中國剩余定理的中外歷史發(fā)展比較
發(fā)布時間：2019-06-01 12:34
【摘要】：《孫子算經(jīng)》中的“物不知數(shù)”問題在中國傳統(tǒng)數(shù)學(xué)史上占有極為重要的地位。至南宋,秦九韶對物不知數(shù)問題做精細研究,最終創(chuàng)造了此題的解法,稱為大衍總數(shù)術(shù)(簡稱大衍術(shù)),著錄于《數(shù)書九章》中�，F(xiàn)今稱此術(shù)為“中國剩余定理”。中國剩余定理是舉世聞名的定理,是中外任何一本基礎(chǔ)數(shù)論教科書中不可或缺的,并被廣泛應(yīng)用于密碼學(xué)、快速傅里葉變換理論等諸多領(lǐng)域中,但其歷史發(fā)展的研究卻較為稀少。本論文在前人研究的基礎(chǔ)上,以中國剩余定理發(fā)展的歷史為研究對象,將相關(guān)文獻進行系統(tǒng)地梳理,尤其對南宋秦九韶《數(shù)書九章》,清代張敦仁《求一算術(shù)》、黃宗憲《求一術(shù)通解》,以及印度婆什迦羅二世的《麗羅娃底》,日本關(guān)孝和《括要算法》,德國高斯《算術(shù)探索》等數(shù)學(xué)原典;以及日本三上義夫《中國和日本的數(shù)學(xué)發(fā)展》和《中國算學(xué)之特色》、法國巴歇《數(shù)學(xué)趣味》中所記載的相關(guān)資料進行深入研究。主要完成了以下工作:首先,從大衍術(shù)產(chǎn)生的背景出發(fā),以歷代數(shù)學(xué)家對其的貢獻為主線,梳理出國內(nèi)中國剩余定理的歷史發(fā)展,并結(jié)合錢寶琮的相關(guān)文獻,作出了中國剩余定理在中國的歷史發(fā)展演進路線簡圖;分印度、日本、歐洲三大板塊,依次梳理出國外中國剩余定理的歷史發(fā)展。其次,從研究的時間與成果、問題的起源與傳播、符號的產(chǎn)生與使用等多個角度,將國內(nèi)外對中國剩余定理的相關(guān)研究作對比。其中,國內(nèi)重點討論秦九韶和黃宗憲的研究工作,國外以歐洲且主要以高斯時期的數(shù)學(xué)家為研究對象。希望能夠以多種視角全面的呈現(xiàn)國內(nèi)外中國剩余定理研究的差異。中國剩余定理是一個曠世之作,但秦九韶在運用時出現(xiàn)了錯誤。因此,本論文還分析了秦九韶運用大衍術(shù)計算“古歷會積”算題時出現(xiàn)的錯誤及其修正情況。最后,本論文參考李倍始《13世紀中國數(shù)學(xué)》中對一次同余式組解法的十種水平的分類,及其所呈現(xiàn)的15個有代表性的數(shù)學(xué)家或著作所達到的水平的表格,結(jié)合本論文的相關(guān)內(nèi)容,按照其分類方法,補充了秦九韶之前(主要是印度)以及其后(中國、日本、歐洲)的數(shù)學(xué)家所達到的水平(中國至清末黃宗憲、日本主要是關(guān)孝和與三上義夫、歐洲至比利時赫師慎),并作出了相對完善的列表。發(fā)現(xiàn)印度普遍水平較低,到了婆什迦羅二世才有提升。日本關(guān)孝和僅達到印度的最高水平,但比其晚了500多年。中國清末的黃宗憲是同時代水平最高的,且最早達到十種水平。而對于歐洲,李倍始的列表中有所遺漏,早在1612年,法國巴歇便達到了高斯的水平�？傊�,“物不知數(shù)問題”的解法要義不明,或許是一種“缺憾”。但正是如此,才導(dǎo)致了秦九韶對其算法原意的探析,進而得出大衍總數(shù)術(shù)。一道數(shù)學(xué)問題最終成為了數(shù)學(xué)史上的華麗篇章,因此探究其解法背后隱藏的原理——中國剩余定理的演變源流、梳理該原理的中外歷史發(fā)展,無疑是具有積極意義的。
[Abstract]:The problem of knowing the number of things in Sun Tzu's Sutra occupies a very important position in the history of traditional Chinese mathematics. To the Southern Song Dynasty, Qin Jiushao made a detailed study of the problem of unknown matter, and finally created a solution to this problem, called the total number of derivatives (Da Yan for short), which is recorded in the Nine chapters of the Book of numbers. At present, this technique is called "Chinese remainder Theorem". Chinese remainder theorem is a world-famous theorem, which is indispensable in any basic number theory textbook at home and abroad, and is widely used in cryptography, fast Fourier transform theory and many other fields. However, the study of its historical development is relatively rare. On the basis of previous studies, this paper takes the history of the development of Chinese surplus theorem as the research object, and systematically combs the relevant literature, especially for Qin Jiushao in the Southern Song Dynasty and Zhang Dunren in the Qing Dynasty. Huang Zongxian's General solution of "seeking a skill", as well as the original mathematical scriptures such as Lerova II of Indian Boruscharo II, Guan Hsiao and "including the algorithm" of Japan, Gao Si of Germany, "arithmetic Exploration", etc. As well as the mathematical development of China and Japan and the characteristics of Chinese arithmetic, the relevant data recorded in Bayer's Mathematical interest, France, are deeply studied. The main work has been completed as follows: first of all, starting from the background of the great derivative, taking the contributions of mathematicians of the past dynasties as the main line, this paper combs the historical development of the Chinese surplus theorem in China, and combines the relevant literature of Qian Baocong. A schematic diagram of the historical development and evolution of the Chinese remainder theorem in China is made. Divided into India, Japan and Europe, the historical development of foreign Chinese surplus theorem is sorted out in turn. Secondly, from the point of view of the time and achievement of the study, the origin and propagation of the problem, the generation and use of symbols, and so on, this paper compares the relevant research on the surplus theorem in China at home and abroad. Among them, the research work of Qin Jiushao and Huang Zongxian is discussed in China, and the mathematicians of Gao Si period are taken as the research objects abroad. It is hoped that the differences in the study of Chinese residual theorem at home and abroad can be presented comprehensively from a variety of perspectives. The Chinese remainder theorem is an extensive work, but Qin Jiushao made a mistake in his application. Therefore, this paper also analyzes the errors and correction of Qin Jiushao in calculating the problem of "ancient calendar confluence" by using Da Yan technique. Finally, this paper refers to the classification of ten levels of the solution of one congruence group in Li Beizhi's 13th Century Chinese Mathematics, and the table of the level reached by 15 representative mathematicians or works. Combined with the relevant contents of this paper, according to its classification method, this paper supplements the level reached by mathematicians before Qin Jiushao (mainly India) and then (China, Japan, Europe) (Huang Zongxian from China to the end of Qing Dynasty. Japan is mainly Guan Xiaohe and Sanshang Yifu, Europe to Belgium Shi Shen), and made a relatively perfect list. It was found that India was generally low and did not improve until Bashgaro II. Japan's Guan Hsiao and only reached India's highest level, but more than 500 years later. Huang Zongxian in late Qing Dynasty was the highest at the same time and reached ten levels at the earliest. For Europe, Li Bizhi's list is missing, as early as 1612, France reached the level of Gao Si. In a word, the solution of the problem of "thing does not know the number" is unclear, which may be a kind of defect. However, it was precisely this that led to Qin Jiushao's analysis of the original meaning of his algorithm, and then came to the conclusion of the total number of derivatives. A mathematical problem has finally become a gorgeous chapter in the history of mathematics, so it is undoubtedly of positive significance to explore the evolution of the Chinese surplus theorem, which is hidden behind its solution, and to sort out the historical development of the principle at home and abroad.
【學(xué)位授予單位】：四川師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O156
本文編號：2490276
[Abstract]:The problem of knowing the number of things in Sun Tzu's Sutra occupies a very important position in the history of traditional Chinese mathematics. To the Southern Song Dynasty, Qin Jiushao made a detailed study of the problem of unknown matter, and finally created a solution to this problem, called the total number of derivatives (Da Yan for short), which is recorded in the Nine chapters of the Book of numbers. At present, this technique is called "Chinese remainder Theorem". Chinese remainder theorem is a world-famous theorem, which is indispensable in any basic number theory textbook at home and abroad, and is widely used in cryptography, fast Fourier transform theory and many other fields. However, the study of its historical development is relatively rare. On the basis of previous studies, this paper takes the history of the development of Chinese surplus theorem as the research object, and systematically combs the relevant literature, especially for Qin Jiushao in the Southern Song Dynasty and Zhang Dunren in the Qing Dynasty. Huang Zongxian's General solution of "seeking a skill", as well as the original mathematical scriptures such as Lerova II of Indian Boruscharo II, Guan Hsiao and "including the algorithm" of Japan, Gao Si of Germany, "arithmetic Exploration", etc. As well as the mathematical development of China and Japan and the characteristics of Chinese arithmetic, the relevant data recorded in Bayer's Mathematical interest, France, are deeply studied. The main work has been completed as follows: first of all, starting from the background of the great derivative, taking the contributions of mathematicians of the past dynasties as the main line, this paper combs the historical development of the Chinese surplus theorem in China, and combines the relevant literature of Qian Baocong. A schematic diagram of the historical development and evolution of the Chinese remainder theorem in China is made. Divided into India, Japan and Europe, the historical development of foreign Chinese surplus theorem is sorted out in turn. Secondly, from the point of view of the time and achievement of the study, the origin and propagation of the problem, the generation and use of symbols, and so on, this paper compares the relevant research on the surplus theorem in China at home and abroad. Among them, the research work of Qin Jiushao and Huang Zongxian is discussed in China, and the mathematicians of Gao Si period are taken as the research objects abroad. It is hoped that the differences in the study of Chinese residual theorem at home and abroad can be presented comprehensively from a variety of perspectives. The Chinese remainder theorem is an extensive work, but Qin Jiushao made a mistake in his application. Therefore, this paper also analyzes the errors and correction of Qin Jiushao in calculating the problem of "ancient calendar confluence" by using Da Yan technique. Finally, this paper refers to the classification of ten levels of the solution of one congruence group in Li Beizhi's 13th Century Chinese Mathematics, and the table of the level reached by 15 representative mathematicians or works. Combined with the relevant contents of this paper, according to its classification method, this paper supplements the level reached by mathematicians before Qin Jiushao (mainly India) and then (China, Japan, Europe) (Huang Zongxian from China to the end of Qing Dynasty. Japan is mainly Guan Xiaohe and Sanshang Yifu, Europe to Belgium Shi Shen), and made a relatively perfect list. It was found that India was generally low and did not improve until Bashgaro II. Japan's Guan Hsiao and only reached India's highest level, but more than 500 years later. Huang Zongxian in late Qing Dynasty was the highest at the same time and reached ten levels at the earliest. For Europe, Li Bizhi's list is missing, as early as 1612, France reached the level of Gao Si. In a word, the solution of the problem of "thing does not know the number" is unclear, which may be a kind of defect. However, it was precisely this that led to Qin Jiushao's analysis of the original meaning of his algorithm, and then came to the conclusion of the total number of derivatives. A mathematical problem has finally become a gorgeous chapter in the history of mathematics, so it is undoubtedly of positive significance to explore the evolution of the Chinese surplus theorem, which is hidden behind its solution, and to sort out the historical development of the principle at home and abroad.
【學(xué)位授予單位】：四川師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O156
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