A Study On Fibonacci's Liber Abacci

Fibonacci is an Italian mathematician in the 13th century, whose famous Liber Abaci (1202 first edition, 1228 reprint) is the Encyclopedia of Mathematics in Europe in the late Middle Ages. From the 13th to the 16th century, Liber Abaci affected the development of arithmetic mathematical, the revolution in business Mathematics, and the mathematics education in Europe. However, Liber Abaci was written in the Latin language of literature, so it is difficult to understand for researchers, specifically for Chinese scholars, to whom more difficult plus the lack of the original documentation. Based on the translation of Liber Abaci from English to Chinese, and studying the large number of material, this dissertation is mainly intended to comprehensively analyze the contents, explore the algorithm, summarize the methods and thinking of mathematics from Fibonacci.This paper is composed of eight chapters.Chapter One is the preface to introduce the background, contents, and method of research, the current situation of the research of Liber Abaci.In Chapter Two, through the preface of Liber Abaci, the rare description from Fibonacci's other works and a considerable amount of research literature, it introduces the historical background, Fibonacci's life and experience, the existing works, and the common introduction of Liber Abaci, which are the basis to accurately appraise Fibonacci and his Liber Abaci.Chapter Three analyzes Fibonacci's understanding of the concept of numbers and arithmetic. He mainly introduced the India-Arabic numeral system, the concept and arithmetic of fraction, negative number, and irrational number.1,Fibonacci fully accepted to India Arabic decimal place value system and its system of calculation rules. Zero has its own symbol and particularity as well.2,The concept and theory of fraction has absorbed the characteristics of several civilizations, such as the habit of Arabs in writing fraction, i.e., the integral to the right place at the parts; the composed fraction—carry is determined by the measurement or the currency conversion according to actual needs; the second method of fraction calculated is the same as the rule for addition of fraction in today; and Fibonacci focused on the method for the general fraction broken down into several units fraction.3,Fibonacci often used negative, and proved the rules for algorithms. When negative is the answer of the equation, he said that it meant"debt"in practical, though he did not negative in his number system, and ignored the negative solution of the quadratic equation. However, taking into account the 16th and 17th centuries when European mathematicians still held the view that negative is "ridiculous number", Fibonacci's awareness of the negative undoubtedly is of great significance.4,Fibonacci's understanding of the irrational depends on the geometry, and the arithmetic of Fibonacci for square root was the tendency of algorithm.Chapter Four is the business math in Liber Abaci, Fibonacci is good to use mathematical knowledge to solve various practical problems. This paper's studies shows that in Liber Abaci the problem is solved by different algorithms, algorithm rules unified, the content of selection and arrangement meet the needs of the community, the chapter titles reflect the range of the practical application in the social life, and the typical examples of the problem often includes the application of the formulas and the description of algorithms principle, which is the foundation of the mathematical theory of business mathematics in Liber Abaci.In Chapter Five, through the study and analysis of three types of indeterminate linear equation, it is found that: 1," buy the birds" , similar to"hundred fowls", is solved by the method of alloys. 2,"find wallet"and"buy horses"are one-degree indeterminate linear equations, which can be solved by removed parameter, making no significant impact in the history of development of the indeterminate linear equation. 3,the problem of remainder arises with divining number. In sum, indeterminate problem borrowed from the eastern countries is to show Fibonacci's mathematical charm. He did not make in-depth study of the problem, but it is fascinating to the later European scholars, because these issues are interesting, and Fibonacci's solution is creative.Chapter Six discusses"the method tree"and"the method Elchataym". In this chapter, through the analysis of"the method Elchataym"and"the method tree", the study of the problems in chapter thirteen in Liber Abaci, and the comparison of"the method of Excess and Deficit"in Chinese mathematics, it argues that"the method Elchataym"is not equal to"the method of Excess and Deficit", and statement that"Elchataym"is"Khitan"lacks convincing support. Through further analysis, it is found that the two kinds of"the method Elchataym"have different origins. The first kind of"the method Elchataym"like"the method tree"has the same thinking of ratio, and the second kind of"the method Elchataym"is the same as"the method of Excess and Deficit". However, the formula of first kind of"the method Elchataym"and the second kind of"the method Elchataym"can be unified.In Chapter Seven, This chapter established on the comparison of the Fibonacci's Liber Abaci, al-Khwārizmī'algebra and al-Karajī's al-Fakhrī. It analyzes the differences in their skills in algebra, proof method for the effectiveness of algebra method, unknown settings, quadratic equation classification, the form of problem and so on. As a result, Fibonacci's and the Arab mathematicians'are the same rhetoric of algebra, and geometry methods are used to prove the effectiveness of algebraic methods. Fibonacci's method is between al-Khwārizmī's and al-Karajī's, but more inclined to al-Karajī's. Like al-Karajī, Fibonacci was adept at the application of the Euclidean theorem. In addition, the form of problems in Liber Abaci is richer than that in al-Khwārizmī's algebra, and more problems were directly quoted from the al-Fakhrī.Chapter Eight concludes the achievement of this dissertation, and some problems will be answered through more discussions. To sum up, this dissertation argues that Liber Abaci has three kinds of features, i.e., a rich content Mathematic, a variety of mathematics cultural exchange, and the dissemination diversify. Fibonacci focused on solving the problem, as far as possible to design a simple algorithm in order to obtain answers. When he used the geometry method to prove algebraic method, the perfect logical proof was not pursued, but simplified the calculation process and thereby got the answer to his question by structure geometry figure. Liber Abaci emphasizes summed-up algorithm, but does not pay attention to the form of propositions derived, and it mainly reflects the tendency of practical algorithms, such as principal method from India and the algebraic knowledge from the Arab. In Liber Abaci it is found some problems similar to ancient Chinese mathematics like pursuit problems, drainage problems, business problems, hundred fowls problems, remainder problems, indeterminate problems, as well as some Algorithm—fraction addition and subtraction,the compound rule of three, the method Elchataym and the method of Excess and Deficit, and so on. these issues have contributed to in-depth study of the dissemination and exchange of ancient Chinese Mathematics in Europe.