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Proofs from the Book
From the Reviews:
..". Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: ..". all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999
..". the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." LMS Newsletter, January 1999
This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals."
..". Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. Some of the proofs are classics, but many are new and brilliant proofs of classical results. ...Aigner and Ziegler... write: ..". all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... " Notices of the AMS, August 1999
..". the style is clear and entertaining, the level is close to elementary ... and the proofs are brilliant. ..." LMS Newsletter, January 1999
This third edition offers two new chapters, on partition identities, and on card shuffling. Three proofs of Euler's most famous infinite series appear in a separate chapter. There is also a number of other improvements, such as an exciting new way to "enumerate the rationals."
240 pages, Hardcover
First published December 31, 1998
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Displaying 1 - 20 of 20 reviews
February 11, 2014
The late mathematician and philosopher Gian-Carlo Rota referring to the age old (and perhaps unanswerable) question - "are mathematical ideas invented or discovered?" would say that mathematics led a double life. One of these lives dealt with facts, that mathematicians communicate and study much like taxonomists. Mathematical facts are as useful as the facts of any other Science and no matter how abstruse invariably find applications as well. Quoting Rota: "The facts of today's mathematics are the springboard for the Science of tomorrow.".The second life deals with proofs: the process of discovering those facts. A mathematical proof consists of starting with a set of axioms/definitions; well established mathematical facts, and logically deduce, by pure reason the said fact. However there is a hidden circularity here, to come up with "elegant" and "englightening" proofs one must first start with the right definitions to work with or risk end up getting lost or constructing a contraption for a proof from which almost no understandin about the problem might be salvaged. Much of the art of mathematical proof lies in the leap of imagination needed to come up with the right starting definitions. One might say that the simplest, most elegant and beautiful proofs start with just the right definition, thus also maximizing the insight that can be gained into the problem.
One can not learn this art without living with it for years. This book is one of those gems that can help you learn and appreciate this art and help develop a good nose for the "bare minimum" that fits. It is not a book to be read cover to cover. It is a book to be kept by your bedside and read time in and time again, keeping with you a certain proof for weeks till you have absorbed and appreciated its genesis completely. This book is filled with some of the most elegant proofs known and the background needed to appreciate them is minimal too. I personally bought this book on reading Furstenberg's very elegant and unusual topological proof of the infinitude of primes.
The origin of the name of this book is a part of mathematical lore. The legendary mathematician Paul Erdos, sort of as a half joke, talked about this "book" of God that contained all the beautiful and elegant proofs and that the job of mathematicians was only to try to discover proofs from the BOOK. This book was suggested by the authors as a first approximation to this BOOK, an idea that Erdos was very enthusiastic about and made many suggestions toward. Unfortunately he passed away before the book was published and could not be listed as a co-author. Nevertheless this book, by its eclectic choice of proofs and crazy ideas perhaps comes close to describing and teaching what mathematicians mean when they say that a proof is beautiful. It's hard to define, but one knows it when one sees it. This book will train your eye and you would be glad for having bought it.
One can not learn this art without living with it for years. This book is one of those gems that can help you learn and appreciate this art and help develop a good nose for the "bare minimum" that fits. It is not a book to be read cover to cover. It is a book to be kept by your bedside and read time in and time again, keeping with you a certain proof for weeks till you have absorbed and appreciated its genesis completely. This book is filled with some of the most elegant proofs known and the background needed to appreciate them is minimal too. I personally bought this book on reading Furstenberg's very elegant and unusual topological proof of the infinitude of primes.
The origin of the name of this book is a part of mathematical lore. The legendary mathematician Paul Erdos, sort of as a half joke, talked about this "book" of God that contained all the beautiful and elegant proofs and that the job of mathematicians was only to try to discover proofs from the BOOK. This book was suggested by the authors as a first approximation to this BOOK, an idea that Erdos was very enthusiastic about and made many suggestions toward. Unfortunately he passed away before the book was published and could not be listed as a co-author. Nevertheless this book, by its eclectic choice of proofs and crazy ideas perhaps comes close to describing and teaching what mathematicians mean when they say that a proof is beautiful. It's hard to define, but one knows it when one sees it. This book will train your eye and you would be glad for having bought it.
November 3, 2016
I hesitate to call this a textbook (and I make it a point not to add them to my list) because every person I know who has read this has done it solely for pleasure and we all remember the lines not because we tried, but because it took us briefly to another world. This book is a comic, a Bible and a whodunnit - makes you smile at the world even in times of trouble, makes you subscribe to a lifestyle and makes you stay at the edge of your seat, perhaps even after you've read the whole thing.
December 7, 2007
What a delight! There's plenty of distinguished results from various fields here that were new to me, and the proofs are as sweet and succulent as advertised. Definitely a fine selection for anyone in love with math and beautiful results.
November 15, 2010
Che cos'è Il Libro? Forse la Bibbia? In un certo senso sì. Secondo Paul Erdős, Dio (o The Great Fascist, come lo chiamava lui) aveva un libro con tutte le dimostrazioni matematiche più belle ed eleganti, e ogni tanto qualche mortale riusciva a dargli un'occhiata e trovarne una... che suscitava ammirazione e invidia negli altri matematici. Occhei, il 99,9% degli esseri umani non capisce la differenza, ma da quando in qua questo è motivo per lasciare perdere?
Questo libro raccoglie alcune dimostrazioni che a detta degli autori possono stare nel Libro. Attenzione! Non è affatto detto che queste dimostrazioni siano "facili", diciamo alla portata di uno studente delle superiori; spesso gli autori hanno preferito lavorare sull'inaspettatezza, e sull'usare tecniche che a prima vista non abbiano nulla a che fare con i problemi di partenza. Però ci sono delle chicche davvero belle, che meritano davvero la lettura e l'ammirazione almeno di chi apprezza la matematica come un'arte (quale in effetti è...) e non come una scienza.
Questo libro raccoglie alcune dimostrazioni che a detta degli autori possono stare nel Libro. Attenzione! Non è affatto detto che queste dimostrazioni siano "facili", diciamo alla portata di uno studente delle superiori; spesso gli autori hanno preferito lavorare sull'inaspettatezza, e sull'usare tecniche che a prima vista non abbiano nulla a che fare con i problemi di partenza. Però ci sono delle chicche davvero belle, che meritano davvero la lettura e l'ammirazione almeno di chi apprezza la matematica come un'arte (quale in effetti è...) e non come una scienza.
December 28, 2010
This is a collection of beautiful mathematical proofs. I now know two proofs of the divergence of the harmonic series of primes, Euler's and Erdős's; it is easy to see where Euler's proof is coming from (given modern mathematical notation, which was not available to Euler and which he helped invent), but how Erdős came up with his proof, I cannot comprehend.
February 27, 2011
This is a very enjoyable read. Each chapter is devoted to a mathematical result that can either be proved simply or elegantly, or can be used in such a proof. For the most part, the chapters are self-contained. Elementary definitions and concepts, such as scalar products and the pigeon-hole principle, are introduced, but it is hard to imagine that the book would be very meaningful for a reader who was not already familiar with such topics.
my favorite quote: "But to tell the truth, what they really want to prove, once in their lifetime, is a Lemma, like the one by Fatou in analysis, the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in combinatorics."
my favorite quote: "But to tell the truth, what they really want to prove, once in their lifetime, is a Lemma, like the one by Fatou in analysis, the Lemma of Gauss in number theory, or the Burnside-Frobenius Lemma in combinatorics."
Want to read
October 19, 2011Sitting on my shelf at he lab. Was rather inaccessible to quick skimming. Looking forward o sitting down with pape, pen, and time.
January 10, 2016
such a great book is worth marking..
February 3, 2016
Elegance, beauty, inspiration and -- why not? -- passion. Truly a devotional book for those who worship the God of the Proofs.
September 15, 2021
Disclaimer: I didn't understand most of this book. I "read" it, meaning that I went from the first page to the last one and my eyes took a look at the symbols inside... but not much else was going on. I probably know way more math than the average person, though unfortunately I also know way less math than the average person who reads books about mathematical proofs.
For the few parts of the book that I did understand, they... didn't seem that impressive? There were a few results that were nice (many I was already familiar with), but I didn't really have any "aha!" moments of mathematical awe. Again, I imagine this would not be the case for a more educated reader, but I can only write about my experience.
To further enrage the imaginary elitist math PhD strawman reading this review, let me add this: at times, I couldn't avoid thinking that some of the ideas here be so much easier to visualize and undertsand in a video format... I'm certain that if God has a "BOOK" with the best proofs, then it's surely matched by their equally impressive "PLAYLIST".
In conclusion, this book is probably very interesting and inspiring for a specific group of people. Non-formally-educated quasi-zoomers like me will find more enjoyment elsewhere.
For the few parts of the book that I did understand, they... didn't seem that impressive? There were a few results that were nice (many I was already familiar with), but I didn't really have any "aha!" moments of mathematical awe. Again, I imagine this would not be the case for a more educated reader, but I can only write about my experience.
To further enrage the imaginary elitist math PhD strawman reading this review, let me add this: at times, I couldn't avoid thinking that some of the ideas here be so much easier to visualize and undertsand in a video format... I'm certain that if God has a "BOOK" with the best proofs, then it's surely matched by their equally impressive "PLAYLIST".
In conclusion, this book is probably very interesting and inspiring for a specific group of people. Non-formally-educated quasi-zoomers like me will find more enjoyment elsewhere.
December 9, 2021
This book is horrible. It is a complete failure from its stated purpose. “…everything in this book is supposed to be accessible to readers whose backgrounds include a modest amount of technique from undergraduate mathematics.” Everything in this book is advanced mathematics suitable for maybe those that have advanced degrees in math. The explanations are atrocious. Listening to Khan academy videos is more enlightening and more accessible than anything in this book.
October 16, 2023
I loaned this to a friend in their 80's after reading half of it and I think they forgot and now I'm waiting for them to die so that I can get it back because I miss it.
It's a great book for ex-math academics who like textbooks but only the fun parts. Or gimmicky/weird combinatorial arguments. This may only be me.
It's a great book for ex-math academics who like textbooks but only the fun parts. Or gimmicky/weird combinatorial arguments. This may only be me.
January 5, 2023
I love the concept of this book and the authors chose some very elegant proofs to include in it. A little less approachable than I would have liked (as a second year mathematics undergraduate) but still very readable.
January 2, 2019
Good book
February 24, 2019
This is a must have for anyone who wants to write proofs better.
June 21, 2022
Yass LOL
August 22, 2014
This book gives relatively elegant proofs of theorems from many different fields of mathematics, and often gives multiple proofs for the same theorem. It does require some calculus and linear algebra background, but if you have that, you should be able to follow most of the proofs in this book. There were a couple of times where I thought there was a more elegant proof that was not included (such as Kempe's proof of the 5 color theorem), but still, this collection of proofs is as good as advertised.
April 10, 2010
A collection of some beautiful results from number theory, combinatorics, and geometry.
March 31, 2015
The book-in Erdos's sense- is not completed, but it is still a wonderful one!
March 7, 2023
Excellent book!!!
Displaying 1 - 20 of 20 reviews