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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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