A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

Very Interesting, 17 Nov 2007
This is an excellent dissection of a number of sports, in terms of mathematics and probability. Highly recommended.

A very interersting read, 06 Dec 2005
With a subtitle 'the hidden mathematics of sport' I thought this might be dry, but in fact I read it in one sitting. I'm not particularly a mathematician, but the maths in here is accessible, and the harder stuff is stuck away in the appendix. Lots of sports included, though the most commonly referred to are football, cricket, tennis, rugby and athletics. Great chapter on darts. A very interesting read.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

Very Interesting, 17 Nov 2007
This is an excellent dissection of a number of sports, in terms of mathematics and probability. Highly recommended.

A very interersting read, 06 Dec 2005
With a subtitle 'the hidden mathematics of sport' I thought this might be dry, but in fact I read it in one sitting. I'm not particularly a mathematician, but the maths in here is accessible, and the harder stuff is stuck away in the appendix. Lots of sports included, though the most commonly referred to are football, cricket, tennis, rugby and athletics. Great chapter on darts. A very interesting read.

It worths exponentially much more than its price, 11 Mar 2003
One could buy this book for different reasons: interests in combinatorial optimization, of course; interests in what Papadimitriou has to say, since his thoughts on this subject are definitely invaluable; perhaps the price is a good reason alone.
Whatever the reason, however, I think that would be a rare event to remain duped.

I was preparing my exam in Computability and Complexity when I first used it. I've been wonderfully surprised by the amount of definitions, algorithms, concepts I've found in this book. I think one could use this book for a simple course on Algorithms, on Computability and/or Complexity, on the whole Combinatorial Optimization, and the book would be always and costantly useful.

The chapters on algorithms and complexity, or those on NP completeness have proved to be gems. The chapters on Approximation and Local Search are great, and they feature a bunch of detailed and excellent quality stuff (e.g. there is a detailed treatment of Christofides' algorithm to approximate the TSP, that is quite an idiosyncratic topic).

All in all, a very great book, with a value exponentially greater than the very insignificant price.

Excellent., 19 Jun 1998
Every programmer should have read this book. It is complete, detailed and makes a great reference for the engineer's bookshelf. It goes beyong the enumeration of cookie-cutter algorithms , by providing enough theory, to let you create solutions to your own optimization problems.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

Very Interesting, 17 Nov 2007
This is an excellent dissection of a number of sports, in terms of mathematics and probability. Highly recommended.

A very interersting read, 06 Dec 2005
With a subtitle 'the hidden mathematics of sport' I thought this might be dry, but in fact I read it in one sitting. I'm not particularly a mathematician, but the maths in here is accessible, and the harder stuff is stuck away in the appendix. Lots of sports included, though the most commonly referred to are football, cricket, tennis, rugby and athletics. Great chapter on darts. A very interesting read.

It worths exponentially much more than its price, 11 Mar 2003
One could buy this book for different reasons: interests in combinatorial optimization, of course; interests in what Papadimitriou has to say, since his thoughts on this subject are definitely invaluable; perhaps the price is a good reason alone.
Whatever the reason, however, I think that would be a rare event to remain duped.

I was preparing my exam in Computability and Complexity when I first used it. I've been wonderfully surprised by the amount of definitions, algorithms, concepts I've found in this book. I think one could use this book for a simple course on Algorithms, on Computability and/or Complexity, on the whole Combinatorial Optimization, and the book would be always and costantly useful.

The chapters on algorithms and complexity, or those on NP completeness have proved to be gems. The chapters on Approximation and Local Search are great, and they feature a bunch of detailed and excellent quality stuff (e.g. there is a detailed treatment of Christofides' algorithm to approximate the TSP, that is quite an idiosyncratic topic).

All in all, a very great book, with a value exponentially greater than the very insignificant price.

Excellent., 19 Jun 1998
Every programmer should have read this book. It is complete, detailed and makes a great reference for the engineer's bookshelf. It goes beyong the enumeration of cookie-cutter algorithms , by providing enough theory, to let you create solutions to your own optimization problems.

a catalogue of charts, 20 Sep 2002
This is a train spotters approach to recognising every conceivable way of charting data - the detail is painstaking, though not painful.

Be clear, however - this is a book that almost exclusively focuses on visualising quantitative information - there's no 'signage' type concepts here, and there isn't even any colour, which is actually a bonus as colour would only introduce even more distraction. You will never have believed how many different ways you can chart a string of data points until you leaf through this tome.

The book, in attempting to catalogue charting from so many different dimensions, ends up repeating itself a lot - it could have been a third of its size and still conveyed the same volume of information.

It's a book that's great to flick through when you're looking for inspiration to show that piece of boring statistics in a more engaging form.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

Very Interesting, 17 Nov 2007
This is an excellent dissection of a number of sports, in terms of mathematics and probability. Highly recommended.

A very interersting read, 06 Dec 2005
With a subtitle 'the hidden mathematics of sport' I thought this might be dry, but in fact I read it in one sitting. I'm not particularly a mathematician, but the maths in here is accessible, and the harder stuff is stuck away in the appendix. Lots of sports included, though the most commonly referred to are football, cricket, tennis, rugby and athletics. Great chapter on darts. A very interesting read.

It worths exponentially much more than its price, 11 Mar 2003
One could buy this book for different reasons: interests in combinatorial optimization, of course; interests in what Papadimitriou has to say, since his thoughts on this subject are definitely invaluable; perhaps the price is a good reason alone.
Whatever the reason, however, I think that would be a rare event to remain duped.

I was preparing my exam in Computability and Complexity when I first used it. I've been wonderfully surprised by the amount of definitions, algorithms, concepts I've found in this book. I think one could use this book for a simple course on Algorithms, on Computability and/or Complexity, on the whole Combinatorial Optimization, and the book would be always and costantly useful.

The chapters on algorithms and complexity, or those on NP completeness have proved to be gems. The chapters on Approximation and Local Search are great, and they feature a bunch of detailed and excellent quality stuff (e.g. there is a detailed treatment of Christofides' algorithm to approximate the TSP, that is quite an idiosyncratic topic).

All in all, a very great book, with a value exponentially greater than the very insignificant price.

Excellent., 19 Jun 1998
Every programmer should have read this book. It is complete, detailed and makes a great reference for the engineer's bookshelf. It goes beyong the enumeration of cookie-cutter algorithms , by providing enough theory, to let you create solutions to your own optimization problems.

a catalogue of charts, 20 Sep 2002
This is a train spotters approach to recognising every conceivable way of charting data - the detail is painstaking, though not painful.

Be clear, however - this is a book that almost exclusively focuses on visualising quantitative information - there's no 'signage' type concepts here, and there isn't even any colour, which is actually a bonus as colour would only introduce even more distraction. You will never have believed how many different ways you can chart a string of data points until you leaf through this tome.

The book, in attempting to catalogue charting from so many different dimensions, ends up repeating itself a lot - it could have been a third of its size and still conveyed the same volume of information.

It's a book that's great to flick through when you're looking for inspiration to show that piece of boring statistics in a more engaging form.

Rewarding, but not easy, reading, 17 Dec 2000
Watts' innovative study of the small world phenomena has helped to revitalise this field of research, which had until recently been considered trivial in academic circles, material for anecdotes, rather than an important feature of network organisation. Watts shows how and why networks can be organised along small world principles, with examples as diverse as the spread of diseases (or gossip) through a population, the connectivity of worm's neural structures, and, infamously, the Kevin Bacon Game. While the book starts at a gentle pace, the mathematical detail soon becomes fairly dense, especially for those with little post-school mathematical training. However, the reader's perseverance is rewarded by Watts, who has provided a range of applications of small world theory, making this a must for anyone planning to study network organisation.

A pleasant read, 22 Aug 2007
I read this book whilst on holiday (had my laptop with me, but managed not to turn it on).

The first part is a gentle intoduction to algorithms. There's little or no mathematics, but the concepts are well covered. The second part is the catalog, which seemed to live up to its name.

This is a must for programmers, 29 Jun 1999
This book is very well organized. It really helps identifying and solving problems. Highly recommended.

The hitch-hiker's guide to Algorithms., 28 Jul 1998
The Catalog was my main reason for buying the book. It's an invaluable reference base for people whose boss 'needs an answer by tomorrow'.

+ : The War Stories are fun reading, and do a good job of explaining how theory relates to practice. - : Restating the obvious at times, while deliberately vague elsewhere. Net : if you use a greedy heuristic to select your reading, this book probably comes ahead of the pack.

Quite a useful repository of algorithms, 06 May 1998
This book has some excellent information about writing and selecting algorithms, step by step, as well as plenty of pointers to outside information. Chapter 8 in particular is an invaluable reference for quickly implementing a solution to any of many varied problems.

However, the textual explanations are sometimes confusing, with significant "jumps" between concepts that could throw off the beginning algorist. Furthermore, the author discounts entire paradigms of computer programming, giving the text a biased and unbalanced feel.

Review from co-developer of the CD-ROM and website, 07 Oct 1997
As an unbiased reviewer :^) I feel that this is the most useful algorithms text written for the real-world algorist. The CD-ROM contains a wealth of information (including the entire repository of implementations found on the affiliated website, and hours of audio lectures from the author's own algorithms course). The catalog of algorithms is also invaluable.

Disappointed, 15 Dec 2008
If you are familiar with R and with Bayesian Computation this book may be a good introduction to using R packages for Bayesian Computation, but I didn't feel it is a good introduction to Bayesian Computation or to R if you are unfamiliar with either or both.

quite excellent, 31 Jan 2003
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.

Proofs from THE BOOK, 14 Jan 2002
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.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

MUST OWN!!!, 20 Jul 1999
I first purchased this book two years ago and became really excited when I found that I had it in me to write proofs. The truth be told, I have NOT STOPPED writing proofs since being inspired by this book. It is a masterpiece and I think ANY student approaching the writing of proofs for the first time couldn't possibly go wrong with an investment in a copy of this book. I also think that professors teaching a first class in abstract mathematics would be doing a service to their students by either requiring the book or giving it serious recommendation. AWESOME!!

Wonderful into to rigorous mathematics, 29 Dec 1998
I agree with Usispaul's comments.

I only want to add that this is a wonderful introduction to mathematical thinking. It is completely engaging, and not like other textbooks. [This is a rigorous math book (not a book about math) and covers the material of first course for mathematics majors, logic, sets, relations, functions.] There are exercises (do them!) and examples.

I took math in college, but this book made me want to know MORE about mathematics.

Engaging book on producing mathematical proofs, 18 Sep 1997
The emphasis of Vellemans book on the difference between manufacturing a proof and the proof's final presentation speaks directly to the confusion of the uninitiated to proofs. It meets the (perhaps frequent) naive expectation of an invariable and immediate recognition of a polished proofs rhyme and reason. It consequently points to the often necessary autonomous efforts of the student to independantly unravel the proof of a theorem or definition. The book moves rapidly from the necessary setential logic and truth tabels (a Wittgensteinian invention) to the chapters on proof writing and follows with chapters on functions, relations, closures, and more.

Very Interesting, 17 Nov 2007
This is an excellent dissection of a number of sports, in terms of mathematics and probability. Highly recommended.

A very interersting read, 06 Dec 2005
With a subtitle 'the hidden mathematics of sport' I thought this might be dry, but in fact I read it in one sitting. I'm not particularly a mathematician, but the maths in here is accessible, and the harder stuff is stuck away in the appendix. Lots of sports included, though the most commonly referred to are football, cricket, tennis, rugby and athletics. Great chapter on darts. A very interesting read.

It worths exponentially much more than its price, 11 Mar 2003
One could buy this book for different reasons: interests in combinatorial optimization, of course; interests in what Papadimitriou has to say, since his thoughts on this subject are definitely invaluable; perhaps the price is a good reason alone.
Whatever the reason, however, I think that would be a rare event to remain duped.

I was preparing my exam in Computability and Complexity when I first used it. I've been wonderfully surprised by the amount of definitions, algorithms, concepts I've found in this book. I think one could use this book for a simple course on Algorithms, on Computability and/or Complexity, on the whole Combinatorial Optimization, and the book would be always and costantly useful.

The chapters on algorithms and complexity, or those on NP completeness have proved to be gems. The chapters on Approximation and Local Search are great, and they feature a bunch of detailed and excellent quality stuff (e.g. there is a detailed treatment of Christofides' algorithm to approximate the TSP, that is quite an idiosyncratic topic).

All in all, a very great book, with a value exponentially greater than the very insignificant price.

Excellent., 19 Jun 1998
Every programmer should have read this book. It is complete, detailed and makes a great reference for the engineer's bookshelf. It goes beyong the enumeration of cookie-cutter algorithms , by providing enough theory, to let you create solutions to your own optimization problems.

a catalogue of charts, 20 Sep 2002
This is a train spotters approach to recognising every conceivable way of charting data - the detail is painstaking, though not painful.

Be clear, however - this is a book that almost exclusively focuses on visualising quantitative information - there's no 'signage' type concepts here, and there isn't even any colour, which is actually a bonus as colour would only introduce even more distraction. You will never have believed how many different ways you can chart a string of data points until you leaf through this tome.

The book, in attempting to catalogue charting from so many different dimensions, ends up repeating itself a lot - it could have been a third of its size and still conveyed the same volume of information.

It's a book that's great to flick through when you're looking for inspiration to show that piece of boring statistics in a more engaging form.

Rewarding, but not easy, reading, 17 Dec 2000
Watts' innovative study of the small world phenomena has helped to revitalise this field of research, which had until recently been considered trivial in academic circles, material for anecdotes, rather than an important feature of network organisation. Watts shows how and why networks can be organised along small world principles, with examples as diverse as the spread of diseases (or gossip) through a population, the connectivity of worm's neural structures, and, infamously, the Kevin Bacon Game. While the book starts at a gentle pace, the mathematical detail soon becomes fairly dense, especially for those with little post-school mathematical training. However, the reader's perseverance is rewarded by Watts, who has provided a range of applications of small world theory, making this a must for anyone planning to study network organisation.

Dry and outdated, but cheap, 27 Apr 2004
This book offers a lot of content for a low price. That's the good thingabout it.
The bad thing is that it's difficult to read - it's very dry and boring.Of course, pretty much everything in mathematics is dry and boring, but Iknow lots of math books which teach far more efficiently than this one. Itgives too many examples and often doesn't go straight to the point, and itdidn't make it clear why the hell one would want to learn about grouptheory.
Also, the some notation used in this book is outdated, which is slightlyannoying at times as it just adds to the confusion of a confusing subject.

Great book for an introduction to graph theory., 11 Jun 2001
This book has all the information to get you through any graph theory course. The theory is explained well but the most helpfull is the hundreds of examples. There are a few examples on every topic in the book that are simple enough for anyone to read.