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Concepts of Modern Mathematics: 8 [Paperback]

Ian Stewart
4.0 out of 5 stars  See all reviews (4 customer reviews)
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Book Description

1 Feb 1995
In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts underlying "new math": groups, sets, subsets, topology, Boolean algebra, and other subjects. No advanced mathematical background is needed to follow thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, and more. 200 illustrations.

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Concepts of Modern Mathematics: 8 + What Is Mathematics?: An Elementary Approach to Ideas and Methods (Oxford Paperbacks) + Mathematics: A Very Short Introduction (Very Short Introductions)
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Product details

  • Paperback: 352 pages
  • Publisher: Dover Publications Inc.; Revised edition edition (1 Feb 1995)
  • Language: English
  • ISBN-10: 0486284247
  • ISBN-13: 978-0486284248
  • Product Dimensions: 21.3 x 13.2 x 1.9 cm
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 75,202 in Books (See Top 100 in Books)
  • See Complete Table of Contents

More About the Author

Professor Ian Stewart is the author of many popular science books. He is the mathematics consultant for the New Scientist and a Professor of Mathematics at the University of Warwick. He was awarded the Michael Faraday Medal for furthering the public understanding of science, and in 2001 became a Fellow of the Royal Society.

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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Customer Reviews

4.0 out of 5 stars
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Most Helpful Customer Reviews
7 of 7 people found the following review helpful
5.0 out of 5 stars sound accumulation of early degree math syllabus 20 May 2010
Format:Paperback|Verified Purchase
This book is a 'Dover' paperback which, if your have not bought from this publisher before, they tend to re-publish math books previously sold from around the world. Their budget-priced books tend to be in b&w and graphs in grayscale.

First Impressions

What stands out the most is the VERY attractive, organised exploration of linked math topics from the earliest chapters and throughout the book. Its clear the author encourages some theoretical issues before 'labouring' with calculations.

For example, i am attracted at the quick and early linking between 'Sets', 'Functions', 'Group Theory' and (MOD), and 'axiom' mathematics.The reason being if you are up to understanding these connections with the first three areas, then the way the axiom stuff is handled gives a VERY clear way in answering many math problems that are a model for throughout your future studies! The other linear algebraic stuff is (i.m.h.o) less attractive - compaired to other authors - but still well-worth the effort to digest it.

Harder topics

To me, the areas that seemed rather difficult and laboured being the 'Topology' aress. But i can say that the many graphs give a 'gut-feeling' of the topic. The graphs did help and i read the whole book over a long week-end.


This book has a very pleasent and stimulating style in its explanations, that will be of use for whatever math studies you may continue to follow.
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5 of 5 people found the following review helpful
5.0 out of 5 stars Still a Valuable Introduction 5 Jun 2010
I read the first edition of this book some thirty years ago and can continue to recommend it today as an appetiser to anyone contemplating mathematical studies both at A-level and beyond. In addition, it remains a first rate exposition of the nature of present day mathematics for the enquiring non-specialist reader.
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I was slightly dubious about buying this book based on some of the previous reviews. Mainly ones that suggested the explanations weren't complete or rigorous enough. After reading the book I now consider those views unfounded, I feel in any given deduction or proof there is ample explanation to follow his logic and arrive at the same conclusions.

Overall I found the book very enjoyable, especially the broad content as I had never dealt with topics like topology before. I was glued to it in a similar way to when I read a good novel. I would like to add that IMO a higher level of prerequisite knowledge than is quoted in the description is needed, as there is very little background information/techniques covered.
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6 of 16 people found the following review helpful
2.0 out of 5 stars too clever by half 30 Oct 2009
Format:Paperback|Verified Purchase
This book claims to require only basic algebra and a smattering of trigonometry. I've completed first year university maths and I can quite honestly say, what rubbish. Often you can work out his reasoning, after a good ten minutes thinking through what he could possibly mean, but just as often it is quite unclear how he gets from one statement to the other. Like many of the so-called "popular" mathematicians, Prof Stewart relies on reviews by other mathematicians who think he has a clever way of expressing familiar concepts. If you are not very fluent indeed in algebra and trigonometry, you will not understand much of what this book says. I think it is quite funny thinking of a class load of young mothers keen to help their kids at school sitting through any one of these lectures. No doubt they would not be there if they did not have some sort of feel for maths, but I cannot see ordinary people getting much out of this.
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Most Helpful Customer Reviews on (beta) 4.7 out of 5 stars  34 reviews
154 of 155 people found the following review helpful
5.0 out of 5 stars outstanding guide to higher math for the layman 18 Oct 2000
By Michael Vanier - Published on
This book is very much in the same spirit as more recent books such as Keith Devlin's "Mathematics, the New Golden Age" (which I also recommend). It explains various subjects in pure mathematics in order to make them accessible and interesting to non-mathematicians. A great variety of subjects are covered, including abstract algebra, group theory, number theory, and especially topology, to which the author devotes several chapters. The links between different branches of mathematics (e.g. topology and group theory) are given special attention, and one of the central themes of the book is the fundamental unity of mathematics. I strongly recommend this book to anyone with a serious interest in mathematics. Plus, the price is definitely right!
118 of 119 people found the following review helpful
5.0 out of 5 stars A classic - the first version of this book appeared in 1975. 8 Mar 2001
By Randall Raus - Published on
This charming book was written by a man who knows how to teach, and how to have fun. For example, as each successive topic is discussed, Mr. Stewart is careful to furnish the reader with an intuitive grasp of its main points. Only then, does he delve into the topic's details. However, what really makes this book readable is the author's wit, and sense of delight, as he illuminates--one-by-one--the abstract concepts of modern mathematics. Amazingly, this book can be read by almost anyone, and they will come away with an understanding of the why, and the wherefore, of modern math.
In theory at least, having a degree in pure math meant that I had insights that most engineers don't have. In reality, it meant I was more aware of what I didn't understand. When I got this book, I went straight to the topics I'd never gotten the point of: set theory, topology, and hyperspace. I was not disappointed, but it was not until I settled down and read the whole book that I really got the point. Modern mathematics (modern meaning the late 1800s on) provides a framework for all math. That is why it is--of necessity--more abstract, generalized, and rigorous.
Interestingly, the figures in this book are hand drawn. Perhaps its because this book has a way of transporting the reader to a university classroom - somewhere. It wouldn't have seemed right if the figures were anything but hand drawn.
148 of 155 people found the following review helpful
5.0 out of 5 stars Absolutely brilliant! 25 Dec 1998
By D. C. Carrad - Published on
Format:Paperback|Verified Purchase
Deserves 10 stars. Here is an author who understands so many advanced concepts and who can write smoothly, clearly and convincingly, bearing the reader along with his keen and interesting mind. Convincingly demonstrates the interrelationships between different areas of modern mathematics. Great mathematics for the layman without being in the slightest bit condescending. I have had an amateur's interest in mathematics since high school but was never able to follow it up professionally. This book is the best I have read in the 30 years I have had this interest. A delight to read, educational and informative.
60 of 61 people found the following review helpful
5.0 out of 5 stars for serious non-mathematicians 2 Jun 2001
By Ken Braithwaite - Published on
This is a serious book. Stewart explains clearly and concisely for a non-mathematician some of the central ideas of mathematics. Perfect for those willing to put in some thought. I'd also recommend it to anyone in first year pure math. And especially to anyone who teaches math.
33 of 33 people found the following review helpful
5.0 out of 5 stars A Must Read 19 Aug 2008
By Dylan - Published on
This book is by far the best book on mathematics I have ever read. It teaches the concepts in an intuitive, exciting way, and yet it is able to remain fun and engaging throughout. Technical material is tackled, in depth, without there seeming to be any work done. There are no exercises to be done, you simply follow Stewart along for a tour through modern mathematics. Ian Stewart's writing is flawless and almost turns this book into a thriller. I read this book in one night- I could not put it down! I stayed up until 4 in the morning reading and rereading passages; it is truly a masterpiece. The chapters are as follows:

Chapter 1- Mathematics in General: Here Stewart describes certain aspects of mathematics, and discusses their purpose and implications. He talks about abstractness and generality, intuition vs. formalism, and pure vs. applied mathematics. He tells the reader the importance of understanding WHY a theorem is true, not simply that it is. He ends with a collection of anecdotes.

Chapter 2- Motion without Motion: This is an example of thinking a bit outside the box. The chapter is devoted to overturning Euclid's proof that the base angles are congruent, and making a new one based on rigid motions. It doesn't sound too engaging, but, somehow, Stewart manages to make it quite exciting!

Chapter 3- Short Cuts in the Higher Arithmetic: A basic introduction to number theory- prime numbers, moduli, congruences, etc. The informal tone makes this the easiest and most understandable read on number theory I've yet encountered.

Chapter 4- The Language of Sets: Throughout the rest of the book, Stewart uses the language of set theory, so he introduces that here in an easy to understand way (using some imagery like bags of items, etc).

Chapter 5- What is a function?: Here Stewart addresses some of the historical problems of defining a function, and then uses the set theory from the previous chapter to define a general function, and the different types of functions.

Chapter 6- The Beginnings of Abstract Algebra: An introduction to groups, fields, rings, etc. Stewart uses the rigid motions from Ch. 2 as an example of the group concept, and then goes on to make a proof about the game solitaire (the British version) using groups. Also an explanation of the proofs about constructibility (trisecting an angle, etc) are given here.

Chapter 7- Symmetry: The Group Concept: This is where we begin to see that Ian Stewart may have a bit of a bias towards abstract algebra and group theory, as that is his specialty. That is perfectly fine, but definitely something to be aware of. The chapter on Real Analysis is certainly less in-depth than this one, but there are many hundreds of books on that you can use to fill the gaps. (Also, Real Analysis is difficult to make accessible to those without a background in calculus, whereas algebrais concepts are fairly natural). In this chapter Stewart discusses groups, subgroups, and isomorphisms with great passion.

Chapter 8- Axiomatics: This is one of my favorite chapters, and it centers on Euclidean geometry and the importance of axiomatics. It discusses models, the parallel postulate, alternate geometries, consistency, and completeness.

Chapter 9- Counting: Finite and Infinite: This is the standard treatment of Cantor and his amazing discovery. I mostly skimmed this chapter, because I had just completed a book specializing in the subject.

Chapter 10- Topology: From Mobius strips, to Klein Bottles, to orientability, to the Hairy Ball Theorem. This chapter keeps to its title. I especially love the last line about the Hairy Ball Theorem (which is a theorem that seems entirely useless at face value). "It has one application in algebra: it can be used to prove that every polynomial equation has solutions in complex numbers (the so-called 'fundamental theorem of algebra')."

Chapter 11- The Power of Indirect Thinking: This is a foray into graph theory and Euler's Formula. A lovely discussion at the end about coloring, as well.

Chapter 12- Topological Invariants: Continues the discussion of topology and proves Euler's generalized formula. Also classifies surfaces, and proves some more coloring theorems.

Chapter 13- Algebraic Topology: You can see that topology is an incredibly important tool in modern mathematics. Here he discusses Holes, Paths, and Loops.

Chapter 14- Into Hyperspace: A short treatment of polytopes and higher dimensions.

Chapter 15- Linear Algebra: A bit on the geometrical, set-theoretic, and matrix views of solving simultaneous linear equations.

Chapter 16- Real Analysis: A light treatment of infinite series, limits, completeness, continuity, and proving analytical theorems.

Chapter 17- The Theory of Probability: Random walks, binomial distibution, etc. Treated informally.

Chapter 18- Computers and Their Uses: Programming and how it works on a mathematical level.

Chapter 19- Applications of Modern Mathematics: A very interesting read about optimization and catastrophe theory.

Chapter 20- Foundations: The best treatment of Godel's proof I have yet to see. It is surprisingly rigorous, but easy to follow.

Appendix- And still it moves...: This was added 5 years after the book was written, and is an absolute gem! Stewart addresses the proof of the four-color theorem, he talks about polynomials and primes, he talks about chaos and attractors, and he ends with a reflection on real mathematics. A great end to a masterpiece.

This book is for everyone and anyone- a modest background in high school algebra and an appreciation for mathematics is all you need. Buy this book! Give it to your friends!
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