Customer Reviews

A Concise Introduction to Pure Mathematics (Chapman Hall/Crc Mathematics) (Chapman Hall/CRC Mathematics Series)

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I'm not sure how many of us there are out there, but I am one of a breed of consumers of applied mathematics who learned some pretty sophisticated mathematical technology without the rigor of pure mathematics. Although this book is aimed at freshmen entering mathematics programs who need to be inculturated into the world of pure mathematics, I found it to be the crucial link I needed to advance my own applied mathematical training. I reached a point where I was ready to move from applied texts to the more cryptic world of math texts written for graduate mathematicians. Unfortunately, I was not properly trained to decypher their special language and way of doing things, particularly that of the formal proof. I found reading introductory texts in analysis to be like trying to learn Japanese from books which were themselves written in Japanese. Then I found this wonderful little text. It made things much more accessible to me and helped me crack enough of the code where I could find my way around those analysis texts, which in turn allowed me to move on to the graduate math texts containing the methods I am studying now. The book is a bit high priced for its size, but for me it was well worth it.

This is a review of the (2005) 2nd ed. The number of texts covering the transition from secondary school to college mathematics has grown considerably in recent years. This is one of the better-written and well-organized texts. Its greatest concentration is on important concepts from pure mathematics, such as sets and numbers, real and complex, and some interesting topics from number theory. Explanations are clear and the in-text examples and proofs are well chosen and explained. The emphasis here is primarily on proofs rather than on the solution of applied problems. The author uses only the minimum level of mathematical rigor required, and this is supplemented by clear discussions. I enjoyed the gentle introduction to set theory and the in-text questions, followed by solutions. The proofs of propositions are clear and complete.

The Forward says this book can "be read by a student on his or her own". The Preface restates this slightly differently, by saying that as "well as being designed for use in a first university course, the book is also suitable for self-study". However, debatably, this text does not serve both purposes equally well, as it seems less suitable for a self-study target audience.

A " Solutions Manual for a Concise Introduction to Pure Mathematics" is listed on-line. The Solutions Manual described is about 70 pages in length. If this is correct, it's contents could easily have been included with this text, while still keeping the text relatively concise at less than 300 pages. At the time of this review, this manual was not available from Amazon or other on-line sellers.

The lack of fully-worked solutions to exercises is typical of many books designed for classroom use. This allows faculty to assign problems that students must work out on their own, as solutions are not readily available. While this approach is, arguably, appropriate for a classroom environment, the lack of detailed exercise solutions considerably reduces the value of this text for self-study. Mathematics is not a spectator sport, so the opportunity to work through a considerable variety of problems and check results against detailed solutions is quite important, particularly for self-study. The lack of fully-worked exercise solutions is perhaps the key deficiency of this text. However, it is enjoyable to read, with explanations that are very well done. Thus, although not self-contained, it could be excellent for self-study if supplemented appropriately with a problems book with fully-worked solutions.

I encountered this book in my first proofs course, and it was a delight to read. My less dedicated classmates found the book too difficult, but I think it's just right for a student who has been curious about methods of proof and some of the more elementary parts of pure mathematics. Some of the problems are easy and some are pretty challenging, but I think they can all be solved especially if you've got access to people who know mathematics (and everyone on the internet has this is they look).

Now in an updated and expanded third edition, "A Concise Introduction To Pure Mathematics" by Martin Liebeck provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics including the theory of solving cubic equations, the use of Euler's formula to study the five Platonic solids, the use of prime numbers to encode and decode secret information, the theory of how to compare the sizes of two infinite sets, the limits of sequences and continuous functions, the use of the intermediate value theorem to prove the existence of nth roots, and so much more. Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, 'student friendly', and highly recommended choice for classroom textbooks for highschool and college level mathematics curriculums, "A Concise Introduction To Pure Mathematics" is further enhanced with a selective bibliography, and index of symbols, and a comprehensive index.

This short little book is very thorough and challenging, I am still reading it so I can't be more conclusive than that