Numbers and Functions: Steps to Analysis [Paperback]

R. P. Burn (Author)

Book Description

Publication Date: August 28, 2000 | ISBN-10: 0521788366 | ISBN-13: 978-0521788366 | Edition: 2

The transition from studying calculus in high school to studying mathematical analysis in college is notoriously difficult. In this new edition of Numbers and Functions, Dr. Burn invites the student to tackle each of the key concepts, progressing from experience through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis. The problems, with all solutions supplied, draw readers into constructing definitions and theorems. This novel approach to rigorous analysis will enable students to grow in confidence and skill and thus overcome traditional difficulties in learning this subject.

Editorial Reviews

Review

' ... an excellent guide through the basic course of mathematical analysis at university.' EMS

Book Description

In this new edition of Numbers and Functions, the reader is invited to tackle each of the key concepts of mathematical analysis in turn, progressing from experience through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis.

Product Details

• Paperback: 350 pages
• Publisher: Cambridge University Press; 2 edition (August 28, 2000)
• Language: English
• ISBN-10: 0521788366
• ISBN-13: 978-0521788366
• Product Dimensions: 8.9 x 6.1 x 1.1 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #722,461 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

17 of 17 people found the following review helpful:

4.0 out of 5 stars Interesting and refreshing approach, May 5, 2000

By 

Alastair Farrugia (Waterloo, Ontario, Canada) - See all my reviews

This review is from: Numbers and Functions: Steps to Analysis (Paperback)

I worked through this book several years ago and I remember enjoying its style of pointing out an interesting property of a particular function, and then showing, step by step, that a whole class of functions have that property; that is, the theorems are built up from examples, instead of the other way round. I also think each step was quite manageable - there were no big gaps where I was left scratching my head not knowing what to do. It is not meant as a reference book, as you're more likely to find sketches or hints to parts of proofs, rather than complete proofs. I don't know if it's ever been used as a text book, but if it were, students couldn't just sit back and absorb knowledge - they would have to figure things out (with the benefit of lots of pointers to tell them what needs figuring out, and hints about how to do it), which is just what the book is intended to encourage.

15 of 15 people found the following review helpful:

5.0 out of 5 stars Numbers and Functions, Steps into Analysis, February 28, 2005

By 

Marc K. Pestana (Los Angeles) - See all my reviews
(REAL NAME)   

This review is from: Numbers and Functions: Steps to Analysis (Hardcover)

If you follow the study plan laid out in the preface of this book, you'll get a background in Calculus and Undergraduate Analysis second to none. After this truly thorough coarse of study, 'little Rudin' will present few difficulties and 'Big Rudin' will be much more approachable. Depending on your background, you may have to buy the references that are conveniently listed at the beginning of each chapter. But the effort and expense is well worth it. If you're just starting out or you're looking for an indepth review of Calculus and basic Analysis, this book is THE best. This book will also benefit those studying applied math and engineering because it provides the mathematical background for many of the more complicated functions and approximations used in mathematical applications.

The poor review found above is not reliable. There is abundant background material provided in the excellant (and mostly cheap) references. Each chapter ends with historical insights very helpful in comprehending the material. The whole approach is based the presentation of fundemental concepts and their origins and implications. The title is exactly right. This book combines problem solving with the historical sequence that led to the development of the major IDEAS behind the mathematics. Thus, each problem is a stepping stone to the next problem. Each section is a stepping stone to the next section and so on. Also, each problem is broken down into conceptual components that allow the student to understand not only the raw mathematics but also the strategy used to solve the problem. Each problem set is followed by an exposition of the big picture of just what concepts are being illucidated. I love the very well done and useful approach to teaching. If you have no or a weak background in the subject of Calculus or Analysis, you must do the preliminary reading indicated at the beginning of each chapter. For those with a stronger background the concurrent and further reading suggestions are very helpful though not necessary in all cases. Be patient and give yourself a chance, it's well worth it.

11 of 11 people found the following review helpful:

5.0 out of 5 stars Best Undergraduate Single Variable Real Analysis Text by Far, December 22, 2005

By 

Sandy Lemberg - See all my reviews

This review is from: Numbers and Functions: Steps to Analysis (Paperback)

This beautiful book is by far the best undergraduate single variable real analysis text I have seen. It covers all the basic topics in impeccable detail. Each chapter opens by listing a few references, labelled "Preliminary", "Concurrent", and "Further" Reading. The main part of each chapter consists of "questions" which guide the student through a complete theoretical development of the material and which the student is invited to work through. This part of the chapter contains definitions, statements of theorems, and informal discussion. This section is followed by a brief summary, outlining the previous material. Next comes a "Historical Note" which is very illuminating and fun to read. The last part of the chapter contains a complete working out of all the "questions". At the end of the book is an extensive bibliography, containing all books mentioned at the beginning of the chapters and many others. There is also an accurate and detailed index.

All in all, the text contains an exhaustive and perspicuous treatment of material which often is presented in a less transparent way in other texts such as Rudin. I also prefer it by far to other excellent recent books such as those by Ross or Abbott. The format engages the reader in a unique way that other books don't. This book was developed for use in the math program at the University of Warwick and as far as I know, it is still in use there. Unfortunately, it is less well known in the US. I cannot recommend this book highly enough. Once you see a copy for yourself, I think you will understand why.