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Jia Xian,Northern Song DynastyAbout 1050, he completed the Yellow Emperor's Nine Chapters of Suanjing Fine Grass. The original book was lost, but its main content wasYang Hui(about the middle of the 13th century), so it was handed down from generation to generation.Yang Hui《Explain the nine chapter algorithm in detail》(1261) contains a diagram of "the origin of the method of making prescriptions", indicating "Jia Xian uses this method".This is the famous“Jia Xian Triangle”, or“Yang Hui Triangle”。Detailed Interpretation of the Nine Chapter Algorithm also recorded Jia Xian's high power derivation“Multiplication opening method”。
Main works
Yellow Emperor's Nine Chapters of Algorithmic Grass (Volume 9) and Ancient Collection of Algorithmic (Volume 2)
Jia Xian, Northern Song Dynasty of China in the first half of the 11th centurymathematician。Jia Xian was an outstanding mathematician in the first half of the eleventh century (Northern Song Dynasty) in China《Nine Chapters of the Yellow EmperorAlgorithmic Grass (Volume 9) and《Ancient collection of algorithms》(Volume II), all have been lost.According to《History of the Song Dynasty》According to records, Jia Xian studied under mathematiciansChu YanStudied astronomy, calendar calculation, and wrote《Yellow Emperor Nine Chapters Algorithm Grass》《Lock release calculation book》Wait for books.Jia Xian's works have been lost, but his important contribution to mathematics wasYang HuiReferences are saved.
Jia Xian's main contribution is to create“Jia Xian Triangle”And“Multiplication opening method”。The method of multiplication and opening is to findHigher powerThe positive root method of.At present, in middle school mathematicsComprehensive divisionIts principle and procedure are similar to it.Compared with the traditional method, the method of increasing and multiplying is neat, simple and more procedural, so it shows its advantages especially when it is used to open the high power.The calculation procedure of the method of multiplication and expansion is roughly the same as that of European mathematician Horner (1819 AD), but 770 years earlier than him.
Jia Xian was the first to discoverJia Xian Triangle。Written by Yang Hui《Explain the nine chapter algorithm in detail》In the chapter of "The Foundation of Prescription Methods", Jia Xian made a diagram of prescription methods and explained that "Jia Xian used this method to release the lock calculation book".Jia Xian's square drawing is Jia Xian's triangle.Yang Hui also explained in detail the lock release and leveling method, lock release and leveling method, multiplication and leveling method and multiplication and leveling method invented by Jia Xian.
Achievements in mathematics
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Jia Xian's teacherChu YanHe was a famous astronomer and mathematician in the early Northern Song Dynasty, "especially in the Jiuzhang, Jigu, Zhuozhu, and Haidao.".People at that timeWang Zhu(997-1057) It is recorded that "the world minister of heaven, Chu, is the leaderSenility, Yes, Zi Jia Xian and Zhu Ji are famous.Xianjin is the left classDianzhi, Jili Taishi.The constitutional calculation is also wonderful. Some books have been handed down to the world. "According to《History of the Song Dynasty · Art and Literature Annals》Record Jia Xian's worksYellow Emperor's Nine Chapters of Suanjing GrassNine volumes,[1]According to Ming DynastyJiao HongIt is recorded in the National History · Art and Culture Records, and has written two volumes of the Ancient Collection of Algorithm Boards[1]And "Release Lock", unfortunately, they have been lost.Yang HuiJia Xian's "Origin of Prescription" figure (i.e., the index is a positive integerbinomialExpansion coefficient table, now known as "Yang Hui Triangle") and "Multiplication and Expansion Method" (positive root method for higher power).Former ratioPascal(Pascal Blaise, 1623-1662) The triangle was 600 years earlier, and the latter was 770 years earlier than William George Horner's method (1819).In addition, the presentation of the "method of establishing, releasing, locking and opening", the perfection of the "thirteen diagrams of Pythagorean transformation", and the "increase ofPowerThe establishment of the "law of seeking honesty" shows that Jia Xian has made important contributions to the abstraction, programming and mechanization of algorithms.
Mathematical method
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Although the data about Jia Xian are not completeYang HuiWe can still find some unique things in the fine grass capturedMathematical thoughtAnd methods mainly include the following two points.
Abstract analysis
In the process of studying Nine Chapters, Jia Xian used abstract analysis, especially in solving the problem of Pythagorean. He first proposed“Thirteen Diagrams of Pythagorean Evolution”。He said, "Pythagorean chords merge into sum, subtract into comparison, change into equality, become multiplication, become segments, self multiplication is product, and become power." Thirteen names refer to tick (a), share (b), chord (c), Pythagorean comparison (b-a), tick (c-a), tick (c-b), tick (a+b), tick and (a+c), tick and (b+c), chord and (c+(b-a)), chord and (c+(a+b))Chord sum comparison ((a+b) - c), chord comparison (c - (b-a)).He has perfected all the relations between Pythagorean chord and its sum and difference, saying that these relations "take what is useful, not what is useless, and test it by drawing a picture", which shows that he has put aside the problem itself in the Nine Chapters and made an abstract analysis of Pythagorean problem.
For example, "go out of the north and south gate to test the city", the method of the Nine Chapters is as follows: multiply the number of steps out of the north gate by the number of westward steps, and the number of steps out of the south gate is the subordinate method. The square root is divided into the city side.Jia Xian's method is as follows: the saying goes: more than one hook is multiplied by shares, double it is true, and two more hooks are subordinate, and the formula is not excluded.It was only by mastering this method that he was able to use pure mathematical methods to rewrite the text of Nine Chapters, leaving behind formulaic examples of problem solving for future generations.He also widely used this method in other chapters such as Equation Theory.
Procedural methods
Jia Xian
Procedural method mainly refers to the thinking procedure, process and steps of exploring problems.It is applicable to solving the same kind of problems under the same theoretical system.Jia Xian's“Multiplication opening method”And“The Method of Increasing the Power and Seeking Honesty”This method is particularly concentrated, such asShaoguangThe seal says: "Today's product is 1.86867 million feet. Ask: Is it cubic geometry?" This is a pair of 1860867CubicProblems.Jia Xian's method was as follows: (1) In fact, the number one place in the business was 100.(2) If the above business is taken by the lower method, the price will be 100 yuan, and if the price is taken by the lower method, the price will be ten thousand yuan.(3) A total of 200 people will be paid by the lower method and 30000 people will be paid by the lower method.(4) And then he took the lower Dharma and joined the Lian Dynasty, totaling 300.(5) The first, the second and the third will be ten.(6) Then, after the first quotient, the second to resume the business got 20, and the total amount was 320 by multiplying the lower method, and the total amount was 36400 by multiplying the lower method. The business was ordered to eliminate the truth, and the balance was 1328867.(7) The total amount of incorruptible goods is 340, and the total amount of incorruptible goods is 43200 feet.(8) And then the following law was added to the incorruptible, totaling 360.(9) The first, the second and the third are the first.(10) The third place in Shangshang got three feet, three hundred and sixty-three times the lower law, and forty-four thousand two hundred and eighty-nine times the lower law. He ordered Shangshang three feet to remove the solid, and when it was enough, he got one side of the cube.
It can be expressed in a modern way as follows:
Lower the law, be honest, square and honest
（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 noticed thisOpen cubeThe process has formed a fixed procedure.Contemporary scholars have found that the programmed mathematical thinking method isAncient Chinese MathematicsJia Xian's work made the prescription procedure systematic and standardized.Jia Xian'sMathematical MethodologyyesSong Yuan mathematicianHas had a profound impact“Four Masters of Song and Yuan Dynasties”We must draw the essence from it.
Educational ideology
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We don't know whether Jia Xian was engaged in mathematics teaching, but as far as the private school was active and mathematics status was concerned in the early Song Dynasty, we can't rule out the possibility that he could teach mathematics knowledge. "Constitution calculation 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' writing books is to educate the world, so we have reason to explore Jia Xian's mathematical education thought.A careful study of the fine grass shows thatMathematical educational thoughtWhere the light shines.
Attach importance to abstraction
Jia Xian
“Multiplication opening method”In the two examples of grass, it can be clearly seen that the algorithm is obtained after removing the number.Moreover, this way of fine grass runs through his writings (as far as the existing ones are concerned).Jia Xian's reason for doing so should be deeply influenced by the educational thought of "giving people fish is better than giving them fish" that has long existed in ancient China.As far as we know,《Yellow Emperor's Nine Chapter Suanjing GrassThe book was completed around 1050. After its publication, it spread widely in the society and gradually replaced《Chapter Nine Arithmetic》。In the Southern Song Dynasty, Bao Huanzhi said in 1200, "Since the time when I went south in dress, this school of mathematics has been abandonedJiuzhang SuanjingAlmost disappearedNo transmissionYes.It is the foundation of modern folk, and its title is "Nine Chapters of the Yellow Emperor"... "This is also the recognition of his mathematical education thought by the society at that time.
Focus on generalization
Jia Xian is giving“Vertical unlocking method”Later, he proposed“The Method of Increasing the Power and Seeking Honesty”And giveSixth order Jia Xian TriangleExplain the relationship between the parties.When discussing the Pythagorean problem, we first discuss the "thirteen diagrams of Pythagorean growth and change", and then discuss the solution of the problem, giving a clear sense of system.All his attempts reflect the importance he attaches to the outline of knowledge.Guo ShuchunMr. Jia believes that this is Jia Xian's "list of general theories" and "a development of deductive logic".It is also a good teaching method to focus on the generalization of knowledge outline in mathematics education.
Systematic thinking
Jia Xian
According to the existing data, Jia Xian did not get involvedLiu HuiScore andDifferential(i.eLimit theory）Field, his martial brotherZhu JiHe has also been criticized for "abandoning the remaining points and never giving up the reason".In addition, the formula problem discussed by him in the Yellow Emperor's Nine Chapters of Suanjing Fine Grass does not involve the inexhaustible situation, and he even changed the problem with fractional solution in the Nine Chapters to get integer solution.These signs indicate that his work is based onInteger setAbove.On this basis, he briefly summarized the Pythagorean and square root problems, and gave general solutions to many other problems, from which we can vaguely see the traces of systematic methods.With Jia Xian's level of mathematical knowledge, it is impossible for him not to be familiar with fractions, nor not to understandLiu HuiHis idea of seeking differential numbers was just that he did not study thoroughly the problem of endless sources.Therefore, he evaded fractions in his writings in order to systematically transmit his mathematical knowledge to the world, which is invaluable in the history of ancient mathematical education.
Jia Xian did not stick to the ideas and algorithms of predecessors when he discussed various problems in Chapter Nine, and found many new computing methods.stayUniform transmissionIn the chapter, he proposed the "class score method", "score reduction method", and the "equation method" to calculate the difference rate;In the chapter of surplus and deficit, he proposed“There are skills today”、“Confluent operation”、“Fractionation”、“Equation technique”、“Double deficiency technique”And other methods;On“Pythagorean Party”When asked, he proposed“Pythagorean side method”Wait.It can be seen that Jia Xian not only focused on generalizing theoretical research methods, but also committed himself toDivergent thinkingIt is beneficial to the innovation of knowledge.Jia Xian has a set of original views on the systematization, programmatic, generalization (abstraction) and diversification of thinking of mathematics education, and many aspects can be used for reference.
Status impact
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《Chapter Nine Arithmetic》It is the most famous mathematical work in China before the 11th century. During its spread, many people made grass for it.Three mathematicians have made outstanding contributions to mathematical theory----Liu Hui(The establishment of theoretical basis), Jia Xian (the improvement of theoretical level) and Yang Hui (the basic improvement of theory), Jia Xian plays a role of connecting the past and the future.On the other hand, Jia Xian'sMathematical MethodologyIt also stimulated the upsurge of mathematical research in the Song and Yuan Dynasties, and he played a role in fuelling the flames.This is reflected in the following two aspects.
Mathematical thought
Nine Chapters of Arithmetic
Jia Xian abstractly analyzed the problems raised in Nine Chapters of Arithmetic to reveal the essence of mathematics;Explain the principle of the method with the help of programming;Summarize and sort out the context of knowledge;Pay attention to systematization of knowledge and avoid paradox.These methods of thinking had a profound impact on mathematicians in the Song and Yuan Dynasties.Yang HuiBy《Explain the nine chapter algorithm in detail》Drawing on Jia Xian's abstraction and exploration achievements, the topics of the Nine Chapters were re classified.Li YeBy《Circular borescope》It inherits and develops these mathematical methods, and establishes a rigorous logical deduction system.Zhu ShijieBy《Siyuan Jade Mirror》They also used these methods of thinking to achieve the peak in the history of mathematics in ancient China.Qin JiushaoIn his book A Brief Introduction to Mathematical Skills (Mathematical Nine Chapters), he learned from Jia Xian by writing skills without mentioning specific figures, which shows the vitality of his methodology.Of course, these mathematical thinking methods are not unique to Jia Xian, but also the results of mathematical writings, research and accumulation of all ages, which Jia Xian refined and inherited.
Achievements in mathematics
First of all, Jia Xian's "method of increasing, multiplying and driving" created a research topic of high powerQin Jiushao“Positive and negative square cutting”Perfect it to makeHigher order equationThe problem of finding the positive root is solved.Plus from Li YeTianyuan Skill(unary linear or higher order equation) to Zhu Shijie'sQuaternionThe establishment of (quaternion linear or higher order equations) finally established a complete set of equation theory at the beginning of the 14th century, making it the most successful subject of the Song and Yuan mathematics.secondly,Jia Xian TriangleAnd created a higher orderIsochromatic seriesZhu Shijie discovered the research direction of the sum problem from each diagonal line of the "triangle"“Triangle I”、“Scatter Stars I”EtcSummation formula of higher order equal difference series。Third, the "method of increasing, multiplying and opening" actually simplifies the calculation procedure and makes the procedure more reasonable, which is instructive to the improvement of later generations' calculation, quick calculation and even calculation tools.Fourth, the writing form of "Fine Grass" has created a mathematical research method, which has been widely used for reference by later mathematicians.QianjiaWhen the school saved and sorted out mathematical works, it once explained a batch of calculation books, annotations or fine sketches.
