A Research On The Expression Methods Of Infinite Series Of Ming An-tu,Dong You-cheng And Xiang Ming-da

After the introduction of the du's three formulas in the Qing dynasty,there existed an infinite series of expression problems,without algebraic symbols,how to express the infinite series? This is an important problem in the Qing dynasty.Ming An-tu first studied the three formulas,and give the other six and its proof,and his original knowledge can not satisfactorily explain and express infinite series,the urgent need for some new knowledge to provide new methods,so that the existing knowledge constitutes the main driving force for the search for information,it make the research of infinite series on a higher level.The different degrees of Dong You-cheng and Xiang Ming-da are inspired by the ideas and methods of Ming An-tu,which constitute the mainstream of the infinite series research in the Qing dynasty,and many experts have called it the“Ming An-tu school”.The research of this paper concludes as follows:Ming An-tu based on the traditional cyclotomic,expanded the cyclotomic surgery geometric method,absorbed the recursive addition method in Mei Wen-ding's Ji He Tong Jie(??????),constructs the proportional relation,borrowed from the method of borrowing root in the Shu Li Jing Yun(??????).Ming An-tu started the first research ofinfinite series in the history of the Chinese mathematics,and the first created a set of unique infinite series representation in his Ge Yuan Mi LüJie Fa(????????Quick methods for trigonometry and for determining the precise ratio of the circle).Dong You-cheng absorbed the proportional four-rate method in the Shu Li Jing Yun,and proposed the addition and subtraction of different orders of trigonometric stacking,established the corresponding expression.He had not seen Ming An-tu's representations and proofs,but had been affected by the spread of nine formulas,and the proofs of nine formulas are completed independently,and the nine formulas are simplified as the original four formulas of the legislation,research on infinite series and its representation by using the technique of stacking.The calculation of the coefficients in the expansion form is based on the triangular stack,which establishes the relation between the cyclotomic and the stacking operation.Xiang Ming-da inherited the method of Dong You-cheng's stacking operation,popularized the recursion of Dong You-cheng,the original four formulas of legislation were simplified into two formulas,but his infinite series notation did not use the method of Dong You-cheng,but instead transplanted the representation and operation methods of Mei's Shao Guang Shi Yi(??????)into the expression of infinite series.The infinite series representation of the Chinese mathematicians,Ming An-tu,Dong You-cheng and Xiang Ming-da,each is not unified,each has its own characteristics,language narration,schema expression,each schema has a specific expression method,and the underneath of schema is attached with operation methods and related annotations.In the history of Chinese mathematics,their infinite series notation shows great superiority,it can be visualized to show the operation object,operation rule,operation order and the principle of place value.It can improve the interoperability between the constructed systems,and also can reveal the intrinsic relationship between the infinite series expressions,which has positive effect on the popularization of the spread of mathematics.This article is divided into five parts to discuss.The first part discusses the method foundations of expressing infinite series in Ming An-tu's Ge Yuan Mi Lü Jie Fa: The expansion of the cyclotomic geometry method,the construction of proportional relation and the reference of the method of borrowing root.The second part analyzes the expression method of infinite series in Ming's Ge Yuan Mi LüJie Fa,the paper holds that Ming An-tu draw lessons from the Tong Wen Suan Zhi(??????)is refers to the representation method of three ratio method in,the expression method and the operation method are transplanted into the addition,subtraction,multiplication,multiplication and squared of infinite series by means of individual and polynomial representations.From his representation,the appearance of the Catalan numbers is inevitable,which was the result ofoperation,and the inverse problem of infinite series is to solve the inverse function.The expression of the Leibniz series absorbs the western method.The expression and processing of the odd mantissa is a new method adopted in new problems.The third part expounds the expression method of infinite series in Dong's Ge Yuan Lian Bi Li Shu Tu Jie(??????????),Dong You-cheng used the proportional four-rate method in Shu Li Jing Yun and apply the stacking technique to the study of infinite series,but the expression method of infinite series is different from that of Ming An-tu.The forth part discusses the expression method of infinite series in Xiang's Xiang Shu Yi Yuan(??????),and think that Xiang Ming-da reaches the way of the stacking method of Dong You-cheng,but the expression of infinite series is an alternative.He uses the Di Jia Tu,combined with Mei's method in Shao Guang Shi Yi to express the infinite series,which is different from the predecessors.The fifth part,the conclusion of this article,gives a detailed summary of the similarities and differences of their infinite series representations.This paper starts from the point of view of the present international mathematical practice,that is,how to express,what is expressed,and why did ancient Chinese mathematicians expressed the problem of the expansion of infinite series under situation then.This paper starts fromthe case study,and finally tries to grasp the whole thread from the macroscopic.