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Proofs from "The Book" (Hardcover)

by Martin Aigner (Author), Gunter M. Ziegler (Author)

5.0 out of 5 stars See all reviews (2 customer reviews)

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Product details

Hardcover: 224 pages
Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K; 2nd edition (Jan 2001)
Language English
ISBN-10: 3540678654
ISBN-13: 978-3540678656
Product Dimensions: 24.8 x 19.7 x 2 cm

Average Customer Review: 5.0 out of 5 stars See all reviews (2 customer reviews)

Amazon.co.uk Sales Rank: 972,864 in Books (See Bestsellers in Books)

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Product Description

The (mathematical) heroes of this book are "perfect proofs": brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erd/s, who himself suggested many of the topics in this collection. The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For this revised and expanded second edition several chapters have been revised and expanded, and three new chapters have been added. According to the great mathematician Paul Erd/s, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, sis, com binatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

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39 of 40 people found the following review helpful:

5.0 out of 5 stars Proofs from THE BOOK, 14 Jan 2002

By A Customer

If you are at all interested in mathematics, and your background is anything from undergraduate to Fields medallist, then this is the one book you really must have. Paul Erdos, the most prolific and creative mathematician of the 20th century, had no need for the hypothesis of God. He nevertheless liked to believe that this non-existent deity kept a book in which were recorded all the most perfect and beautiful proofs in mathematics, and that, just once in a while, mere mortals were permitted a brief glance at some of the pages. As a tribute to Erdos, who died in 1996, Aigner and Ziegler have made a guess at what those pages might contain. They have compiled an A to Z of what they regard as the most elegant proofs from a wide range of mathematics, including Number Theory, Geometry, Analysis, Combinatorics and Graph Theory (all subjects close to Paul's heart). Many of the results are remarkably easy to state, for instance the theorem that there are infinitely many prime numbers, but all require some ingenuity to prove them; this particular example is given six different proofs, three of them direct, two involving Analysis and Topology, and one based on a neat counting argument. Similarly, in Combinatorics we get four totally different proofs of Cayley's formula for the number of trees on n vertices, each of them a real classic, and one (perhaps the best) very recent.

This book is superbly written and produced. Each chapter has a brief but clear historical introduction to the problem at hand, some excellent illustrative diagrams, a discussion of the proof or proofs, and a selection of references for further reading. There are portraits of many of the mathematicians, ancient and modern, whose ideas appear here, while Karl Hofmann's witty cartoons are an extra and unexpected joy.

Even if you have only a modest mathematical training, you can dip into this book, almost at random, and be guaranteed to come away a few minutes later wondering "Where on earth does an idea like that come from?". You cannot read this book and fail to be inspired to create mathematics yourself. If there were six stars available for reviewed books, this would get them. Buy it.

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11 of 11 people found the following review helpful:

5.0 out of 5 stars quite excellent, 31 Jan 2003

By	D. Bevan - See all my reviews (REAL NAME)

If you have ever been excited by the beauty of pure mathematics - or wondered at people who were - this book is for you. Not only do the authors exhibit a variety of the most elegant proofs around, but their presentation is of the highest quality - and in a very attractively designed book too. Recommended for even the most half-hearted maths lover.

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