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Concepts of Modern Mathematics Reviews

Customer Rating:
Summary: Great intro to elementary concepts in math
Comment: This well-written book introduces some of the most elementary concepts in math in a very simple language. The text is quite engaging and interesting, so even the more knowledgeable people will enjoy reading the book.

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Customer Rating:
Summary: Cheap and Wonderful Read
Comment: Very well written, and reads almost like a good novel. Quite an achievement for a maths book!

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Customer Rating:
Summary: A Must Read
Comment: 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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Customer Rating:
Summary: how to get one's bearings
Comment: I picked up this book partially from the reviews and partially from "Justin Bond"'s listmania who wisely advised to read it slow but don't get bogged down.

I have been stuck in this very uncomfortable stage between lower division and upper division math. I knew that I want to take more math but I had no idea what I wanted and where it led. Who's going to sit you down and explain in practical terms what 'topology' is so you know what it's about and whether or not it will do you any good? The syllabus won't tell you. Wikipedia won't tell you.

In this context, this book was very useful to me. It provides a very casual and friendly overview of upper-division math. It gives you a taste and a place to start from, some inkling of the topic, its relevance, and connections to other fields of math.

Between the fact that I'm not a native math speaker and that the material had to be simplified, there were definitely a few times where I was a bit lost, but I, and you, should not read it to learn specific concepts, so it did not phase me at all. Yet on the other hand, he provides some insights that I had gone for many years without realizing. Even if a particular part may not be interesting, the overall presentation has a lot to offer.

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Customer Rating:
Summary: Advanced Mathematical Concepts - Simply & Elegantly Explained
Comment: If you are interested in learning some advanced mathematical concepts, this is a great book. Even if you are not interested in mathematics, this book has additional rewards beyond the mathematical concepts: it will provide you with insight into approaching non-mathematical problems -you will be able to use most of the mathematical concepts contained in it, for unrelated but analogous problems. I especially enjoyed the simple explanation of modularity. You don't have to be mathematically inclined to enjoy and gain from the reading of this book. It is excellent. It should be in the collection of any person who is interested in learning how to think better and more rigorously.

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