Limits, Limits Everywhere: The Tools of Mathematical Analysis
Format: PDF / Kindle (mobi) / ePub
A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series.
Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and ?, continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject.
A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.
numbers can be added by the rule: (a + ib) + (c + id) = (a + c) + i(b + d), and also multiplied together by expanding brackets in the usual way, remembering that i2 = −1 so (a + ib)(c + id) = ab + ibc + iad + i2 bd = (ac − bd) + i(bc + ad). 9 The representation of complex numbers as points in two-dimensional space is often called an Argand diagam after Jean-Robert Argand (1768–1822). 135 8 INFINITE PRODUCTS We can even discuss convergence of sequences of complex numbers, so if (zn ) is such a
by one of the great mathematical minds of the early twentieth century, which was first published in 1925, is reprinted in From Frege to Gödel: A Sourcebook in Mathematical Logic ed. J. van Heijenoort, Harvard University Press (1967). 2 See http://en.wikipedia.org/wiki/Georg_Cantor 145 10 HOW INFINITE CAN YOU GET? In this procedure each symbol is mapped to a number between 1 and 6 and no symbol is mapped to more than one number. This is the essence of counting. We would implement the same
exist and x indeed this is the case for many important functions. To be precise, we say that a function f is differentiable at the point x if this limit exists and in this case we deﬁne f (x + h) − f (x) . h→0 h f (x) = lim 7 I’m not going to comment on the priority dispute that so obsessed many of their followers and remains the subject of scholastic enquiry to this day. 8 Of course, Newton and Leibniz did not know the precise definition of a limit but they had sufficient insight to be able to
Remarks About the History of Analysis T he history of mathematics is itself a vast subject. To isolate just one topic – analysis – and try to outline its history is a somewhat dangerous enterprise and the reader should be aware that the author is not a professional historian and is a lot less sure of his footing in this territory. Analysis did not really emerge as an area of mathematics with its own identity until the nineteenth century. Although we can trace the development of some important
worked entirely with whole numbers. Now we must split these apart and come to an understanding of how every point on the line in Figure 1.1 can represent a number. That is the task of the next chapter. 1.4 Exercises for Chapter 1 The first six questions in this section are designed to help you practise simple proof techniques and the remainder focus on prime numbers. 1. Prove that if n is an odd number then n2 and n3 are also odd. [Hint: Use the fact that if n is odd then n = 2m − 1 for m = 1,