When Cambridge mathematician Andrew Wiles announced a solution for Fermat's last theorem in 1993, it electrified the world of mathematics. After a flaw was discovered in the proof, Wiles had to work for another year--he had already laboured in solitude for seven years--to establish that he had solved the 350-year-old problem. Simon Singh's book is a lively, comprehensible explanation of Wiles's work and of the colourful history that has build up around Fermat's last theorem over the years. The book contains some problems that offer a taste for the maths, but it also includes limericks to give a feeling for the quirkier side of mathematicians.

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Bit of a guilty pleasure, 21 Dec 2008
A book basically consisting of a list of numbers and their interesting properties can only appeal to a very small percentage of readers, but those readers will love it to the hilt.

Wells starts from 0 and reaches Graham's Number (so big it can't be written down) and all manner of each number's unique quirks are displayed. For example, 10213223 is a self-describing number (one zero, two ones, three twos and two threes), and 153 is the only number to be the sum of the cubes of its digits. If you find such things interesting, this book is a treasure trove.

By design, it's not really the kind of thing you can read through in one sitting (like a novel), but is something to be dipped into in idle moments. As a previous reviewer said, it's the perfect toilet read, and that's most definitely a complement.

Doesn't read like a dictionary, 08 May 2003
David Wells has assembled an unique and readable collection of facts about numbers, arranged in numerical order! Entries are fascinating, for the most part, though they can be frustrating, too. For example, when first encountering the notion of automorphic numbers (numbers whose squares end in the same digits as the original number), it is tempting to discover if there are other related entries -- 'trimorphic numbers', for instance? It is possible to track these down using the small index provided and quite fun to do.
Unlike other dictionaries this is best read from front to back though it can be used as a reference, once one is familiar with it.
Many concepts are briefly explained as they are encountered, and some merely referred to in passing, and the frustration here is that there need not be full information in the text. However, this is most enjoyably resolved by brushing up one's own skills and trying to demonstrate the simpler claims for oneself. There is plenty here for the dabbling amateur to try, though the serious mathematician can also enjoy the book.
I have one qualification: David Wells identifies 51 as the least uninteresting number (no, not a contradiction: it is simultaneously interesting and uninteresting, he claims) -- because he cannot find an interesting fact about it. He fails to notice that it is the fourth trimorphic (and non-automorphic) number: 4, 9, 49, 51 and 75 being the first five cases. This means that it is mildly more interesting than at first supposed.
I look forward to a revised edition -- with readers' contributions and comments.

Takes pride of place in the loo, 23 Oct 2001
Books which are great for dipping into for a few minutes take pride of place in the loo - this one included. It is just tremendous - full of interesting stuff for any geeks who like numbers and maths. You'll come back to this book time and time again - the loo becomes a more inviting place with this book.

No recreational mathematician should be without it, 10 Dec 2000
In the foreword to G.H. Hardy's book A Mathematician's Apology, C.P. Snow tells an anecdote about Hardy and his collaborator Srinavasa Ramanujan. Hardy, perhaps the greatest number theorist of 20th century, took a taxi from London to the hospital at Putney where Ramanujan was dying of tuberculosis, Hardy noticed its number, 1729. Always inept about introducing a conversation, he entered the room where Ramanujan was lying in bed and, with scarcely a hello, blurted out his opinion about the taxi-cab number. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy! No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Usually it takes a great deal of insight as well as considerable mathematical training to discover a yet unknown properties of some number. Only recognizing the beauty of a number pattern is much easier, though, especially with a friendly book like this one on hand. Wells, a long-time mathematics popularizer, has collected over 1000 numbers he considers interesting. Each of them is given a short explanation, often accompanied with a bibliographic reference. Celebrities among the numbers, like i, e or Pi, are given a more comprehensive treatment. Included are also several sequences, like Fibonacci's, Mersenne's, Fermat's, Carmichael's or Kaprekar's, each accompanied with its explanation. So are cyclic, amicable, untouchable or lucky numbers, and many more sequences you probably didn't know about.

While Wells' dictionary certainly gives the impression of a well-researched work, the list of numbers is by no means exhaustive. Anyone familiar with chaos theory will notice the absence of Feigenbaum constant; prime hunters would probably be interested in discussion on Woodall primes, Sophie-Germain primes, or Proth primes. But they are better off with Paulo Ribenboim's book on primes, anyway, while Wells' book, with its easily understandable explanations and accessible price is probably more suited for the "recreational mathematics" audience.

Bedside reading, 30 Nov 2000
This is the kind of thing you read before going to bed, especially if you have a geeky gene. The pseudo-victorian title alone is lovable, although the book itself is a bit disappointing if, like me, you aren't into mathematics. The anecdotes and personal stories of mathematicians are interesting, but the majority of entries simply describe the number in question as being 'the start of a remarkable chain of amicable numbers', or consist of formulae. And it might well give you traumatic memories of school.

It's worth it for the entry on 48 (I think), which states it to be the first uninteresting number, and thus interesting for being so.

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Bit of a guilty pleasure, 21 Dec 2008
A book basically consisting of a list of numbers and their interesting properties can only appeal to a very small percentage of readers, but those readers will love it to the hilt.

Wells starts from 0 and reaches Graham's Number (so big it can't be written down) and all manner of each number's unique quirks are displayed. For example, 10213223 is a self-describing number (one zero, two ones, three twos and two threes), and 153 is the only number to be the sum of the cubes of its digits. If you find such things interesting, this book is a treasure trove.

By design, it's not really the kind of thing you can read through in one sitting (like a novel), but is something to be dipped into in idle moments. As a previous reviewer said, it's the perfect toilet read, and that's most definitely a complement.

Doesn't read like a dictionary, 08 May 2003
David Wells has assembled an unique and readable collection of facts about numbers, arranged in numerical order! Entries are fascinating, for the most part, though they can be frustrating, too. For example, when first encountering the notion of automorphic numbers (numbers whose squares end in the same digits as the original number), it is tempting to discover if there are other related entries -- 'trimorphic numbers', for instance? It is possible to track these down using the small index provided and quite fun to do.
Unlike other dictionaries this is best read from front to back though it can be used as a reference, once one is familiar with it.
Many concepts are briefly explained as they are encountered, and some merely referred to in passing, and the frustration here is that there need not be full information in the text. However, this is most enjoyably resolved by brushing up one's own skills and trying to demonstrate the simpler claims for oneself. There is plenty here for the dabbling amateur to try, though the serious mathematician can also enjoy the book.
I have one qualification: David Wells identifies 51 as the least uninteresting number (no, not a contradiction: it is simultaneously interesting and uninteresting, he claims) -- because he cannot find an interesting fact about it. He fails to notice that it is the fourth trimorphic (and non-automorphic) number: 4, 9, 49, 51 and 75 being the first five cases. This means that it is mildly more interesting than at first supposed.
I look forward to a revised edition -- with readers' contributions and comments.

Takes pride of place in the loo, 23 Oct 2001
Books which are great for dipping into for a few minutes take pride of place in the loo - this one included. It is just tremendous - full of interesting stuff for any geeks who like numbers and maths. You'll come back to this book time and time again - the loo becomes a more inviting place with this book.

No recreational mathematician should be without it, 10 Dec 2000
In the foreword to G.H. Hardy's book A Mathematician's Apology, C.P. Snow tells an anecdote about Hardy and his collaborator Srinavasa Ramanujan. Hardy, perhaps the greatest number theorist of 20th century, took a taxi from London to the hospital at Putney where Ramanujan was dying of tuberculosis, Hardy noticed its number, 1729. Always inept about introducing a conversation, he entered the room where Ramanujan was lying in bed and, with scarcely a hello, blurted out his opinion about the taxi-cab number. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy! No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Usually it takes a great deal of insight as well as considerable mathematical training to discover a yet unknown properties of some number. Only recognizing the beauty of a number pattern is much easier, though, especially with a friendly book like this one on hand. Wells, a long-time mathematics popularizer, has collected over 1000 numbers he considers interesting. Each of them is given a short explanation, often accompanied with a bibliographic reference. Celebrities among the numbers, like i, e or Pi, are given a more comprehensive treatment. Included are also several sequences, like Fibonacci's, Mersenne's, Fermat's, Carmichael's or Kaprekar's, each accompanied with its explanation. So are cyclic, amicable, untouchable or lucky numbers, and many more sequences you probably didn't know about.

While Wells' dictionary certainly gives the impression of a well-researched work, the list of numbers is by no means exhaustive. Anyone familiar with chaos theory will notice the absence of Feigenbaum constant; prime hunters would probably be interested in discussion on Woodall primes, Sophie-Germain primes, or Proth primes. But they are better off with Paulo Ribenboim's book on primes, anyway, while Wells' book, with its easily understandable explanations and accessible price is probably more suited for the "recreational mathematics" audience.

Bedside reading, 30 Nov 2000
This is the kind of thing you read before going to bed, especially if you have a geeky gene. The pseudo-victorian title alone is lovable, although the book itself is a bit disappointing if, like me, you aren't into mathematics. The anecdotes and personal stories of mathematicians are interesting, but the majority of entries simply describe the number in question as being 'the start of a remarkable chain of amicable numbers', or consist of formulae. And it might well give you traumatic memories of school.

It's worth it for the entry on 48 (I think), which states it to be the first uninteresting number, and thus interesting for being so.

The most superb introduction to Sacred Geometry ever!, 12 Jul 2005
I cannot rate this book too highly. When I ordered it, I presumed from the cover artwork (and the title) that it would be a kind of "Sacred Geometry for Dummies". I was, in fact, looking for a relatively easy primer to help me grasp the basic principles of this hugely deep and densely arcane subject, but what I did not anticipate was that it would be both exceptionally accessible as well as intellectually satisfying. Our culture has no innate understanding of "the mathematical archetypes of nature, art, and science" (the sub-title) and in fact, our culture, being so hamstrung, so crippled by left-brain dogma, does not even consider that nature, art and science could have any possible common denominator to even discuss! That the author could express ideas that are really so beyond our normal Western rationale, and moreover, does so with such intelligence and yet with such a light touch, is quite an extraordinary feat. The text is profusely illustrated with diagrams and drawings that precisely explain all that is required, and is also littered with hundreds (this I really appreciated) of superbly chosen quotations from all the great minds of history, from all cultures. I do not believe that there exists a better introduction to this deep and wonderful wisdom, and I would gladly give it 9 stars.

Even after an hour browsing through it, I felt that I had absorbed levels of knowledge, of perception, that were not there previously. This book is about a level of mathematics that, shamefully, is not taught in our educational system. So much for our so-called "progressive' modern culture, that the real pearls have been disregarded in favour of something that has had all knowledge, life and magic stripped away. All spiritual qualities, in fact. What does mathematics and geometry mean to anybody today, other than for the most common-place pedestrian purposes? I did not know, prior to this book, that the Greek "mathema" signifies "learning in general" and was the root of the Old English "mathein", "to be aware" and the Old German "munthen", meaning "to awaken". Today the word "maths" has, for most people, constricted its scope to emphasise mundane measurement and mere manipulation of quantities.

The only, and I mean only, quibble I have with the book, is that it is printed on paper of a quality that does not do justice to the content. This is a shame, but perhaps it's only applicable to the HarperCollins paperback publication that I bought. According to an inner page giving the ISBN details, there was a hard cover version published in 1994, also HarperCollins, and I would assume that that would be printed on better stock. But don't let this put you off. If you have any interest whatsoever in numbers, mathematics, higher knowledge, or any curiosity whatsoever about why the world is designed the way it is, buy this book. It contains the entire universe!

An Esoteric Feast, 30 Aug 2002
The title is grandiose, the book's layout makes you travel sick when you read it, Michael's brain is clearly on over drive. I loved every line, enjoyed every quote, was intrigued by all the images. It helped me unify many thoughts I had in this field. He has presented sacred geometry in a non-pretentious and easy-to-understand way. He will stimulate you to look further in the subject and you will surely start to see the world in a different way. Why can I not find any other books by this man?

Sensational!, 13 Sep 2000
Once every few years you come across a book which is genuinely life-changing, and this is one of them. In its simplest form it's a book which makes numbers interesting - completely different from the usual dull stuff you did at school. But it goes way beyond that, to open up a whole world of cosmic geometry which you'd never noticed before. It explains the principles of number and sacred geometry with extraordinary clarity and applies them to architecture, religion, ancient art, modern design, science, the natural world, etc, on the basis that all forms of construction and creation work on those same principles. It covers everything, from the reason why manhole covers are round to the divine emanations of the Qabalah, and links the physical with the spiritual in ways you would never have thought of. The book has a wide format and very clear layout, with lots of illustrations, and assumes no previous knowledge of - or interest in - the subject.

I notice in Mr Schneider's comments that it took him 20 years to research the book. Well, it certainly shows, and I can assure him it was very well worth it. This is truly a revelation, an amazing book which will completely change the way you look at the world.

A quintessential presentation., 12 Aug 1999
I have always been mesmerized by mathematics and its infinte implications; one could call it impassioned. Many people do not share my zeal. Some of these people are my family, friends, and associates. FINALLY I have a book to recommend that can open each and every one of them to unsuspected dimensions of this absolutely fascinating subject; the presentation alone will, I am sure, compel them to read on and on and on. I am awed at the scientific authenticity and gentleness with which Schneider creates such a sensible, spiritual, and harmonious synthesis. Utterly awed...and so very grateful. I'd love to see an index in the next edition.

It was difficult to put the book down long enough to type these comments!

An addictive adventure in the history of natural math., 28 Nov 1998
It was very difficult to put this book down. Not only does Schneider evince a love and profound knowledge of his material, but he communicates his passion to the reader with a clarity uncharacteristic of many math exposeurs. It is a perfect teaching vehicle for parents and their children to foster a heartfelt respect for the mathematical majesty of nature, using examples from cultural history across the globe. His discussion of music and symbolic geometry is especially enlightening. This book should be required reading for any and all educators. A wonderful read!

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Bit of a guilty pleasure, 21 Dec 2008
A book basically consisting of a list of numbers and their interesting properties can only appeal to a very small percentage of readers, but those readers will love it to the hilt.

Wells starts from 0 and reaches Graham's Number (so big it can't be written down) and all manner of each number's unique quirks are displayed. For example, 10213223 is a self-describing number (one zero, two ones, three twos and two threes), and 153 is the only number to be the sum of the cubes of its digits. If you find such things interesting, this book is a treasure trove.

By design, it's not really the kind of thing you can read through in one sitting (like a novel), but is something to be dipped into in idle moments. As a previous reviewer said, it's the perfect toilet read, and that's most definitely a complement.

Doesn't read like a dictionary, 08 May 2003
David Wells has assembled an unique and readable collection of facts about numbers, arranged in numerical order! Entries are fascinating, for the most part, though they can be frustrating, too. For example, when first encountering the notion of automorphic numbers (numbers whose squares end in the same digits as the original number), it is tempting to discover if there are other related entries -- 'trimorphic numbers', for instance? It is possible to track these down using the small index provided and quite fun to do.
Unlike other dictionaries this is best read from front to back though it can be used as a reference, once one is familiar with it.
Many concepts are briefly explained as they are encountered, and some merely referred to in passing, and the frustration here is that there need not be full information in the text. However, this is most enjoyably resolved by brushing up one's own skills and trying to demonstrate the simpler claims for oneself. There is plenty here for the dabbling amateur to try, though the serious mathematician can also enjoy the book.
I have one qualification: David Wells identifies 51 as the least uninteresting number (no, not a contradiction: it is simultaneously interesting and uninteresting, he claims) -- because he cannot find an interesting fact about it. He fails to notice that it is the fourth trimorphic (and non-automorphic) number: 4, 9, 49, 51 and 75 being the first five cases. This means that it is mildly more interesting than at first supposed.
I look forward to a revised edition -- with readers' contributions and comments.

Takes pride of place in the loo, 23 Oct 2001
Books which are great for dipping into for a few minutes take pride of place in the loo - this one included. It is just tremendous - full of interesting stuff for any geeks who like numbers and maths. You'll come back to this book time and time again - the loo becomes a more inviting place with this book.

No recreational mathematician should be without it, 10 Dec 2000
In the foreword to G.H. Hardy's book A Mathematician's Apology, C.P. Snow tells an anecdote about Hardy and his collaborator Srinavasa Ramanujan. Hardy, perhaps the greatest number theorist of 20th century, took a taxi from London to the hospital at Putney where Ramanujan was dying of tuberculosis, Hardy noticed its number, 1729. Always inept about introducing a conversation, he entered the room where Ramanujan was lying in bed and, with scarcely a hello, blurted out his opinion about the taxi-cab number. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy! No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Usually it takes a great deal of insight as well as considerable mathematical training to discover a yet unknown properties of some number. Only recognizing the beauty of a number pattern is much easier, though, especially with a friendly book like this one on hand. Wells, a long-time mathematics popularizer, has collected over 1000 numbers he considers interesting. Each of them is given a short explanation, often accompanied with a bibliographic reference. Celebrities among the numbers, like i, e or Pi, are given a more comprehensive treatment. Included are also several sequences, like Fibonacci's, Mersenne's, Fermat's, Carmichael's or Kaprekar's, each accompanied with its explanation. So are cyclic, amicable, untouchable or lucky numbers, and many more sequences you probably didn't know about.

While Wells' dictionary certainly gives the impression of a well-researched work, the list of numbers is by no means exhaustive. Anyone familiar with chaos theory will notice the absence of Feigenbaum constant; prime hunters would probably be interested in discussion on Woodall primes, Sophie-Germain primes, or Proth primes. But they are better off with Paulo Ribenboim's book on primes, anyway, while Wells' book, with its easily understandable explanations and accessible price is probably more suited for the "recreational mathematics" audience.

Bedside reading, 30 Nov 2000
This is the kind of thing you read before going to bed, especially if you have a geeky gene. The pseudo-victorian title alone is lovable, although the book itself is a bit disappointing if, like me, you aren't into mathematics. The anecdotes and personal stories of mathematicians are interesting, but the majority of entries simply describe the number in question as being 'the start of a remarkable chain of amicable numbers', or consist of formulae. And it might well give you traumatic memories of school.

It's worth it for the entry on 48 (I think), which states it to be the first uninteresting number, and thus interesting for being so.

The most superb introduction to Sacred Geometry ever!, 12 Jul 2005
I cannot rate this book too highly. When I ordered it, I presumed from the cover artwork (and the title) that it would be a kind of "Sacred Geometry for Dummies". I was, in fact, looking for a relatively easy primer to help me grasp the basic principles of this hugely deep and densely arcane subject, but what I did not anticipate was that it would be both exceptionally accessible as well as intellectually satisfying. Our culture has no innate understanding of "the mathematical archetypes of nature, art, and science" (the sub-title) and in fact, our culture, being so hamstrung, so crippled by left-brain dogma, does not even consider that nature, art and science could have any possible common denominator to even discuss! That the author could express ideas that are really so beyond our normal Western rationale, and moreover, does so with such intelligence and yet with such a light touch, is quite an extraordinary feat. The text is profusely illustrated with diagrams and drawings that precisely explain all that is required, and is also littered with hundreds (this I really appreciated) of superbly chosen quotations from all the great minds of history, from all cultures. I do not believe that there exists a better introduction to this deep and wonderful wisdom, and I would gladly give it 9 stars.

Even after an hour browsing through it, I felt that I had absorbed levels of knowledge, of perception, that were not there previously. This book is about a level of mathematics that, shamefully, is not taught in our educational system. So much for our so-called "progressive' modern culture, that the real pearls have been disregarded in favour of something that has had all knowledge, life and magic stripped away. All spiritual qualities, in fact. What does mathematics and geometry mean to anybody today, other than for the most common-place pedestrian purposes? I did not know, prior to this book, that the Greek "mathema" signifies "learning in general" and was the root of the Old English "mathein", "to be aware" and the Old German "munthen", meaning "to awaken". Today the word "maths" has, for most people, constricted its scope to emphasise mundane measurement and mere manipulation of quantities.

The only, and I mean only, quibble I have with the book, is that it is printed on paper of a quality that does not do justice to the content. This is a shame, but perhaps it's only applicable to the HarperCollins paperback publication that I bought. According to an inner page giving the ISBN details, there was a hard cover version published in 1994, also HarperCollins, and I would assume that that would be printed on better stock. But don't let this put you off. If you have any interest whatsoever in numbers, mathematics, higher knowledge, or any curiosity whatsoever about why the world is designed the way it is, buy this book. It contains the entire universe!

An Esoteric Feast, 30 Aug 2002
The title is grandiose, the book's layout makes you travel sick when you read it, Michael's brain is clearly on over drive. I loved every line, enjoyed every quote, was intrigued by all the images. It helped me unify many thoughts I had in this field. He has presented sacred geometry in a non-pretentious and easy-to-understand way. He will stimulate you to look further in the subject and you will surely start to see the world in a different way. Why can I not find any other books by this man?

Sensational!, 13 Sep 2000
Once every few years you come across a book which is genuinely life-changing, and this is one of them. In its simplest form it's a book which makes numbers interesting - completely different from the usual dull stuff you did at school. But it goes way beyond that, to open up a whole world of cosmic geometry which you'd never noticed before. It explains the principles of number and sacred geometry with extraordinary clarity and applies them to architecture, religion, ancient art, modern design, science, the natural world, etc, on the basis that all forms of construction and creation work on those same principles. It covers everything, from the reason why manhole covers are round to the divine emanations of the Qabalah, and links the physical with the spiritual in ways you would never have thought of. The book has a wide format and very clear layout, with lots of illustrations, and assumes no previous knowledge of - or interest in - the subject.

I notice in Mr Schneider's comments that it took him 20 years to research the book. Well, it certainly shows, and I can assure him it was very well worth it. This is truly a revelation, an amazing book which will completely change the way you look at the world.

A quintessential presentation., 12 Aug 1999
I have always been mesmerized by mathematics and its infinte implications; one could call it impassioned. Many people do not share my zeal. Some of these people are my family, friends, and associates. FINALLY I have a book to recommend that can open each and every one of them to unsuspected dimensions of this absolutely fascinating subject; the presentation alone will, I am sure, compel them to read on and on and on. I am awed at the scientific authenticity and gentleness with which Schneider creates such a sensible, spiritual, and harmonious synthesis. Utterly awed...and so very grateful. I'd love to see an index in the next edition.

It was difficult to put the book down long enough to type these comments!

An addictive adventure in the history of natural math., 28 Nov 1998
It was very difficult to put this book down. Not only does Schneider evince a love and profound knowledge of his material, but he communicates his passion to the reader with a clarity uncharacteristic of many math exposeurs. It is a perfect teaching vehicle for parents and their children to foster a heartfelt respect for the mathematical majesty of nature, using examples from cultural history across the globe. His discussion of music and symbolic geometry is especially enlightening. This book should be required reading for any and all educators. A wonderful read!

A very useful book for anyone thinking of doing Mathematics at university., 31 Oct 2006
I am a first year student at Imperial College (where Professor Liebeck lectures) and I have to say that this book has really helped me. It was on the reading list that the university gave to me, so over the summer I used he book as a study aid.

Liebeck writes clearly and concisely, presenting the mathematics in an easy to understand way. At the same time the material covered is more challenging than at A-Level (which I found to be a bit repetitive) and will stimulate all students, regardless of their ability.

An Excellent Introduction to Pure Mathematics, 17 Sep 2006
I had a privilege of attending a first-year course at Imperial College, based on Prof. Martin Liebeck's book. The book, as well as the course (then taught by Prof. Kevin Buzzard), are superb. They are readily accessible to first-year university students and provide an easy transition from A-level to undergraduate mathematics. Moreover, the language is clear and concise, the examples instructive, and the book is generally fun to read. Liebeck selects some of the most interesting topics in elementary pure mathematics and stimulates the student's interest in the subject. Unfortunately, A-level mathematics is taught as a collection of algorithms, and the student may not be able to appreciate its depth and beauty. Whether you are a first-year mathematics undergraduate, or still at school, I would thoroughly recommend you to read this book so that you know what mathematics is really about.

An excellent introduction to university mathematics, 27 Sep 2004
The gap between high school and university mathematics is
quite noticeable. I found this book to be an excellent book
to prep a smooth landing to university mathematics.
(The best one out of a long list of other similar books
I had a look at)

Starts of really easy and clear but still goes beyond the
"surface" when required.
The chapters are structured very short, which I thought
was a good thing.

It has a lot of worked examples.
However, the book does not have solutions to the
end-of-chapter exercises, which I thought was a long minus
since I was reading the book on my own as a self study..

But all in all, a very enjoyable book to read!

Very useful, 11 Sep 2004
This book is ideal for A-level students who are considering doing a numerate degree, particularly maths. It contains lots of useful methods and tricks, with full proofs of every theorem. It isn't highly technical, nor does it go into much depth, but it is an excellent primer and will make you realise some of the amazing things that can be proved quite simply with the right concepts.

Enjoyable and educational, 05 Sep 2000
A concise Introduction to Pure Mathematics is very legible, it is written so that it is absorbed easily, it intoduces many prime topics including a very extensive and clear section on Integration

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Bit of a guilty pleasure, 21 Dec 2008
A book basically consisting of a list of numbers and their interesting properties can only appeal to a very small percentage of readers, but those readers will love it to the hilt.

Wells starts from 0 and reaches Graham's Number (so big it can't be written down) and all manner of each number's unique quirks are displayed. For example, 10213223 is a self-describing number (one zero, two ones, three twos and two threes), and 153 is the only number to be the sum of the cubes of its digits. If you find such things interesting, this book is a treasure trove.

By design, it's not really the kind of thing you can read through in one sitting (like a novel), but is something to be dipped into in idle moments. As a previous reviewer said, it's the perfect toilet read, and that's most definitely a complement.

Doesn't read like a dictionary, 08 May 2003
David Wells has assembled an unique and readable collection of facts about numbers, arranged in numerical order! Entries are fascinating, for the most part, though they can be frustrating, too. For example, when first encountering the notion of automorphic numbers (numbers whose squares end in the same digits as the original number), it is tempting to discover if there are other related entries -- 'trimorphic numbers', for instance? It is possible to track these down using the small index provided and quite fun to do.
Unlike other dictionaries this is best read from front to back though it can be used as a reference, once one is familiar with it.
Many concepts are briefly explained as they are encountered, and some merely referred to in passing, and the frustration here is that there need not be full information in the text. However, this is most enjoyably resolved by brushing up one's own skills and trying to demonstrate the simpler claims for oneself. There is plenty here for the dabbling amateur to try, though the serious mathematician can also enjoy the book.
I have one qualification: David Wells identifies 51 as the least uninteresting number (no, not a contradiction: it is simultaneously interesting and uninteresting, he claims) -- because he cannot find an interesting fact about it. He fails to notice that it is the fourth trimorphic (and non-automorphic) number: 4, 9, 49, 51 and 75 being the first five cases. This means that it is mildly more interesting than at first supposed.
I look forward to a revised edition -- with readers' contributions and comments.

Takes pride of place in the loo, 23 Oct 2001
Books which are great for dipping into for a few minutes take pride of place in the loo - this one included. It is just tremendous - full of interesting stuff for any geeks who like numbers and maths. You'll come back to this book time and time again - the loo becomes a more inviting place with this book.

No recreational mathematician should be without it, 10 Dec 2000
In the foreword to G.H. Hardy's book A Mathematician's Apology, C.P. Snow tells an anecdote about Hardy and his collaborator Srinavasa Ramanujan. Hardy, perhaps the greatest number theorist of 20th century, took a taxi from London to the hospital at Putney where Ramanujan was dying of tuberculosis, Hardy noticed its number, 1729. Always inept about introducing a conversation, he entered the room where Ramanujan was lying in bed and, with scarcely a hello, blurted out his opinion about the taxi-cab number. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy! No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Usually it takes a great deal of insight as well as considerable mathematical training to discover a yet unknown properties of some number. Only recognizing the beauty of a number pattern is much easier, though, especially with a friendly book like this one on hand. Wells, a long-time mathematics popularizer, has collected over 1000 numbers he considers interesting. Each of them is given a short explanation, often accompanied with a bibliographic reference. Celebrities among the numbers, like i, e or Pi, are given a more comprehensive treatment. Included are also several sequences, like Fibonacci's, Mersenne's, Fermat's, Carmichael's or Kaprekar's, each accompanied with its explanation. So are cyclic, amicable, untouchable or lucky numbers, and many more sequences you probably didn't know about.

While Wells' dictionary certainly gives the impression of a well-researched work, the list of numbers is by no means exhaustive. Anyone familiar with chaos theory will notice the absence of Feigenbaum constant; prime hunters would probably be interested in discussion on Woodall primes, Sophie-Germain primes, or Proth primes. But they are better off with Paulo Ribenboim's book on primes, anyway, while Wells' book, with its easily understandable explanations and accessible price is probably more suited for the "recreational mathematics" audience.

Bedside reading, 30 Nov 2000
This is the kind of thing you read before going to bed, especially if you have a geeky gene. The pseudo-victorian title alone is lovable, although the book itself is a bit disappointing if, like me, you aren't into mathematics. The anecdotes and personal stories of mathematicians are interesting, but the majority of entries simply describe the number in question as being 'the start of a remarkable chain of amicable numbers', or consist of formulae. And it might well give you traumatic memories of school.

It's worth it for the entry on 48 (I think), which states it to be the first uninteresting number, and thus interesting for being so.

The most superb introduction to Sacred Geometry ever!, 12 Jul 2005
I cannot rate this book too highly. When I ordered it, I presumed from the cover artwork (and the title) that it would be a kind of "Sacred Geometry for Dummies". I was, in fact, looking for a relatively easy primer to help me grasp the basic principles of this hugely deep and densely arcane subject, but what I did not anticipate was that it would be both exceptionally accessible as well as intellectually satisfying. Our culture has no innate understanding of "the mathematical archetypes of nature, art, and science" (the sub-title) and in fact, our culture, being so hamstrung, so crippled by left-brain dogma, does not even consider that nature, art and science could have any possible common denominator to even discuss! That the author could express ideas that are really so beyond our normal Western rationale, and moreover, does so with such intelligence and yet with such a light touch, is quite an extraordinary feat. The text is profusely illustrated with diagrams and drawings that precisely explain all that is required, and is also littered with hundreds (this I really appreciated) of superbly chosen quotations from all the great minds of history, from all cultures. I do not believe that there exists a better introduction to this deep and wonderful wisdom, and I would gladly give it 9 stars.

Even after an hour browsing through it, I felt that I had absorbed levels of knowledge, of perception, that were not there previously. This book is about a level of mathematics that, shamefully, is not taught in our educational system. So much for our so-called "progressive' modern culture, that the real pearls have been disregarded in favour of something that has had all knowledge, life and magic stripped away. All spiritual qualities, in fact. What does mathematics and geometry mean to anybody today, other than for the most common-place pedestrian purposes? I did not know, prior to this book, that the Greek "mathema" signifies "learning in general" and was the root of the Old English "mathein", "to be aware" and the Old German "munthen", meaning "to awaken". Today the word "maths" has, for most people, constricted its scope to emphasise mundane measurement and mere manipulation of quantities.

The only, and I mean only, quibble I have with the book, is that it is printed on paper of a quality that does not do justice to the content. This is a shame, but perhaps it's only applicable to the HarperCollins paperback publication that I bought. According to an inner page giving the ISBN details, there was a hard cover version published in 1994, also HarperCollins, and I would assume that that would be printed on better stock. But don't let this put you off. If you have any interest whatsoever in numbers, mathematics, higher knowledge, or any curiosity whatsoever about why the world is designed the way it is, buy this book. It contains the entire universe!

An Esoteric Feast, 30 Aug 2002
The title is grandiose, the book's layout makes you travel sick when you read it, Michael's brain is clearly on over drive. I loved every line, enjoyed every quote, was intrigued by all the images. It helped me unify many thoughts I had in this field. He has presented sacred geometry in a non-pretentious and easy-to-understand way. He will stimulate you to look further in the subject and you will surely start to see the world in a different way. Why can I not find any other books by this man?

Sensational!, 13 Sep 2000
Once every few years you come across a book which is genuinely life-changing, and this is one of them. In its simplest form it's a book which makes numbers interesting - completely different from the usual dull stuff you did at school. But it goes way beyond that, to open up a whole world of cosmic geometry which you'd never noticed before. It explains the principles of number and sacred geometry with extraordinary clarity and applies them to architecture, religion, ancient art, modern design, science, the natural world, etc, on the basis that all forms of construction and creation work on those same principles. It covers everything, from the reason why manhole covers are round to the divine emanations of the Qabalah, and links the physical with the spiritual in ways you would never have thought of. The book has a wide format and very clear layout, with lots of illustrations, and assumes no previous knowledge of - or interest in - the subject.

I notice in Mr Schneider's comments that it took him 20 years to research the book. Well, it certainly shows, and I can assure him it was very well worth it. This is truly a revelation, an amazing book which will completely change the way you look at the world.

A quintessential presentation., 12 Aug 1999
I have always been mesmerized by mathematics and its infinte implications; one could call it impassioned. Many people do not share my zeal. Some of these people are my family, friends, and associates. FINALLY I have a book to recommend that can open each and every one of them to unsuspected dimensions of this absolutely fascinating subject; the presentation alone will, I am sure, compel them to read on and on and on. I am awed at the scientific authenticity and gentleness with which Schneider creates such a sensible, spiritual, and harmonious synthesis. Utterly awed...and so very grateful. I'd love to see an index in the next edition.

It was difficult to put the book down long enough to type these comments!

An addictive adventure in the history of natural math., 28 Nov 1998
It was very difficult to put this book down. Not only does Schneider evince a love and profound knowledge of his material, but he communicates his passion to the reader with a clarity uncharacteristic of many math exposeurs. It is a perfect teaching vehicle for parents and their children to foster a heartfelt respect for the mathematical majesty of nature, using examples from cultural history across the globe. His discussion of music and symbolic geometry is especially enlightening. This book should be required reading for any and all educators. A wonderful read!

A very useful book for anyone thinking of doing Mathematics at university., 31 Oct 2006
I am a first year student at Imperial College (where Professor Liebeck lectures) and I have to say that this book has really helped me. It was on the reading list that the university gave to me, so over the summer I used he book as a study aid.

Liebeck writes clearly and concisely, presenting the mathematics in an easy to understand way. At the same time the material covered is more challenging than at A-Level (which I found to be a bit repetitive) and will stimulate all students, regardless of their ability.

An Excellent Introduction to Pure Mathematics, 17 Sep 2006
I had a privilege of attending a first-year course at Imperial College, based on Prof. Martin Liebeck's book. The book, as well as the course (then taught by Prof. Kevin Buzzard), are superb. They are readily accessible to first-year university students and provide an easy transition from A-level to undergraduate mathematics. Moreover, the language is clear and concise, the examples instructive, and the book is generally fun to read. Liebeck selects some of the most interesting topics in elementary pure mathematics and stimulates the student's interest in the subject. Unfortunately, A-level mathematics is taught as a collection of algorithms, and the student may not be able to appreciate its depth and beauty. Whether you are a first-year mathematics undergraduate, or still at school, I would thoroughly recommend you to read this book so that you know what mathematics is really about.

An excellent introduction to university mathematics, 27 Sep 2004
The gap between high school and university mathematics is
quite noticeable. I found this book to be an excellent book
to prep a smooth landing to university mathematics.
(The best one out of a long list of other similar books
I had a look at)

Starts of really easy and clear but still goes beyond the
"surface" when required.
The chapters are structured very short, which I thought
was a good thing.

It has a lot of worked examples.
However, the book does not have solutions to the
end-of-chapter exercises, which I thought was a long minus
since I was reading the book on my own as a self study..

But all in all, a very enjoyable book to read!

Very useful, 11 Sep 2004
This book is ideal for A-level students who are considering doing a numerate degree, particularly maths. It contains lots of useful methods and tricks, with full proofs of every theorem. It isn't highly technical, nor does it go into much depth, but it is an excellent primer and will make you realise some of the amazing things that can be proved quite simply with the right concepts.

Enjoyable and educational, 05 Sep 2000
A concise Introduction to Pure Mathematics is very legible, it is written so that it is absorbed easily, it intoduces many prime topics including a very extensive and clear section on Integration

Number Theory from the roots up, 29 Nov 2008
Very helpfull book, I am having no lessons on number theory, yet the chapters I have been through I feel very confident with. The style of showing you one proof, helping you understand it, and then asking you to adapt it to a similar situation is very confidence building. The last chapter is for the very brave though. It deals with fermat's last theorem in a bit more depth than fans of Simon Singh's "fermat's last theorem" might be familiar with (though he is a very good author).

Buy this if you want a good grounding in the essentials of number theory, but remember that it is most helpful if you give some of the sections more than a "once-over".

Superb introduction to number theory, 25 Jun 2003
This book is one I would have liked to have read when I was an undergraduate. It is quite the best mix of 'old style' and 'new style' number theory that I have ever seen in a book. Compared to, say, Hardy and Wright's classic book it is much more accessible, and uses terminology and techniques that are now commonplace to modern (under)graduate readers to both simplify and demystify the subject. Then again, it covers less ground than H. and W., but as a starting point it succeeds admirably.

I came to this book just as I was myself trying to gather together everything I knew (or could redisover) about the Phi groups (the groups of units of the rings Z_n); this book did it all for me; and mostly in a way that delighted the mind. I was reluctant to read it (I prefer to try myself before 'cheating') but when I did I found all that I needed there: and authors sympathetic to my own viewpoint. What a delightful feeling of coming home.

An excellent introduction to number theory, 19 Nov 2001
This is an excellent textbook. It is very clear and self-contained, making it possible to work through it without the need to refer elsewhere. I found that it was pitched at just the right level, challenging but not overwhelming, and a good mix of exercises, all with full solutions. The structure is very well thought out, so that it is always clear where arguments are heading. Probably the best maths textbook I've ever read; other authors please take note!

A smooth introduction to number theory, 07 Sep 2001
This is a nice little book (290 pages), which can be used as course litterature for an introductory course in number theory or a by-side reading for somebody taking a first course. It's exposition is so pedagogical and clear that I could study the book from the beginning to the end on my own without help. This is pretty rare for a mathematical book. It covers not only the basic subjects likes divisibility, primes and congruences but more advanced subjects like Euler's functions, quadratic residues and Riemann Zeta function as well. There is even a final chapter on Fermat's Last Theorem, which is quite accessible. I would not hesitate to recommend this book to anybody starting to study number theory. Finally it contains complete solutions to all exercises.

A smooth introduction to number theory, 07 Sep 2001
This is a nice little book (290 pages), which can be used as course litterature for an introductory course in number theory or a by-side reading for somebody taking a first course. It's exposition is so pedagogical and clear that I could study the book from the beginning to the end on my own without help. This is pretty rare for a mathematical book. It covers not only the basic subjects likes divisibility, primes and congruences but more advanced subjects like Euler's functions, quadratic residues, Riemann zeta function as well. there is even a final chapter on Fermat's Last Theorem, which is quite accessible. I would not hesitate to recommend this book to anybody starting to study number theory. Finally it contains complete solutions to all exercises.

Promising...but disappointing in the end, 28 Oct 2008
The book looks like the author is just postponing the end of the story just repeating and repeating the same ideas. The part of the proof and the attempts to correct the proof are quite disappointing because they are too much redundant. Moreover Singh is sliding some e-mails which don't add anything to the story and are quite "impenetrable". I do not like this way of writing. The author pretends not to use math symbology and math concepts beyond very basic ones, and then he lets go concepts like Hecke algebra, Euler system, "quasi-automorphic representations", i.e. without giving any clue about what they mean.
I think it leaves too much maths unexplained (and in a book about a math conundrum you understand it is a big problem!); I would have loved to see the same ingenuity Derbyshire put in his wonderful "Prime Obsession".

Mathematics as you've never seen it before, 21 May 2008
I was never a fan of maths at school. It did not come easily to me and I failed to see the relevance of trigonometry to my everyday life.

I say this so you realise I am not some sort of science geek who was best friends with a calculator. That's because I found this book absolutely fascinating. It made me laugh 3 times in the first 20 pages alone!

What Simon Singh does is through Fermat's puzzle describe the history of mathematics from Pythagoras right up to the 1990's. To the layman names like Euclid put in the mind very dull old guys, but they are brought to life with fascinating anecdotes. For example there's the tortured young French mathematician Galois who is dead by 20, his final mathematical theories frantically scribbled down before a dual. Then there's the story that Pythagoras himself drowned a man when he discovered a certain type of number he objected to!

All of this is carefully woven into the story of Andrew Wiles' life long obsession to prove Fermat's last theorem a puzzle that had foxed the whole world for over 350 years!

Everything is explained in a way that it can be digested by someone who has only a passing interest in maths and as a whole is a remarkable book.


Interesting, exciting, challenging; great read, 20 May 2008
What I loved the most about this book was it's timeline-structure. Dating back to the Pythagorean ages to the present; I thought this was a brilliant idea. The book is full of interesting stories of what the most famous mathematicians in the world had experienced during their profession.

The book reaches out to people on many levels:
Women:
The story told about Sophie Germain (born in 1776), the daughter of a merchant whom outside of her work shared a great passion for Mathematics. However during this age, female mathematicians were frowned upon, and so to study at the Ecole, she took the identity of a former student at the Academy named August Le Blanc. The academy was unaware that he had left Paris and continued to print lecture notes and problems for him. Germain had been submitting the answers to these problems under his name. As her work progressed she had made a remarkable breakthrough in revealing the proof to Fermat's Last Theorem; and with the help of Gauss, one of the most famous mathematicians. They would keep in regular contact regarding mathematical problems until the day where she had submitted this breakthrough to him, she had also revealed that she in fact, is a woman; and received an astonishing response from Gauss's overwhelming reaction (In the best way possible) - Germain had become an icon for female mathematicians.

Mathematicians/People who love maths:
Appendixes located in the back of the book where readers with a higher level of mathematical knowledge can read further into the problem with more examples.

Musicians:
mathematical properties of plucking a string to achieve different tones.

Etc.

I remember particularly being shocked about Pythagoras's shame. Where one of his students had discovered the concept of irrational numbers, and as Pythagoras failed to understand this concept, he had sent for the student to be drowned, and claimed irrational numbers as the devil's work; absolutely shameful of such a famous and respected mathematician. Again, this could possibly reach the interest of historians.

There are so many aspects of the book to talk about but I need to keep this short and sweet. Overall the book was a huge success and covered enough of mathematical history to engage the reader in the problem and allow them to enjoy it at the same time. However I did notice that a lot of other significant people in mathematics were not mentioned, like Muhammad bin Mks al-Khwrizm+ - who discovered Algebra mathematics. I also feel that towards the end of the book where the story of Andrew Wiles's steps to solving the theorem was slightly lengthy, and to be completely honest, started to bore me (Hence my 4 star rating).

I highly recommend this book to anyone interested in mathematics, history, or simply like mysteries and puzzles.

Better than the Da Vinci Code!, 20 Mar 2008
This is a very well-written book: high-level mathematics made accessible to all. It is a true adventure story - and if you are also interested in finding out what exactly it is about mathematics that motivates mathematicians - then this is the book to read. Highly recommended.

Fermat's Last Theorem, 10 Jan 2008
an interesting book about Mathematics and about mathematicians both the famous and not so famous

Bit of a guilty pleasure, 21 Dec 2008
A book basically consisting of a list of numbers and their interesting properties can only appeal to a very small percentage of readers, but those readers will love it to the hilt.

Wells starts from 0 and reaches Graham's Number (so big it can't be written down) and all manner of each number's unique quirks are displayed. For example, 10213223 is a self-describing number (one zero, two ones, three twos and two threes), and 153 is the only number to be the sum of the cubes of its digits. If you find such things interesting, this book is a treasure trove.

By design, it's not really the kind of thing you can read through in one sitting (like a novel), but is something to be dipped into in idle moments. As a previous reviewer said, it's the perfect toilet read, and that's most definitely a complement.

Doesn't read like a dictionary, 08 May 2003
David Wells has assembled an unique and readable collection of facts about numbers, arranged in numerical order! Entries are fascinating, for the most part, though they can be frustrating, too. For example, when first encountering the notion of automorphic numbers (numbers whose squares end in the same digits as the original number), it is tempting to discover if there are other related entries -- 'trimorphic numbers', for instance? It is possible to track these down using the small index provided and quite fun to do.
Unlike other dictionaries this is best read from front to back though it can be used as a reference, once one is familiar with it.
Many concepts are briefly explained as they are encountered, and some merely referred to in passing, and the frustration here is that there need not be full information in the text. However, this is most enjoyably resolved by brushing up one's own skills and trying to demonstrate the simpler claims for oneself. There is plenty here for the dabbling amateur to try, though the serious mathematician can also enjoy the book.
I have one qualification: David Wells identifies 51 as the least uninteresting number (no, not a contradiction: it is simultaneously interesting and uninteresting, he claims) -- because he cannot find an interesting fact about it. He fails to notice that it is the fourth trimorphic (and non-automorphic) number: 4, 9, 49, 51 and 75 being the first five cases. This means that it is mildly more interesting than at first supposed.
I look forward to a revised edition -- with readers' contributions and comments.

Takes pride of place in the loo, 23 Oct 2001
Books which are great for dipping into for a few minutes take pride of place in the loo - this one included. It is just tremendous - full of interesting stuff for any geeks who like numbers and maths. You'll come back to this book time and time again - the loo becomes a more inviting place with this book.

No recreational mathematician should be without it, 10 Dec 2000
In the foreword to G.H. Hardy's book A Mathematician's Apology, C.P. Snow tells an anecdote about Hardy and his collaborator Srinavasa Ramanujan. Hardy, perhaps the greatest number theorist of 20th century, took a taxi from London to the hospital at Putney where Ramanujan was dying of tuberculosis, Hardy noticed its number, 1729. Always inept about introducing a conversation, he entered the room where Ramanujan was lying in bed and, with scarcely a hello, blurted out his opinion about the taxi-cab number. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy! No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."

Usually it takes a great deal of insight as well as considerable mathematical training to discover a yet unknown properties of some number. Only recognizing the beauty of a number pattern is much easier, though, especially with a friendly book like this one on hand. Wells, a long-time mathematics popularizer, has collected over 1000 numbers he considers interesting. Each of them is given a short explanation, often accompanied with a bibliographic reference. Celebrities among the numbers, like i, e or Pi, are given a more comprehensive treatment. Included are also several sequences, like Fibonacci's, Mersenne's, Fermat's, Carmichael's or Kaprekar's, each accompanied with its explanation. So are cyclic, amicable, untouchable or lucky numbers, and many more sequences you probably didn't know about.

While Wells' dictionary certainly gives the impression of a well-researched work, the list of numbers is by no means exhaustive. Anyone familiar with chaos theory will notice the absence of Feigenbaum constant; prime hunters would probably be interested in discussion on Woodall primes, Sophie-Germain primes, or Proth primes. But they are better off with Paulo Ribenboim's book on primes, anyway, while Wells' book, with its easily understandable explanations and accessible price is probably more suited for the "recreational mathematics" audience.

Bedside reading, 30 Nov 2000
This is the kind of thing you read before going to bed, especially if you have a geeky gene. The pseudo-victorian title alone is lovable, although the book itself is a bit disappointing if, like me, you aren't into mathematics. The anecdotes and personal stories of mathematicians are interesting, but the majority of entries simply describe the number in question as being 'the start of a remarkable chain of amicable numbers', or consist of formulae. And it might well give you traumatic memories of school.

It's worth it for the entry on 48 (I think), which states it to be the first uninteresting number, and thus interesting for being so.

The most superb introduction to Sacred Geometry ever!, 12 Jul 2005
I cannot rate this book too highly. When I ordered it, I presumed from the cover artwork (and the title) that it would be a kind of "Sacred Geometry for Dummies". I was, in fact, looking for a relatively easy primer to help me grasp the basic principles of this hugely deep and densely arcane subject, but what I did not anticipate was that it would be both exceptionally accessible as well as intellectually satisfying. Our culture has no innate understanding of "the mathematical archetypes of nature, art, and science" (the sub-title) and in fact, our culture, being so hamstrung, so crippled by left-brain dogma, does not even consider that nature, art and science could have any possible common denominator to even discuss! That the author could express ideas that are really so beyond our normal Western rationale, and moreover, does so with such intelligence and yet with such a light touch, is quite an extraordinary feat. The text is profusely illustrated with diagrams and drawings that precisely explain all that is required, and is also littered with hundreds (this I really appreciated) of superbly chosen quotations from all the great minds of history, from all cultures. I do not believe that there exists a better introduction to this deep and wonderful wisdom, and I would gladly give it 9 stars.

Even after an hour browsing through it, I felt that I had absorbed levels of knowledge, of perception, that were not there previously. This book is about a level of mathematics that, shamefully, is not taught in our educational system. So much for our so-called "progressive' modern culture, that the real pearls have been disregarded in favour of something that has had all knowledge, life and magic stripped away. All spiritual qualities, in fact. What does mathematics and geometry mean to anybody today, other than for the most common-place pedestrian purposes? I did not know, prior to this book, that the Greek "mathema" signifies "learning in general" and was the root of the Old English "mathein", "to be aware" and the Old German "munthen", meaning "to awaken". Today the word "maths" has, for most people, constricted its scope to emphasise mundane measurement and mere manipulation of quantities.

The only, and I mean only, quibble I have with the book, is that it is printed on paper of a quality that does not do justice to the content. This is a shame, but perhaps it's only applicable to the HarperCollins paperback publication that I bought. According to an inner page giving the ISBN details, there was a hard cover version published in 1994, also HarperCollins, and I would assume that that would be printed on better stock. But don't let this put you off. If you have any interest whatsoever in numbers, mathematics, higher knowledge, or any curiosity whatsoever about why the world is designed the way it is, buy this book. It contains the entire universe!

An Esoteric Feast, 30 Aug 2002
The title is grandiose, the book's layout makes you travel sick when you read it, Michael's brain is clearly on over drive. I loved every line, enjoyed every quote, was intrigued by all the images. It helped me unify many thoughts I had in this field. He has presented sacred geometry in a non-pretentious and easy-to-understand way. He will stimulate you to look further in the subject and you will surely start to see the world in a different way. Why can I not find any other books by this man?

Sensational!, 13 Sep 2000
Once every few years you come across a book which is genuinely life-changing, and this is one of them. In its simplest form it's a book which makes numbers interesting - completely different from the usual dull stuff you did at school. But it goes way beyond that, to open up a whole world of cosmic geometry which you'd never noticed before. It explains the principles of number and sacred geometry with extraordinary clarity and applies them to architecture, religion, ancient art, modern design, science, the natural world, etc, on the basis that all forms of construction and creation work on those same principles. It covers everything, from the reason why manhole covers are round to the divine emanations of the Qabalah, and links the physical with the spiritual in ways you would never have thought of. The book has a wide format and very clear layout, with lots of illustrations, and assumes no previous knowledge of - or interest in - the subject.

I notice in Mr Schneider's comments that it took him 20 years to research the book. Well, it certainly shows, and I can assure him it was very well worth it. This is truly a revelation, an amazing book which will completely change the way you look at the world.

A quintessential presentation., 12 Aug 1999
I have always been mesmerized by mathematics and its infinte implications; one could call it impassioned. Many people do not share my zeal. Some of these people are my family, friends, and associates. FINALLY I have a book to recommend that can open each and every one of them to unsuspected dimensions of this absolutely fascinating subject; the presentation alone will, I am sure, compel them to read on and on and on. I am awed at the scientific authenticity and gentleness with which Schneider creates such a sensible, spiritual, and harmonious synthesis. Utterly awed...and so very grateful. I'd love t