Fundamental Concepts of Mathematics

PREFACE

PREFACE Fundamental Concepts of Mathematics is the first of several volumes of a proposed series on fundamental mathematics which serve primarily as textbooks for students in preparation for the A-Level or other public examinations. However they go into more depth than what is required by these examinations, and contain topics that are useful but often omitted for lack of time in the undergraduate courses. Therefore the books of this series can also be used by university students as reference books for supplementary reading. Further volumes on topics in algebra, geometry and calculus are forthcoming. It is sincerely hoped that in their entirety they will ease the dire shortage of suitable textbooks for ALevel mathematics that has persisted over decades in Hong Kong. The present volume gives a straightforward account of the various number systems of fundamental mathematics. Chapter 1 is a brief account of the informal set language. The reader may find this more accessible and easier than the presentation in Elementary Set Theory by D..L.C. Chen and the author. Natural numbers are studied in Chapters 2 and 3. No attempt has been made to define natural numbers in terms of primitive concepts and axioms. Only the principle of mathematical induction is singled out for a detailed scrutiny in Chapter 2. The well-ordering property of natural numbers is assumed, and the principle of mathematical induction is derived and then studied from different points of views. Natural numbers are used in counting and Chapter 3 is devoted to counting the different ways in which an event can take place. A balls-into-boxes model is used to study the various problems of permutations and combinations. This unified approach to the topics is probably new at this level of school mathematics. Chapter 4 deals with elementary number theory. At present, bits and pieces of this subject are scattered throughout the 13-year curriculum from class primary one to form upper six. No effort has been made anywhere to present it in a systematic manner. It is not unusual to find beginners at the university who could cope with complicated integration but had no idea that the greatest common divisor can be written as a linear combination. The present chapter is an attempt to redress this shortcoming and to introduce the subject up to the unique factorization theorem and the Chinese remainder theorem. Students in the upper-forms will have no difficulty in working through this chapter and undergraduate students may find it useful to read the chapter before they take up a course on algebra or number theory. The next two chapters on the real numbers are the most difficult part of the book. Chapter 5 brings out the similarities and the essential differences between the rational numbers and the real numbers. It also viii Preface leads to the density theorem and the postulate of contintlity. Powers, roots and logarithms of real numbers are then given rigorous definitions. The postulate is put to further use in Chapter 6 where the fundamental notion of convergence is discussed. The idea of limit is introduced first informally by a series of definitions and examples. A precise definition is then formulated after the reader is sufficiently familiar with the main theme. Undue formalism and abstraction are avoided, and proofs that demand more maturity are put in the appendix. The material of these two chapters exceeds the requirement of the A-Level examination. Undergraduate students will find in them many useful detailed discussions that may be omitted in the lectures. Chapter 7 introduces complex numbers as the ultimate result of a series of attempts to remove the inadequacy of the previous number systems in the provision of solutions to polynomial equations. The system of complex numbers is viewed geometrically as a twodimensional extension of the one-dimensional system of real numbers. The reader may find the approach to the imaginary unit by rotations both natural and interesting. In the concluding sections, the analytic geometry of straight lines and circles appears in a new form as geometry of complex numbers. This may be taken as an illustration of the adaptability of mathematical ideas and of the fact that seemingly different branches of mathematics are actually well connected. The appendix to this chapter outlines the further development of the number concept in the last century. All material of the chapter is within the reach of upper-form students. Several hundred exercises are included in the book, with the more difficult ones marked by an asterisk...