and steps of exploring problems. It is suitable for solving the same kind of problems under the same theoretical system. Jia Xian's "method of increasing Multiplicity" and "method of increasing multiplicity to seek honesty" embody this method in particular. For example, Shao guangzhang said: "today there is a product of 1.86867 feet, ask: is it cubic geometry?" This is a cubic problem for 1860867. Jia Xian's method is as follows: Cao said: (1) in fact, the number one in Shang is 100. (2) The above business takes the law to set the low price 100, takes the low price as the square 10000, divides the solid, ends. (3) If the Shang Dynasty is more than 100, the lower law will bring in 200, and the lower law will bring in 30000. (4) Then he took the law and entered the incorruption, totaling three hundred. (5) The first, the second, the third and the tenth. (6) After the first quotient, the second quotient was 20. The total number was 320, and the total number was 36400. (7) The second Shang was 20 times, the lower law was 3400 feet, and the lower law was 43200 feet. (8) And he took the law and entered incorruption, three hundred and sixty. (9) The first, the second, and the third, as before. (10) The third place in Shang Dynasty was three feet, and the number of the three feet in Shang Dynasty was 363. The number of the three feet in Shang Dynasty was 44289.

It can be expressed in modern way as follows:

To be honest in law and honest in practice

（1）1000000+00-18608671

+1000000+10000001000000

（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 notice that this open cube process has formed a fixed program. Contemporary scholars have found that procedural mathematical thinking is an important feature of ancient Chinese mathematics, and Jia Xian's work makes the prescription procedure systematic and standardized. Jia Xian's mathematical methodology had a profound impact on the mathematicians of song and Yuan Dynasties. Throughout the "four masters of song and Yuan Dynasties", we can not but draw the essence from it.

Educational thought

We don't know whether Jia Xian had ever been engaged in mathematics teaching. However, in terms of the active private learning and the status of mathematics in the early Song Dynasty, we can't rule out the possibility that he imparted mathematics knowledge. "The operation of the constitution is also wonderful, and there are books handed down to the world" should be evidence. We know that one of the purposes of ancient scholars is to educate the world, so we have reason to discuss Jia Xian's mathematics education thought. Through careful study of the fine grass, we can find the flash of his mathematics education thought.

Emphasis on abstraction

In the two examples of "increasing multiplication and opening method", we can clearly see that the algorithm is obtained after eliminating the number. And this kind of fine grass way runs through his works (as far as existing) all the time. The reason why Jia Xian did so should be deeply influenced by the ancient Chinese educational thought that "it is better to teach people to fish than to teach them to fish". As far as we know, Huangdi Jiuzhang Suanjing Xicao was written around 1050. After its publication, it spread widely in society and gradually replaced Jiuzhang suanzi to a certain extent. Bao Huanzhi of the Southern Song Dynasty said in 1200: "since Yiguan's southward journey, this study has been abandoned, and there are few good ones, and Jiuzhang Suanjing has hardly been handed down. It's the root of modern folk culture, and its title is "nine chapters of the Yellow Emperor"... " This is also the social recognition of his mathematics education thought at that time.

Focus on generalization

After giving "the method of establishing, explaining and opening the lock", Jia Xian put forward "the method of increasing power and seeking honesty" and gave the sixth order Jia Xian triangle to explain the relationship between the powers. When we discuss the Pythagorean problem, we first discuss the "Thirteen diagrams of Pythagorean birth and change", and then discuss the nature of the problem