Product Description
In THE BOOK OF NUMBERS, two famous mathematicians fascinated by beautiful and intriguing number patterns share their insights and discoveries with each other and with readers. John Conway is the showman, master of mathematical games and flamboyant presentations; Richard Guy is the encyclopedist, always on top of problems waiting to be solved. Together they show us why patterns and properties of numbers have captivated mathematicians and non-mathematicians alike for centuries. THE BOOK OF NUMBERS features Conway and Guy's favorite stories about all the kinds of numbers any of us is likely to encounter, and many others besides. "Our aim," the authors write, "is to bring to the inquisitive reader. . .an explanation of the many ways the word 'number' is used." They explore patterns that emerge in arithmetic, algebra, and geometry, describe these pattern' relevance both inside and outside mathematics, and introduce the strange worlds of complex, transcendental, and surreal numbers. This unique book brings together facts, pictures and stories about numbers in a way that no one but an extraordinarily talented pair of mathematician/writers could do.Amazon.com Review
The Book of Numbers lets readers of all levels ofmathematical sophistication (or lack thereof) understand the origins,patterns, and interrelationships of different numbers. Whether it is avisualization of the Catalan numbers or an explanation of how theFibonacci numbers occur in nature, there is something in here todelight everyone. The diagrams and pictures, many of which are incolor, make this book particularly appealing and fun. A few of thediscussions may be confusing to those who are not adeptmathematicians; those who are may be irked that certain facts arementioned without an accompanying proof. Nonetheless, The Book ofNumbers will succeed in infecting any reader with an enthusiasmfor numbers. ... Read more

Customer Reviews (17)

Interesting introduction to numbers
Fascinating overview of the many different types of numbers.It made me think, like a book of puzzles.I really like the geometrical interpretations in the early chapters.Unfortunately, the book suffers from too many typos.I also found the discussion on surreal numbers hackenbush, nim, and the infinities rushed and unclear, and the ending too abrupt.It seems the authors, having worked directly on those topics for many years, thought the material in this last chapter is obvious and needs little explanation.I would love to see an updated edition with expanded material covering the more difficult topics.

Confusion! Poor Editing
Horrible - the beginning of Chapter 2 is a prime example of confusion both textually and graphically. Defintately would NOT recommend unless you already know the stuff well enough to overcome the above limitations! But then, what's the point of buying?

The Book of Pure Enjoyment
I own a lot of books on Mathematics and Mathematical topics. This book truly is my favorite. The color illustrations, the other drawings and diagrams make the topics very understandable and tantalizing. The section on infinite and infinitesimal numbers alone, has inspired me to submit a paper on this subject to a Mathematics Journal. This book will cause anyone who is not already a Mathphile, to quikly become one. Take a look at the middle numbers in Pascals Triangle....with a little manipulation of this sequence, you produce CATALAN NUMBERS, which further leads to a number of interesting problems with Friezes and Polygons, none of which are mind bending in their demand of your mental powers. The chapters are all short, and you can open the book at random and find something very nice to dine on. BUY THIS ONE!Dr.Pratt.

Heavier than I thought
I enjoyed reading this book very much. But I do not agree with the reviewers who say it does not require more than a highschool level in mathematics.
I have an M.Sc. degree in electrical engineering, so I can safely say I have a good knowlegde of mathematics, but still there were parts I didn't completely understand.
Maybe we don't have to understand it all at once, maybe this level is needed to keep it interesting. And interesting it was, right up to the last page. And after reading the last page, I would have loved to read more like it.

A Fascination with Numbers
This is a marvelous book written by two very distinquished mathematicians with a knack for teaching and fascinating their readers."The Book of Numbers" is a romantic text - Conway and Guy are truly in love with the subject and their passion is truly contagious. Enter, "The Book of Numbers" and plan on staying a while - not bceasue it is a difficult read but becasue you will have no desire to checkout. Truly enjoyable.
L J Petrillo ... Read more

Product Description
This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable. It treats several topics in geometric function theory as well as potential theory in the plane. In particular it covers: conformal equivalence for simply connected regions, conformal equivalence for finitely connected regions, analytic covering maps, de Branges' proof of the Bieberbach conjecture, harmonic functions, Hardy spaces on the disk, potential theory in the plane. The level of the material is gauged for graduate students. Chapters XIII through XVII have the same prerequisites as the first volume of this text, GTM 11. For the remainder of the text it is assumed that the reader has a knowledge of integration theory and functional analysis. Definitions and theorems are stated clearly and precisely. Also contained in this book is an abundance of exercises of various degrees of difficulty. ... Read more

Product Description

This book is an introductory text in functional analysis. Unlike many modern treatments, it begins with the particular and works its way to the more general.

From the reviews: "This book is an excellent text for a first graduate course in functional analysis....Many interesting and important applications are included....It includes an abundance of exercises, and is written in the engaging and lucid style which we have come to expect from the author." --MATHEMATICAL REVIEWS

Customer Reviews (4)

For Pure Math Courses Only
This book is appropriate for a graduate course in functional analysis in a mathematics department. It assumes a strong background in undergraduate topology, advanced linear algebra (a linear algebra course that covered direct sums and products, dual spaces, quotient spaces, isomorphisms, and universal mapping properties) and complex analysis, along with a graduate level course in real analysis and measure theory. This book should not be used for a graduate course in Applied Functional Analysis (as it was when I took it). Very few applications are discussed, there is only one diagram in the entire book, proofs skip many intermediate steps, and examples are stated with no explanation. If you do not have the equivalent knowledge of a bachelors in pure mathematics, this book will be almost unreadable. The take home message here is that this book is not for quantitative scientists (who use a lot of functional analysis tools without even knowing it) to study the basic theory behind the tools they use. Its designed for exactly what it says, "a graduate course in mathematics". So if you are a Professor about to teach a course in "Applied Functional Analysis" do not use this book. Use one of the many books called Applied functional analysis.

That being said, I did appreciate the order of coverage. Starting with Hilbert spaces and then moving to Banach spaces, made things more clear for me.

A wonderful first course for enthousiasts
The only reason for 4 stars instead of 5 is that I am not a great fan of Conway's style, and have wasted too much time on mangled examples in his Complex Analysis.
That said, the layout here is superb, and the choice of topics just what is needed to get one of the ground. Beginners will find it useful to have a copy of Kreyszig at hand.

A Course in Functional Analysis ... the title is correct!
I learned functional analysis by studying this book. I did this under the direction of a master teacher, John Erdman, who taught via a modified Moore Method. I found this very inspirational and challenging. BEFORE I took thecourse, I did not enjoy browsing the book, BUT I learned that the book,upon combination with the right amount of focus and effort, did aremarkable job of bringing functional analysis alive ... of transmiting thereal essence to young, "sprouting" mathematicians. There is alsoan informality that brings a freshness to the book ... and this in asubject that could easily be studied without encountering this importantingredient in a mathematician's training.

This book has as it's highpoint and goal the spectral theorem for normal operators. I add thisbecause no one book can be all encompassing. If this and the spectraltheorem goal are kept in mind, the omissions and emphasis found in the bookwill be found to be completely natural.

This book should be in thelibrary of anyone teaching functional analysis or who wants a workingmathematician's masterfully developed course on functional analysis (withan eye to the spectral theorem for normal operators).

An Excellent Book !
This book is just excellent. The author decides to do something a little unusual, and starts talking about "Hilbert Spaces" before talking about "Banach Spaces". Conway writes down the matereal in a greatway. He gives proves to almost every proposition, and gives lots ofEXAMPLES and EXCERCISES (which are not given in most of the books aboutthis subject). It's a good book for people who have never read this bookbefore, as well as people who are currently studying the course. Also,conway extends the book's content by writing about advanced subjects (thatare not studied in a first course about the subject), like locally convexspaces, weak topologies and even unbounded operators. ... Read more

Product Description
"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book for classroom use or self-study." --MathSciNet ... Read more

Customer Reviews (8)

Excellent Book
I have used this book back in my college days. I am amazed to see such a low rating on this book. This is one of the best books on complex analysis.
Mind you this is a Graduate Text in Mathematics. So it is intended to cover lot of ground using clear logical path. I agree to some extent that this book does not have lot of examples but a GTM is supposed to be like that. The way I used/read this book is to supplement it with other books on the topic. The proofs of theorems in this book are complete without any errors.

If you are a graduate student and want to learn complex analysis, this is the best book available out there.

Nicebook , good level
This book isa classic.Is goodfor an master degree course in mathematics.

Should be avoided.
I concur with the reviewer Sidhant. I had tons of frustrations with this book this semester - it's so annoying. I just finished and reviewed Munkres' topology book, and Sterling Berberian's Fundamentals of Real Analysis, and the difference between these books and Conway's is (respectively) like the difference between say Bach and pure cacophony: cymbals, screeches, sirens, horns, etc.

Some of my complaints include, but are not limited to:

- No examples whatsoever; there may be one or two per chapter, jammed lamely into the body of the text
- The "expository prose" did nothing to elucidate the underlying mathematics; often Conway babbles for a while, then says something like "the proof is left to the reader". It came to a point last month where I simply just stopped reading the text and started to focus on just the theorems and proofs
- There were errors in some proofs, of omission and of commission. The two ugliest proofs I've ever seen in mathematics lie in this book: (1) a standard composition theorem for analytic functions done by cases (?) which ended with "the general case follows easily", and the argument was built upon sequences (?). In other books, the result is proved in three lines; (2) the Casorati-Weierstrass theorem: same sloppiness, but Wikipedia saved me with an elegant four-line proof. The open mapping theorem was almost incoherent; and a crucial part of it was left as an exercise. I managed to get this part from Adult Rudin with no problems, though.
- the exercises: some are actually fine, but many are obtuse, and obtusely stated. Ultimately - and this is a huge problem - one cannot trust whether or not exercises were written correctly, because of too much general and ubiquitous sloppiness.
- chapter 2 (mapping properties of analytic functions, mobius maps) is so poorly written i had to skip it entirely.

I have a whole list of complaints here on paper, that I collected while reading this book to expose when I reviewed it. It's simply not worth more time and effort to transcribe them.

Not the whole book is bad, the homotopy integral is treated fairly well (i guess), as are the earlier parts of complex integration, and isolated singularities. But all this stuff is elementary - the later chapters are what counts, and the two chapters following integration are a mess. I hate having to clean up SO MUCH of this book.

I recommend looking at Robert B. Ash's book, as it's only 15 dollars (and free online), compared to the 60 dollars which this book is, and more importantly he makes very wise comments regarding math pedagogy on his webpage. In contrast, Conway in his webpage is pictured drinking martinis; he was probably on his twelfth one when he began the writing of this book.

EDIT: i've been working through ash's online book from the start, and i notice the proofs are far more slick, yet far more intuitive. there are many more problems, and better, than in conway. plus there are hints and solutions - don't peek unless you really don't know where to start! ultimately i'm starting over somewhat. i can't pretend i know nothing after conway, but i've abandoned his book completely. i wasted a graduate semester on conway's garbage.

Not recommended
This book was the recommended textbook for a course in Complex Analysis I took at college. I had already done a 1st course on analysis, but that didn't help me too much. This book, littered with loads of proofs and lemmas, is a little too terse, and the author expects students to understand a lot on their own. Concepts in Complex Analysis need to be demonstrated using examples, and diagrams, if possible. Like for eg. the concept of branches in complex functions. The book starts of defining the complex logrithmic function. The author never says what a branch exactly is. He writes down a hell lot of proofs and expects the student to figure out that the complex logarithm is infact a multi-valued function, and that a branch is essentially a "slice" of this multivalued function. Similiar problems crop up when the author discusses fractional linear transforms. Instead of showing whats happening with simple diagrams, the author makes things look extremely complicated with his equations and theorems. This book makes learning complex analysis a very mechanical exercise, devoid of all fun.

A good read
We're using this book for my graduate level complex analysis course, and over all, I'm pleased with it.Aside from some goofy notation (i.e., an empty box to represent the empty set?), it's pretty well written.The pace of the text isn't too fast or too slow, and there are plenty of exercises of a varying degree of difficulty to help you learn the material.Another nice feature is the price; one can find it for less than $50, so it'll make a nice reference book even if it wasn't assigned for a class. ... Read more

Product Description
First inhabited by the LenapeIndians and settled byEuropean colonists in the seventeenthcentury, New York's Sullivan Countyhas experienced several ages ofprosperity and growth over the lasttwo hundred years. Locals conceivedof timber rafting in the eighteenth century to support the shipbuildingindustry, followed by a prosperous tanning boom in the nineteenthcentury that supplied leather to the Union army. Finally, two periods oftourism, known as the Silver Age and Golden Age, capitalized on thearea's fresh air, clean water and magnificent scenery. In this collectionof images, local author and county historian John Conway provides acomprehensive look at this much-celebrated region. ... Read more

Product Description
Working memory--the ability to keep important information in mind while comprehending, thinking, and acting--varies considerably from person to person and changes dramatically during each person's life. Understanding such individual and developmental differences is crucial because working memory is a major contributor to general intellectual functioning. This volume offers a state-of-the-art, integrative, and comprehensive approach to understanding variation in working memory by presenting explicit, detailed comparisons of the leading theories. It incorporates views from the different research groups that operate on each side of the Atlantic, and covers working-memory research on a wide variety of populations, including healthy adults, children with and without learning difficulties, older adults, and adults and children with neurological disorders. A particular strength of this volume is that each research group explicitly addresses the same set of theoretical questions, from the perspective of both their own theoretical and experimental work and from the perspective of relevant alternative approaches. Through these questions, each research group considers their overarching theory of working memory, specifies the critical sources of working memory variation according to their theory, reflects on the compatibility of their approach with other approaches, and assesses their contribution to general working memory theory. This shared focus across chapters unifies the volume and highlights the similarities and differences among the various theories. Each chapter includes both a summary of research positions and a detailed discussion of each position. Variation in Working Memory achieves coherence across its chapters, while presenting the entire range of current theoretical and experimental approaches to variation in working memory. ... Read more

Customer Reviews (1)

What is Intelligence
Variation in Working Memory

Best, and latest definition of the difference between static intelligence (I.Q) and fluid intelligence (reasoning ability).I now understand why I had to fire pHDs for incompetence in the working world! ... Read more

Product Description
An investigation of the geometry of quaternion and octonion algebras, this book is intended for mathematicians, physicists, and crystallographers at any level--from undergraduate to professional--who are interested in the symmetries of low-dimensional space.

The book can also be used as a text for graduate courses in many mathematical fields, including geometry, group theory, algebra, and number theory.

A Selection of Topics Covered:
The geometry of complex numbers
Quaternions and 3-dimensional groups
Quaternions and 4-dimensional groups
The Hurwitz integral quaternions
Moufang loops
Octonions and 8-dimensional geometry
Integral octonions
The octonion projective plane ... Read more

Customer Reviews (5)

People either love Conway or hate him?
This books gives a window into the newer notation in group theory.
Sometimes things that are "obvious" to Conway and his co-arthor,
just aren't to the rest of us.
But in contrast to that he gives concrete examples
of new approaches that are beyond classical Coexter
and Cartan type approaches.
If you are looking for physics applications to quantum mechanics
for modern group theory,
you might want to try another book,
but if you want an idea of what a Moufang loop is or why
octonions are not associative, then you might like this book.
Some time in this century we may even get
a chance to understand Freudenthal's metasymplectic geometry?
This book for me is sort of a study guide to
what i should try to learn for the future?

moronians
If you want a good laugh take a look at chapter six
in this hastily compiled piece of trash.No one who
does not already understand the material on octonians
will be able to penetrate this unannotated formulary.
The same goes for the entire second half of the book.
[The first half is just a rehash of material so familiar
that there is no need to see it in print for the N+1 st time. ]
The only worthwhile entry here is the reference guiding
readers to John Baez's article on octonians which:
a) is available free onlineand
b) actually explains in a reader friendly way the
history, math, and applications involved.

The authors (not to mention editors) should be ashamed
at such a sloppy treatment of this rich and historically
interesting episode in mathematics.

Good, but something is better
Conway is an excellent mathematician and an extremely lucid author.No criticism should be given to any of his writings.In the case of quaternions (and octonians), a much better, more complete, and more powerful view is achieved by seeing them in the larger setting of geometric algebra.The geometric algebra gives direct access to all the results and all the geometry of these algebras, and does so in an intuitive and useful way.I suggest that the new book by Chris Doran and Anthony Lasenby called "Geometric Algebra for Physicists" is a better place, generally, to get acquainted with these issues deeply.It isn't a criticism of Conway.It's just an advantage of seeing things in the right context.

Examining 3- and 4-dimensional Euclidean spaces
On Quaternions And Octonions: Their Geometry, Arithmetic, And Symmetry is a collaboratively presented treatise by John H. Conway and Derek A. Smith on the geometry of the quaternion and octonion algebras. Examining 3- and 4-dimensional Euclidean spaces, enumerating the corresponding finite group of symmetries, analyzing the arithmetics of quaternions and octonions and much more, this impressive presentation sheds new light on the geometry of complex numbers and is a scholarly addition to Advanced Mathematics reference collections and reading lists.

A model of exposition
John Conway's books are always well written, and this could serve as a model for other mathematics authors.I don't need to know that much about quaternions and octonions, but I found myself working through most of the book and the beautiful mathematics it covers.The only thing that disappoints is the dreadful cover and the difficulty getting hold of a copy in a bookstore.But then I guess Amazon.com exists to help people get their hands on stuff they might never see in a bookstore. ... Read more

Product Description
Start with a single shape. Repeat it in some way translation, reflection over a line, rotation around a point and you have created symmetry. Symmetry is a fundamental phenomenon in art, science, and nature that has been captured, described, and analyzed using mathematical concepts for a long time. Inspired by the geometric intuition of Bill Thurston and empowered by his own analytical skills, John Conway, with his coauthors, has developed a comprehensive mathematical theory of symmetry that allows the description and classification of symmetries in numerous geometric environments. This richly and compellingly illustrated book addresses the phenomenological, analytical, and mathematical aspects of symmetry on three levels that build on one another and will speak to interested lay people, artists, working mathematicians, and researchers. ... Read more

Customer Reviews (5)

good summary but tough going
This book provides a very detailed overview of two and three dimensional symmetry types, but for the non-mathematician it can be tough going as the book progresses.

Exquisite and fascinating and informative
Fascinating and thrilling take on finite groups, presented by means of planar and higher-dimensional symmetries.

Presentation will be clear and interesting to laypersons and to mathematicians. Three features distinguish this book from other discussions of symmetry.

First is the inherent interest in the subject. Symmetry is a topic of much greater importance and pervasiveness than most people realize. This book helps us see it in many places we might not think to look, and its classification and analysis is extremely informative and thought-provoking.

Second is the beauty of the examples, which deal with elegant and attractive patterns.

Third is the extraordinarily high production values of the book. The notation for symmetry utilizes different colors in a way that is clear and helpful. The pictures are carefully done. Frankly, this is one of the most beautiful books I've ever owned.

My one complaint is that the corner of the cover of the copy I got from Amazon was a bit bent. It seems like too much trouble to ship it back and get a whole new copy for this, since the text is fine. This is quite unusual for Amazon, however.

In any case, I highly recommend this book for anyone interested in mathematics or even in art, from curious high school students to professional mathematicians.

Come On!?
People selling copies of this for up to $250! Oh Come on! You can buy a new copy here for 1/4 the price. Im all for a nice profit but thats crazy!

Cristal
Very clearly explained and genially illustrated. One has the feeling that one understands the first time in ones life the topics presented -- e. g. the cristalographic groups.

Even the most basic of things can be beautiful
One of the most base concepts of art - symmetry. "The Symmetries of Things" is a guide to this most basic concept showing that even the most basic of things can be beautiful - and addresses why the simplest of patterns mesmerize humankind and the psychological and mathematical importance of symmetry in one's every day life. "The Symmetries of Things" is an intriguing book from first page to last, highly recommended to the many collections that should welcome it.
... Read more

Product Description

Reflecting a rich technical and interdisciplinary exchange of ideas, Water and Life: The Unique Properties of H₂0 focuses on the properties of water and its interaction with life. The book develops a variety of approaches that help to illuminate ways in which to address deeper questions with respect to the nature of the universe and our place within it.

Grouped in five broad parts, this collection examines the arguments of Lawrence J. Henderson and other scholars on the "fitness" of water for life as part of the physical and chemical properties of nature considered as a foundational environment within which life has emerged and evolved. Leading authorities delve into a range of themes and questions that span key areas of ongoing debate and uncertainty. They draw from the fields of chemistry, biology, biochemistry, planetary and earth sciences, physics, astronomy, and their subspecialties. Several chapters also deal with humanistic disciplines, such as the history of science and theology, to provide additional perspectives.

Bringing together highly esteemed researchers from multidisciplinary fields, this volume addresses fundamental questions relating to the possible role of water in the origin of life in the cosmos. It supports readers in their own explorations of the origin and meaning of life and the role of water in maintaining life.

Product Description
Three railroads, a canal, dozens of industries, and hundreds of hotels have come and gone since Sullivan County was formed in 1809. Famous stars of sport an screen have worked and played here, infamous and despicable gangsters have lived and died here. Our history is rich and colorful, and Retrospect is a look back at that history, with reflection, with scrutiny, and, occasionally, with longing. ... Read more

Customer Reviews (3)

Not enough of a good thing
My only complaint is that the book was too short.There is so much interesting information about a topic that is much misunderstood that you find yourself wanting more.By far the best treatment of Sullivan County Catskills history.

Glad I found this book
This book has become harder to find in recent years and that trend is likely to continue.It provides a unique look at the county that gave us the Borscht Belt, Sid Caesar, Jerry Lewis, Danny Kaye, Moss Hart and many other entertainers, and served as a dumping ground for Murder, Inc. in the 1930s.This author has become the most respected authority on Catskills history writing today.Looking forward to his next work.

This book is great!
While in New York this fall I decided to buy this book.I sure am glad I did.I feel the author has really picked up on the way things were then.Things I had forgotten about.This was a good time, the area was buzzing,people were happy and had fun.You couldn't wait for summer to come, thenyou could not wait for it to be over!I loved growing up in the area.Ihave written a story about it too. ... Read more

Product Description
ONAG, as the book is known, is one of those rare publications that sprang to life in a moment of creative energy and has remained influential for over a quarter of a century. Still in high demand, it is being republished with some adjustments and corrections. The original motivation for writing the book was an attempt to understand the relation between the theories of transfinite numbers and mathematical games. By defining numbers as the strengths of positions in certain games, the author arrives at a new class, the surreal numbers (so named by Donald Knuth) that includes at the same time the real numbers and the ordinal numbers.

This new edition ends with an epilogue that sets the stage for further research on surreal numbers. The book is a must-have for all readers with a serious interest in the mathematical foundations of game strategies. ... Read more

Customer Reviews (4)

Mind-blowingly original, side-splittingly funny
This is not a book for mathematical beginners, even though it starts from literally nothing.But readers who have learned enough traditional math to understand the point of set theory and who have a solid grasp of the real number system are in for a wild ride, and will never look at numbers, or games, in the same way again.

Conway is the most original mathematician on the planet, as well as a remarkably witty and vivid writer, who combines wordplay and logic better than anyone since Lewis Carroll. The book is far too densely packed to summarize in a short review. All I can say is that it's practically inexhaustible; like all good math books, what you get out is proportional to the effort you make while reading it, but the amount of effort it will repay is a hundred times as much as for an ordinary book.

This is an all-time classic, a "desert island book".Even though this new edition differs from the old one in very minor ways, I bought it immediately because my 1978 copy was falling apart from extreme overuse.(My other "desert island math book" is Cohen's "Set Theory and the Continuum Hypothesis".)

Math geek heaven
Boy, you wanna talk about your _cool_ books. I read this one twenty years ago and never quite got over it. Georg Cantor sure opened a can of worms with all that infinity stuff.

John Horton Conway is probably best known as the creator/discoverer of the computer game called "Life," with which he re-founded the entire field of cellular automata. What he does in this book is the _other_ thing he's best known for: he shows how to construct the "surreal numbers" (they were actually named by Donald Knuth).

Conway's method employs something like Dedekind cuts (the objects Richard Dedekind used to construct the real numbers from the rationals), but more general and much more powerful. Conway starts with the empty set and proceeds to construct the entire system of surreals, conjuring them forth from the void using a handful of recursive rules.

The idea is that we imagine numbers created on successive "days". On the first day, there's 0; on the next, -1 and +1; on the next, 2, 1/2, -1/2, and -2; on the next, 3, 3/4, 1/4, -1/4, -3/4, and -3; and so on. In the first countably-infinite round, we get all the numbers that can be written as a fraction whose denominator is a power of two (including, obviously, all the whole numbers). We can get as close to any other real number as we like, but they haven't actually been created yet at this point.

But we're just getting started. Once we get out past the first infinity, things really get weird. By the time we're through, which technically is "never," Conway's method has generated not only all the real numbers but way, way, way more besides (including more infinities than you've ever dreamed of). His system is so powerful that it includes the "hyperreal" numbers (infinitesimals and such) that emerge (by a very different route, of course) from Abraham Robinson's nonstandard analysis as a trivial special case.

So there's a lot here to get your mind around, and it's a lot of fun for readers who like to watch numbers being created out of nothing. But wait -- there's more.

See, the _full_ title of the book includes not only "numbers" but also "games". And that's the rest of the story. Conway noticed that in the board game of Go, there were certain patterns in the endgames such that each "game" looked like it could be constructed out of smaller "games". It turns out that something similar is true of all games that have certain properties, and that his surreal numbers tie into such games very nicely; "numbers" (and their generalizations) represent strategies in those games. So in the remainder of the book Conway spells this stuff out and revolutionizes the subject of game theory while he's at it.

Well, there must be maybe two or three people in the world to whom this all sounds very cool and yet who haven't already heard of this book. To you I say: read it before you die, and see how God created math.

A very dense collection of original ideas
We all think we know numbers, and yet every once in awhile something comes along that makes us realize that we actually know very little. I am not talking about facts such as whether a specific large number is prime, but about the fundamental definition of what a number is. The appearance of the surreal numbers is one of those mathematical equivalents of a whack on the side of the head. Suddenly, numbers are defined as the strengths of positions in certain games, something that is at first strange, but it turns out that the class of objects defined this way includes the real and ordinal numbers. It certainly is different, and I had to read the first thirty pages of the book three times before I felt that I truly grasped the concepts behind the definition of the surreal numbers.
From that things move more smoothly. As I read through the book, it was easy to get the impression that most of life can be described as a game, where our day-to-day status in the community can be described as a dynamic set of surreal numbers. I often wondered if that may be an effective approach for artificial intelligence work, as it certainly seems that surreal numbers can be used to model almost any dynamic situation. Furthermore, effective game playing is nothing more than effective decision making.
There are many significant ideas in the book, at times you stop and start mentally jumping through different scenarios, as in "What would be the change if this rule is added, dropped or altered?" It seems that if you took that approach, several lifetimes could be spent in exploring all the possibilities. I have read many books and this one is most likely the densest carrier of new ideas that I have ever encountered.

A Truly Amazing Piece of Work
Conway deals with a certain type of game: games with no element of chance (no dice), the players have complete knowledge about the state of the game (no hidden hand signs like scissors-paper-stone) and where the last player to move wins (though that can be stretched to include Dots and Boxes and endgames from Go - though not in this book).

Conway defines a bunch of mathematical objects.He defines mathematical operations on these objects such as addition and multiplication.The whole work looks suspiciously like a way to define the integers and arithmetic starting from set theory.But we soon see that his construction allows for all sorts of things beyond just integers.We quickly get to fractions and irrationals and we see that he has given us a wonderful new way to construct the real line.Then we discover infinities and all sorts of weird new numbers called nimbers that have fascinating properties.

It all looks a bit abstract until you get to part two (well, he actually starts at part zero so I mean part one).At this point you discover that these objects are in fact positions in games and that the ordinary everyday numbers we know so well are in fact special types of games.Ordinary operations like addition, subtraction and comparison turn out to have interpretations that are game theoretical.So in fact Conway has found a whole new way to think about numbers that is beautiful and completely different to the standard constructions.Even better, you can use this new found knowledge to find ways to win at a whole lot of games.

It's not every day that someone can make a connection like this between two separate branches of mathematics so I consider this book to be nothing less than a work of genius.

BTW This is the Conway who invented (the cellular automaton) the Game of Life and came up with the Monstrous Moonshine Conjectures (whose proof by Borcherds recently won the Fields Medal in mathematics). ... Read more

Product Description
Loomis tells the story of the fight against tuberculosis, believed responsible for a billion deaths worldwide between 1700 and 1900. Dr. Alfred L. Loomis, who was diagnosed with the disease himself in 1867, was an innovative thinker and pioneer of the climatological treatment of the disease in a sanitarium setting. He helped one of his patients, Dr. E. L. Trudeau, to open the first American sanitarium in the Adirondacks in 1883, and realizing the need for a larger facility nearer New York City, purchased land and raised the money to operate such a facility near Liberty in Sullivan County. Opened in 1896, it continued to treat patients through 1942. ... Read more

Customer Reviews (1)

The inspiring story of an eminent physician
Several months ago my son invited me to his new apartment on the western outskirts of the quaint village of Liberty, in Sullivan County, New York. Fearing I'd miss the slightly secluded place, we met by a roadside and I followed him in my car up a steep hill packed with pine and beech trees, along several twisting roads and sharp curves, before stopping in front of a notable four story Victorian building. Being a history buff of the nineteenth-century, I was impressed by its fading beauty but undying spirit. The building was once known as the Babbitt Memorial Medical and Laboratory Building, formerly part of a renowned health resort, I soon learned, named the Loomis Sanitarium for Consumptives. The vista from the top floor apartment was spectacular; rolling hills and trees and a picturesque neo-Romanesque stone building with a central rotunda, named the Edson-Aldrich Library all within view. Soon afterwards I read this wonderful book, Loomis: The Man, The Sanitarium, and The Search for the Cure by the celebrated Sullivan County author and historian John Conway. Readers unfamiliar with or uninterested in medical history will be surprised to discover within its pages a remarkable story of selfless dedication to the betterment of humanity.

Meticulously researched and skillfully written, Mr. Conway's biography of Dr. Alfred L. Loomis (1831-95) is also highly informative, entertaining and superbly illustrated with period photographs. Loomis was a highly respected New York City physician born into a family with a history of tuberculosis. Early in his career he was inspired by the work a German botany student named Hermann Brehmer, who advocated the beneficial effect of country living, sunshine, good food and relaxation. The sickly Loomis soon pitched a tent in the Adirondack Mountains, stopped coughing, ate well and put on badly needed weight. One of Loomis' patients was a colleague named Dr. Edward L. Trudeau, also a victim of tuberculosis, and together they organized the Adirondack Sanitarium in Saranac Lake, NY, where Loomis sent his city patients. In 1896, a year after Loomis' death, Mrs. Loomis, together with philanthropists like Mrs. Mary M. Irvin and financier J.P. Morgan, opened the Loomis Sanitarium in Liberty. Here for $7-$12 a week patients seeking a cure from the devastation of tuberculosis, still a deadly disease today, came by train from nearby cities and states. The fee covered medical care, medicine, room and board, recreational facilities, outdoor activities and laundry. At its peak the institute stood on two-hundred-fifty plus acres with twenty structures: an Administration Building, an infirmary, a nursing school, a daily farm, a horse and buggy barn, cottages, a hotel and casino, a chapel, a library and a laboratory for research scientists seeking to eradicate the tubercule bacillus. It also had its own firehouse and electrical power supply. For nearly a half-century the sanitarium helped thousands of the afflicted until it closed in 1942. Despite the seriousness of Loomis: The Man, The Sanitarium, and The Search for the Cure John Conway manages to entertain us with amusing anecdotes regarding quackery and some colorful characters to round out his richly rewarding and compelling narrative.

-Victor Rodriguez, author of RAVENHALL
... Read more

Product Description
Compiled from the best of John Conway's popular Retrospectcolumn, these articles shine a spotlight on famous faces of the past,from George Suslosky, phenomenal yet feisty diner cook, to the worstwoman on earth, Lizzie Brown Halliday. Enlightening and entertaining,the remarkable historical vignettes in this volume explore the customsand curiosities of the Sullivan County Catskills.

High on a bank in Craig-e-Clare sat the stately Dundas Castle,rumored to house a beautiful woman who lured fishermen from theBeaverkill River into her lair. In the hamlet of De Bruce, every spring amonstrous panther prowled, feasting on trout and tourists. These are nomyths from the dark history of foreign lands, but tales from the colorfulpast of Sullivan County, New York. ... Read more

Customer Reviews (1)

An enjoyable walk down memory lane
About ten years before retiring and moving to Sullivan County, I became an ardent New York City history buff after reading Jack Finney's magnificent novel Time and Again and his lesser known but equally fascinating compilation Forgotten News: The Crime of the Century and Other Lost Stories. It was such an addictive topic that I found myself desperately seeking fragments of old New York wherever possible. Thus I'd walked down Fifth Avenue on foggy nights, hoping to leap into the past and hearing hoofbeats on grey cobblestones, walking under the soft blue and yellow glow of gaslight amid the clank and roll of passing trolley cars. Sounds of muffled laughter, hats tipped, skirts clutched, peanut vender, newsboy and flower girl on the corner. I read many books on the city's remarkable history, gazed dreamily at countless photos and engravings of old New York, marveled at the neo-Romanesque splendor of nineteenth century Manhattan, but saddened and appalled by the poverty and squalor of places like the Lower East Side, as captured by Jacob Riis in his classic How the Other Half Lives. Although I miss those wistful strolls, I've discovered to my delight that Sullivan's history is similarly captivating after reading county historian John Conway's books, in particular his delightful new collection of vignettes, Remembering the Sullivan County Catskills. So once again I find myself strolling down memory lane and stopping and searching for a glimpse of the past.

Mr. Conway divides this slim but informative and enjoyable book, gleaned from his "Retrospective" column for the Sullivan County Democrat, into five parts, thus giving us a brief but well-rounded portrait of a fascinating time and place. We read about famous people, scandals, disasters, early industries, noteworthy towns and villages, resorts and recreation. Meeting along the way bold settlers, town founders, wheeler-dealers, shrewd entrepreneurs, celebrities, artists, anglers, suffragettes, loaded golfers, skiers, eccentric millionaires, a haunted castle with a seductive apparition on the lookout for unwary fishermen, scoundrels and serial killers, and enough disasters and triumphs to enchant readers averse to social history. Charles Atlas, Danny Kaye and D.W. Griffith were just a few of the people that gave the Sullivan Catskills its well earned good repute. Mr. Conway neatly refers to the two major Catskills periods as the Silver Age, nineteenth to early twentieth century, and the Golden Age, which ended in 1965 with the decline of the great upstate resorts. The book is richly illustrated with period pictures that capture the beauty of this once famous and still popular place. Readers unfamiliar with the region will find it helpful having a good map of the county readily at hand to pinpoint the places covered in this fine book.

-Victor Rodriguez, author of Ravenhall ... Read more

Product Description
In the quarter of a century since three mathematicians and game theorists collaborated to create Winning Ways for Your Mathematical Plays, the book has become the definitive work on the subject of mathematical games.

Now carefully revised and broken down into four volumes to accommodate new developments, the second edition retains the original's wealth of wit and wisdom. The authors' insightful strategies, blended with their witty and irreverent style, make reading a profitable pleasure.

In Volume 4, the authors present a Diamond of a find, covering one-player games such as Solitaire. ... Read more

Customer Reviews (2)

The math is Kind of Hidden
This is volume four in the series, and it starts with page 801. Do you need to read the first three volumes first? Well, if you want to. The authors would get more royalties if you buy them. Do you need to in order to understand this volume? Generally speaking -- No. Only in a couple of areas might it help.

So, what do we have here?

A discussion of games, such as Rubik's Cube that you can play, and that they give instructions on how to make it come together. But don't get to thinking that this is all simple. Underneath it all, this is a fairly serious book on game theory, but the mathematics behind it are hidden.

Beyond the cube there are several other games discussed in this volume, some very beiefly, some getting a lot more attention - The last chapter in the book on the Game of Life gets some 35 pages.

As much as anything else, the authors witty writing style is a rare treat on a book like this.

Great reference for one person puzzles and games
The new edition of these classic volumes has been completely reorganized, and this volume now contains mostly one person games or puzzles, such as peg solitaire, Soma, Rubik's Cube, mechanical wire and string puzzles, sliding block puzzles, magic squares, and life.The book is very readable and requires no mathematical background.However, this is no lightweight watered-down book and some sections of the books could take you months to understand completely (try the SOMA map or century puzzle map that appears in the Extras).Fortunately you can just skip over these parts if you don't want to dig down to this level of detail.

I have only looked briefly at the other volumes, but I believe this volume "stands on it's own" more so than volumes 2 & 3.Be warned, however, that there are several concepts (such as "nim addition" that you will need the previous volumes to understand).

Conway's game of Life is the subject of the last chapter, perhaps the most interesting chapter in the book, and that which has probably been most changed since the last edition.Still, they could easily have expanded this chapter into a whole volume, and looking at the internet it is already out of date.

Beware that the figures on the covers of these volumes DO NOT necessarily correspond to what is inside.For example, Volume 3 shows peg solitaire on the cover but the subject itself is all in Volume 4! ... Read more

Product Description
By his thirty-third birthday, Moncure Conway was a Virginian who had abandoned the South, a minister who had rejected Christianity, an aristocrat who had embraced radical abolitionism and feminism, and one of the first American expatriates. He would live another forty-two years as an important transatlantic writer, reformer, and freethought minister, but in his American years he had already lived a lifetime and made his mark. This study of the antebellum South's most radical upper-class white male, whose life--until now--has eluded capture by historians, illuminates the demands of the antebellum Southern gentry, the nature of the abolitionist movement, the boundaries of 19th-century organized Christianity, and the tragic personal impact of the American Civil War. D'Entremont recounts Conway's dramatic career as social reformer, religious radical, and associate of such luminaries as Emerson, Wendell Phillips, Theodore Parker, Walt Whitman, and William Dean Howells.The book climaxes with the Civil War, which saw Conway, an abolitionist with two brothers in the Confederate army, agonized by his conflicting commitments to emancipation and peace. A brilliant portrayal of one of the most intriguing public figures in American history, Southern Emancipator combines important contributions to Southern history, women's history, and the history of antebellum reform and the American Civil War. ... Read more