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Concepts of Modern Mathematics
 
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Concepts of Modern Mathematics [Format Kindle]

Ian Stewart

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Présentation de l'éditeur

Some years ago, "new math" took the country's classrooms by storm. Based on the abstract, general style of mathematical exposition favored by research mathematicians, its goal was to teach students not just to manipulate numbers and formulas, but to grasp the underlying mathematical concepts. The result, at least at first, was a great deal of confusion among teachers, students, and parents. Since then, the negative aspects of "new math" have been eliminated and its positive elements assimilated into classroom instruction.
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 more. According to Professor Stewart, an understanding of these concepts offers the best route to grasping the true nature of mathematics, in particular the power, beauty, and utility of pure mathematics. No advanced mathematical background is needed (a smattering of algebra, geometry, and trigonometry is helpful) to follow the author's lucid and thought-provoking discussions of such topics as functions, symmetry, axiomatics, counting, topology, hyperspace, linear algebra, real analysis, probability, computers, applications of modern mathematics, and much more.
By the time readers have finished this book, they'll have a much clearer grasp of how modern mathematicians look at figures, functions, and formulas and how a firm grasp of the ideas underlying "new math" leads toward a genuine comprehension of the nature of mathematics itself.


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Amazon.com: 4.7 étoiles sur 5  34 commentaires
154 internautes sur 155 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 outstanding guide to higher math for the layman 18 octobre 2000
Par Michael Vanier - Publié sur Amazon.com
Format:Broché
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 internautes sur 119 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 A classic - the first version of this book appeared in 1975. 8 mars 2001
Par Randall Raus - Publié sur Amazon.com
Format:Broché
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 internautes sur 155 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Absolutely brilliant! 25 décembre 1998
Par D. C. Carrad - Publié sur Amazon.com
Format:Broché|Achat vérifié
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 internautes sur 61 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 for serious non-mathematicians 2 juin 2001
Par Ken Braithwaite - Publié sur Amazon.com
Format:Broché
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.
31 internautes sur 31 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 A Must Read 19 août 2008
Par Dylan - Publié sur Amazon.com
Format:Broché
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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