Disquisitiones Arithmeticae

Disquisitiones Arithmeticae
Average customer rating:

understandable to all

Martin Christensen"s Review

Choose your edition carefully

Not bad for 204 years old.

Professor

Disquisitiones Arithmeticae
Carl F. Gauss , and W.C. Waterhouse
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0387962549

Book Description

English translation of standard mathematical work on theory of numbers, first published in Latin in 1801. "Among the greatest mathematical treatises of all fields and periods."--Asger Aaboe.

Customer Reviews:

understandable to all.......2007-03-09

What blows my mind about Gauss's Disquisitiones is that it is understandable to people like me who don't have an advanced background in mathematics. You just need to move through it slowly and carefully. There is a certain joy that comes along with making the discoveries that Gauss walks you through. A book that will change your outlook on life by enhancing what you see as the basic philosophy underlying numbers and mathematics. Mathemtical magicians like Euler and Lagrange will just confuse and dazzle you with their tricks, but Gauss will give you understanding.

Martin Christensen"s Review.......2007-01-08

This book is very good, but not quite excellent. Gauss spent, in my opinion, too much time on the theory of binary forms and too little on the general second-degree equation in two variables that the theory is such a big part of. However, the rest of the book was first-rate, well suited to one of the greatest mathematicians of all time. I considered it money well spent.

Choose your edition carefully.......2006-03-08

This is a great book. It's the place where modern number theory begins. It's also well enough written that it's enjoyable to read today.

There are two editions and you have to choose carefully. I don't know whether my review will appear on both editions or just one. The softcover is only $47. The hardcover is $129, but it is a REVISED translation. A single person, who does not seem to be have been a mathematician, made the first translation from Latin in 1965. That was revised by a team of 4 scholars in 1986.

That turns out to be important, because the original translator got a few things wrong, like the logic of a double negative. So there are some places where the first (cheaper, softcover, Yale) edition is either wrong or unclear. Many of these problems have been fixed in the second (more expensive, hardcover, Springer-Verlag) edition.

I bought the first edition and I have no regrets, but you will get extra value for your extra money if you buy the second edition.

Not bad for 204 years old........2005-07-10

Gauss wrote Disquisitiones between 1795 and 1801. If I can read and understand Clarke's translation in less time than it took Gauss to write the original, I will be doing well. The mathematics has not been superseded and the historical aspect is extremely valuable.

Professor.......2003-09-13

It is the best book for people who takes mathematics seriously

Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics (Mathematics and Its Applications)

Average customer rating:

I am waiting for Dixon 's Octonians...
Mathematics behind physics

Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics (Mathematics and Its Applications)
G.M. Dixon
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 0792328906

Book Description

The four real division algebras (reals, complexes, quaternions and octonions) are the most obvious signposts to a rich and intricate realm of select and beautiful mathematical structures. Using the new tool of adjoint division algebras, with respect to which the division algebras themselves appear in the role of spinor spaces, some of these structures are developed, including parallelizable spheres, exceptional Lie groups, and triality. In the case of triality the use of adjoint octonions greatly simplifies its investigation. Motivating this work, however, is a strong conviction that the design of our physical reality arises from this select mathematical realm. A compelling case for that conviction is presented, a derivation of the standard model of leptons and quarks.
The book will be of particular interest to particle and high energy theorists, and to applied mathematicians.

Customer Reviews:

I am waiting for Dixon 's Octonians..........2007-02-10

I have not yet received my command. P. MERAT

Mathematics behind physics.......1997-09-29

This is an excellent book for those who want to study Hamilton's quaternions, and other algebraic structures, used in modern physics. Dixon believes that octonions and triality of Spin(8) are essential in understanding particle physics. This clear exposition contains many ideas which have gone unnoticed from other researchers. The book is a treasure trove for mathematical physicists. The author also compares the Cayley algebra of octonions to other algebraic systems used in physics: matrices and Clifford algebras, in particular the Dirac algebra.

Average customer rating:

A complicated subject presented in a very uncomplicated manner.
Encompassing and Interesting

The Fabulous Fibonacci Numbers
Alfred S. Posamentier , and Ingmar Lehmann
Manufacturer: Prometheus Books
ProductGroup: Book
Binding: Hardcover

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Book Description

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature—from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, math educators Alfred Posamentier and Ingmar Lehmann take us on a fascinating tour of the many ramifications of the Fibonacci numbers. The authors begin with a brief history of their distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few. Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

Customer Reviews:

A complicated subject presented in a very uncomplicated manner........2007-08-30

The book provides much of the available information on the Fibonacci numbers. It starts with the life of Leonardo da Pisa, the man who first introduced the numbers to the world almost a thousand years ago. It describes the actual sequence, then demonstrates the connection that the numbers have to the natural, as well as to the world of the visual arts and of music. Even the stock market is not immune of the influence of the Fibonacci sequence.
What particularly impressed me about this book is the clarity with which the authors present the subject. Whether you are a mathematician or simply have an inquisitive mind, you will always know the exact meaning of the subject under discussion. In fact, you can skip the (sometimes) long mathematical formulae and still never lose track of the narrative.
A wonderful book that makes one ponder on the origin and significance of the created world. A must for mathematicians, scientists and generally educated individuals. A must also for those who believe that our universe and all its contents are only the product of a series of coincidences. These people may change their minds after becoming familiar with the Fibonacci numbers.

Encompassing and Interesting.......2007-08-23

The book contains many interesting and unpredictable properties of the Fibonacci numbers. I learned a lot of new things about them.

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication

Average customer rating:

Definitive text

Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication
David A. Cox
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Paperback

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ASIN: 0471190799

Book Description

Provides a general solution to the question of which primes p can be expressed in the form x² + ny². Covered first are the special cases considered by Fermat, which involve only quadratic reciprocity and the genus theory of quadratic forms. Further, the book shows how the results of Euler and Gauss can be fully understood only in the context of class field theory. Finally, in order to bring class field theory down to earth, the book explores some of the mignificent formulas of complex multiplication.

Customer Reviews:

Definitive text.......2004-03-02

This is the definitive text on the theory of primes in the form x^2 + ny^2. Problems with simple formulation like this (which primes can be written in that form, and for what n?) make the class field theory interesting and worth understanding. Those with interest in prime numbers would greatly appreciate the topics presented. Recommended for those with a strong background in number theory and algebra.

Factorization and Primality Testing (Undergraduate Texts in Mathematics)

Average customer rating:

Excellent Text on Factorization
Just the facts mam, just the facts.

Factorization and Primality Testing (Undergraduate Texts in Mathematics)
David M. Bressoud
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Book Description

This book focuses on a single problem: how to factor a large integer or prove its prime. From the Sieve of Eratosthenes of ancient Greece to the Multiple Polynomial Quadratic Sieve and the Elliptic Curve Methods discovered in the past few years, this self-contained text provides a survey of the heritage and an introduction to the current research in this field. It can also be used as an introduction to Number Theory and has the advantage over most texts in this area of being built around a unifying theme. With its strong emphasis on algorithms, it encourages learning through computation and experimentation.

Customer Reviews:

Excellent Text on Factorization.......2000-12-12

This is one of the most compact and best organization of material on the subject of factorization and primality.

Written by a promising author, this book explores factorization from the beginning to end, starting with the sieve of Eratosthenes and proceeding to much more complicated material. The book focuses on algorithms, and contains many useful ones, such as how to raise a number a to a power b, mod m. However, the primary focus of the book is factorization, so it contains algorithms for factorizations. They begin with trial division, then progress into Fermat's Algorithm and Pollard Rho. They eventually evolve into one of the strongest methods to date, Quadratic Sieve, and its child, Multiple Polynomial Quadratic Sieve.

This book is certainly a must for amateurs who are exploring the subject.

Just the facts mam, just the facts........1999-07-08

This book gives you the juicy bits you want for factoring and primality testing. It skips all the theory and presents the algorithims before you to observe and play with. For the amateur, this is an excellent first book.

Average customer rating:

Good reference maybe

Bezier and B-Spline Techniques
Hartmut Prautzsch , Wolfgang Boehm , and Marco Paluszny
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Book Description

This book provides a solid and uniform derivation of the various properties Bézier and B-spline representations have, and shows the beauty of the underlying rich mathematical structure. The book focuses on the core concepts of Computer Aided Geometric Design with the intension to give a clear and illustrative presentation of the basic principles, as well as a treatment of advanced material including multivariate splines, some subdivision techniques and constructions of free form surfaces with arbitrary smoothness.
The text is beautifully illustrated with many excellent figures to emphasize the geometric constructive approach of this book.

Customer Reviews:

Good reference maybe.......2006-08-09

I'm not sure what sort of person would find this book useful. If you want to LEARN about Bezier or B-Spline curves, then I don't think this book is for you. The authors' stated goal was to "provide a solid and uniform derivation" of Bezier and B-Spline properties. I believe that they have succeeded, almost every factoid I've seen in other books is included and proved rigorously. What is NOT included is any sort of meaningful explanation of the concepts. The material is presented at a breakneck pace, with everything presented exactly once, and most often in mathematical notation rather than in English. This is a good way to pack a lot of material into a few pages, but it makes for pretty difficult reading. I really don't see how a person could possibly follow the exposition in this book unless they basically already knew the material.

If you already are familiar with Bezier and B-Spline techniques, or perhaps you are an expert in a related field like approximation theory, then you might find this useful.

So if the Authors acheived their stated goal, then why only the 3-star rating? I've recently bought quite a few of these books on splines, in preperation to write a chapter in my own (introductory) 3d math book. Most of the material in this book can be found in other books. Of course, all of the books have significant overlap, since they are covering the same subject. But if you could have a book that covers most of the same material AND also has more exposition, I'd go for the one with the more exposition. I don't think there's enough unique material in this book to trade off the loss of the much more graceful exposition in the other books.

In particular, if you're interested in *learning* about Bezier curves or B-Splines, I found these two books much more accessible:

An Introduction to Splines for use Computer Graphics and Geometric Modeling by Bartels et al. has a much slower pace - paradoxially this means that if you're learning the material, you will be able to read it FASTER since you can maintain a constant pace. It focuses on B-Splines and only stops breifly to mention Bezier curves as a general case.

The other book is Curves and Surfaces for COmputer Aided Geometric Design by Farin. He introduces Bezier curves and fully develops them, at a reasonable pace, before discussing B-splines. I personally found this approach to be better from a teaching perspective, since B-Splines are more abstract and Bezier curves are easier to understand. (Be warned that I have a rather old editition, the 2nd, I believe he's on the 5th now. I don't know what all has changed since then.)

Average customer rating:

This book is amazing!
Fantastic Book!
Review for Mathematical Olympiad Challenges.
Pure delight!
Can I write in Portuguese ?

Mathematical Olympiad Challenges
Titu Andreescu , and Razvan Gelca
Manufacturer: Birkhäuser Boston
ProductGroup: Book
Binding: Paperback

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Book Description

This significantly revised and expanded second edition of Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory from numerous mathematical competitions and journals have been selected and updated. The problems are clustered by topic into self-contained sections with solutions provided separately. Historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on creative solutions to open-ended problems.

New to the second edition:

* Completely rewritten discussions precede each of the 30 units, adopting a more user-friendly style with more accessible and inviting examples

* Many new or expanded examples, problems, and solutions

* Additional references and reader suggestions have been incorporated

Featuring enhanced motivation for advanced high school and beginning college students, as well as instructors and Olympiad coaches, this text can be used for creative problem-solving courses, for professional teacher development seminars and workshops, for self-study, or as a resource for training for mathematical competitions.

-----

From a review of the first edition:

"This [book] is…much more than just another collection of interesting, challenging problems, but is instead organized specifically for learning. The book expertly weaves together related problems, so that insights gradually become techniques, tricks slowly become methods, and methods eventually evolve into mastery…. The book is aimed at motivated high school and beginning college students and instructors. It can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions, and for teacher professional development, seminars, and workshops.

I strongly recommend this book for anyone interested in creative problem-solving in mathematics…. It has already taken up a prized position in my personal library, and is bound to provide me with many hours of intellectual pleasure."

—The Mathematical Gazette

Customer Reviews:

This book is amazing!.......2003-02-19

I remember the first time I touched this book, i fell so in love with it that it was very hard for me to remember how many other things I had to do during my day. It really illustrated how every problem you solve (or at least try really hard) can be an entire lesson you can use later on.
It is very well organized, even the problems in each section are set in a way that each one helps with the previous one in case a more creative solution doesn't show up...
I love this book, and I really recommend it for any student studying for any math contest around the world. It really helped me, and I'm sure it will do the exact same thing to anyone with the desire to spend countless hours solving beautiful math problems. Good luck, God bless you all :)
Pura vida.

Fantastic Book!.......2001-08-06

This is a marvelous book for lovers of mathematical problems. Scattered about are wonderful problems in Geometry, Trig, Algebra and Analysis, Invariants, and Number Theory. A truly delightful read that will have you working on some problems for hours. Each section introduces the reader to the concept or technique needed to solve the problems in each section. The problem sets start off with a few "warm-up" problems that quickly build up to some that require keen (some brilliant!) insight. A true gem among most problem books since this book is not merely a book of problems, but also contains clear presentations and introductions to various concepts in mathematics. The solutions are a true delight, the ingenuity and beauty of mathematical problem solving is captured exquisitely in this fabulous book. Highly Recommended. A++

Review for Mathematical Olympiad Challenges........2001-04-05

The book, Mathematical Olympiad Challenges", is a delightful book on problem solving written by two of the leaders of the craft. Mathematical problem solving is a skill that can be honed like any other and this book is an ideal tool for the job. Problem solving usually involves elementary mathematics; this does not mean "easy mathematics". An elementary mathematical problem is one that is easily stated and can be understood by anyone who has had basic training in the subject (up to calculus). The solution, though, may be quite hard and may require a great deal of ingenuity and thought.

It should be noted that being an exceptional problem solver does not necessarily make one a good mathematician, but it helps. This is certainly true of the second author who is also a renowned mathematician in the field of knot theory and three dimensional topology.

As mentioned the two authors have a sterling record in the arena of problem solving and in coaching would be problem solvers. I am more familiar with Razvan Gelca who led the University of Michigan team to a top five finish in the highly competitive and extremely challenging Putnam exam. This exam is administered yearly and is open to all college students in North America; usually around 430 universities and colleges send teams to compete in the Putnam. The exam has been offered since the thirties and finishing at the top carries a great deal of prestige. Razvan's superior abilities led to the spectacular success of the Michigan team which was no mean feat.

My own experience with the book has been one of revelation with each passing page. I used the book to teach the problem solving course at the University of Michigan, Ann Arbor, and it helped me immensely. The book possesses a variety of topics in elementary mathematics, ranging from algebra to geometry to trigonometry to number theory. Each chapter is divided into sections and each section has a theme. In keeping with the theme, the authors mention some useful formulae and/or facts that may be used in that section. This is followed by a demonstration of some dazzling problem solving techniques applied to a couple of problems. This is then followed by a list of challenging problems of varying levels of difficulty, all related to the theme of the section. There are roughly 18 such sections and many, many problems to think about. The rest of the book, which is the bulk of it, is dedicated to providing elegant solutions to every problem posed in the first part. Occasionally a problem merits more than one solution and sometimes the way is pointed to some interesting mathematics. The authors also acknowledge the source of many of the problems in the book which is a good indicator of the pedigree of the problem. Almost every solution is a gem and each problem demands its own style of solution. As noted earlier, problem solving is a skill and the authors try and succeed in conveying that idea in the problems and solutions they present.

Here is a sample problem from the book; if you can't do it and want to know how, check out the book:

"Show that any cube can be divided into 'n' cubes for any integer 'n' bigger than 54."

In summary if you are interested in figuring out puzzles, if you are a problem solver of elementary mathematical problems, or if you are just plain curious how a large fraction of mathematicians got hooked on mathematics, I would highly recommend you give this book a try. You may learn something and may even enjoy yourself in the process.

Pure delight!.......2001-02-10

I have spent many entertaining hours going through the book only to realize how rusty I became in these last 25 years that passed since I was faighting a hopeless battle in the Romanian solvers' contest against the first author.

The book is centered around a number of tools, tricks, techniques, as you want to name them, which are then used to solve a number of problems depending on them.

The problems are carefully arranged in increasing order of difficulty (maybe that's the reason I was able to solve some of the first entries!), so that the reader is not immediately discouraged by problems too hard to solve.

The actual selection of the problems clearly reflects the taste of the authors. Not a bad taste, I would say, if one notices that Titu Andreescu was the man behind the brilliant success of the USA olimpic team in 1994 (all whose members got then the maximum possible number of points).

I was one of the coaches of an olimpic team myself and I know how fast one goes through a list, I would say through *any* list of problems with these gifted guys. From this point of view, the book is an essential instrument for all who contemplate being involved in problem solving training.

But the book is a good teaching tool for high school teachers who wish to challenge their best students with more interesting problems. As another potential pool of customers, I would say that those old sea wolfs as myself, who get bored from time to time of the technicalities of professional maths will find this book a nice companion.

Yes, I like this book and I warmly recommend it to all lovers of problem solving.

Can I write in Portuguese ?.......2000-05-30

Este é o melhor livro que já vi sobre olimpíadas de matématica, comparando-se com o Winning Solutions e Problems Solving Strategies que também são vendidos pela Amazon.com, que é sem dúvida a maior e melhor livraria do mundo. Este livro cita vários problemas conhecidos de importantes olimpíadas como IMO e ASIAN PACIFIC além de outras, além de abordar interessantes técnicas de como resolver problemas como substituição de variáveis trigonométricas. Pode ser considerado realmente um incomparável e imperdível livro que todos os alunos e professores de olimpíada de matemática deveriam ter... É realmente um tesouro matemático !

Average customer rating:

A Modern Whittaker and Watson, Buy It
A book comes close to " A course of modern analysis "
clean and concise
clean and concise
clean and concise

Special Functions
George E. Andrews , Richard Askey , and Ranjan Roy
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521789885

Book Description

Special functions, which include the trigonometric functions, have been used for centuries. Their role in the solution of differential equations was exploited by Newton and Leibniz, and the subject of special functions has been in continuous development ever since. In just the past thirty years several new special functions and applications have been discovered. This treatise presents an overview of the area of special functions, focusing primarily on the hypergeometric functions and the associated hypergeometric series. It includes both important historical results and recent developments and shows how these arise from several areas of mathematics and mathematical physics. Particular emphasis is placed on formulas that can be used in computation. The book begins with a thorough treatment of the gamma and beta functions that are essential to understanding hypergeometric functions. Later chapters discuss Bessel functions, orthogonal polynomials and transformations, the Selberg integral and its applications, spherical harmonics, q-series, partitions, and Bailey chains. This clear, authoritative work will be a lasting reference for students and researchers in number theory, algebra, combinatorics, differential equations, applied mathematics, mathematical computing, and mathematical physics.

Customer Reviews:

A Modern Whittaker and Watson, Buy It.......2004-10-24

This book is great. It is the best overview I have ever seen of the primary special functions, as seen from a modern viewpoint. Buy it and you will spend many happy hours reading the theorems it contains, and doing the excercizes at the end of each chapter.

A book comes close to " A course of modern analysis ".......2001-08-07

Though this book cannot be compared to Whittaker and Watson's classic book. It comes quite close to it. I just want to comment on the the area covers are too concentrated and the rigorous manner which is the hall mark of " Modern Analysis " is lacking. Anyway, this book deserves 5 stars.

clean and concise.......2001-02-11

It has a very good style of writing for the nature of mathematics. It is clean, no unnecessary explanation or examples. In a way, one can feel something similar to Axler's. It is an excellent reference book. One should keep this book just as Axler's Linear Algebra Done Right, Numerical Recipe, DE Knuth's Art of Programming.

clean and concise.......2001-02-11

Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables

Average customer rating:

Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables
Michel Waldschmidt
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Number Theory: Volume II: Analytic and Modern Tools (Graduate Texts in Mathematics)
The Arithmetic of Dynamical Systems (Graduate Texts in Mathematics)

ASIN: 3540667857

Book Description

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. It includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions and Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is the systematic use, in proofs, of Laurent's interpolation determinants. The most general result is the so-called Theorem of the Linear Subgroup, an effective version of which is also included. It yields new results of simultaneous approximation and of algebraic independence. Two chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

Average customer rating:

Great Book, Where It Sticks To The Topic
A Delightful Read
An interesting history - even when it gets off topic
A page turner
Entertaining and irksome

A History of Pi
Petr Beckmann
Manufacturer: St. Martin's Griffin
ProductGroup: Book
Binding: Paperback

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e: The Story of a Number
The Golden Ratio: The Story of PHI, the World's Most Astonishing Number
An Imaginary Tale: The Story of "i" [the square root of minus one]
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The Joy of Pi

ASIN: 0312381859

Book Description

The history of pi, says the author, though a small part of the history of mathematics, is nevertheless a mirror of the history of man. Petr Beckmann holds up this mirror, giving the background of the times when pi made progress -- and also when it did not, because science was being stifled by militarism or religious fanaticism.

Customer Reviews:

Great Book, Where It Sticks To The Topic.......2007-09-06

First, let me say that this book is a good overview of a persistent mathematical problem; in this case, deriving the value of "pi," or the ratio between the diameter and circumference of a circle. The author begins at the beginning, by going over Stone Age "mathematics" and showing how and when it occurred to early humanity that this ratio existed. Second, Beckmann is very good when it comes to explaining the mathematics of pi and how it was analyzed historically in mathematical fashion. He also has a good handle on the primary-source material (i.e. historical treatments of pi), and can explain them in modern terms. All told, this makes it useful to someone who is new to the history of mathematics and wants to learn about one of its foremost problems.

Having said that, Beckmann clearly has some faults:

1. He frequently diverges into anti-communist rhetoric, not only tangentially, but at times when it's completely irrelevant and superfluous.

2. He views the past anachronistically; specifically his hatred of the Romans, and his contempt for Aristotle, are obvious.

He is technically correct on many of these scores; the Romans were, in fact, brutes compared to the Greeks, Carthaginians, etc. Aristotle also was also overrated. He is also correct in that, even in recent times, the Romans and Aristotle are given too much credit for things. None of that is in doubt.

What is troubling is that he arrives at these conclusions anachronistically. He see Aristotle as overrated, simply because Aristotle did not emphasize quantitative analysis over qualitative. But as a scientist he should realize that qualitative analysis has its place in the long process of learning. That Aristotle did not do what Beckmann personally (as a 20th century scientist) wished he had done, does not mean Aristotle contributed nothing useful to human knowledge.

Beckmann similarly laughs at other historical figures, implying that their lack of (modern-day) mathematical ability makes them contemptible. In his forward he even mentions that people took him to task for this, and appears amused at this critique. I'm not sure he understands the problem with anachronistic thinking, however. He clearly sees himself as "more clever" than figures of the past, as well as his colleagues who think he went too far in condemning the Romans and Aristotle; this shows a certain amount of hubris which is probably not penetrable.

His anti-communist rhetoric is, perhaps, more understandable, since he lived and studied under a communist regime, and later escaped from it. Even so, much of what he says about communism has no place at all in this book ... it's more or less irrelevant to the topic. Beckmann clearly could have written an "insider's" account of the faults and dangers of communist ideology, and perhaps he should have done so; but not here, in the guise of a treatise on pi.

To sum up: This is a valuable read, but only if you filter out Beckmann's anachronistic, personal biases.

A Delightful Read.......2007-05-06

Beckmann's analysis of the history of pi is not the dry text that many would expect after hearing the title. The story that he lays out is not simply about the quest to figure out the number pi to as many decimal places as possible, but it is a comprehensive examination of the history of mathematics and science.

I admit, the mathematics in this book was beyond my knowledge. If you aren't already acquainted with geometry, trigonometry, complex algebra, and calculus, you might have some trouble keeping up with the proofs that pepper many of the pages. I assure you, the book is worth reading even if you skim over the mathematical proofs without understanding them. (As a side note, the only reason this book gets four stars is because the non-expert in math cannot read the book and understand the proofs that Beckmann provides).

There are many great chapters in this book that contain excellent expositions about various historical figures and their impacts. Beckmann delights in the exhaltation of great mathematical thinkers and in the putting down of what he thinks of as lesser minds (for example, Aristotle). In one part of the book he makes fun of Aristotle: "Aristotle used his lofty intellect to deduce that heavier bodies fall to the ground more rapidly; that men have more teeth than women; that the earth is the center of the universe; that heavenly bodies never change; and much more of such wisdom, for he was a very prolific writer." The chapters about Euclid, "The Roman Pest", Archimedes, Pascal, Newton, and Euler were all splendid, but I especially loved the chapter about the struggles between the superstitious church and the intellectual mathematicians and scientists.

Parts of this book were even amusing. Towards the end of the book, Beckmann recounts the stories of a few more modern "thinkers" who have claimed to "square the circle" (that is, who have found a value of pi that is rational). The author wastes no time to expose the weak-mindedness of these men, who were using values of pi that were used thousands of years earlier.

Overall, I loved the book and I intend to read it again in the future. If you're looking for a great read, pick this book up (but don't expect to be able to understand the math unless you are already an expert). When a friend told me to read this book four years ago (I finally got around to it), he told me that I would learn something that might save my life someday. I can only think of two possibilities: (1) Do not contradict the Bible by declaring your own beliefs to be more intellectual (because ignorant superstitious people often take matters into their own hands and torture/kill those who do), or (2) Do not waste your life trying to square the circle, because it cannot be done (as Michael Stifel put it: "Futile is the labor of those who fatigue themselves with calculations to square the circle."). Get a copy of this book and enjoy!

An interesting history - even when it gets off topic.......2007-04-03

It's very difficult to to summarize my opinion of this book. I did enjoy reading it, that much is true. But I can imagine it not being for everyone. The book has a strange mixture of math and history which I really enjoy, however the math is sometimes hard to follow and Beckmann often goes on long rambles in the book about his personal opinion about certain elements of history. I actually enjoyed those rambles, even if I didn't always agree; he has a talent for stating his complaints in a very humorous manner. I would suggest it to people but only those who are willing to think about mathematics in their free time and those willing to put up with rants about personal opinions. I did encounter some minor printing errors in my copy of the book, like the printing of "Chapter Four" at the top of one of the pages in Chapter Five. These did not, however, inhibit my reading in the least.

Beckmann is fascinated by history and it shows. He has a true enthusiasm for the subject and it makes the book more interesting to read. It does cause him to get off track at times, talking about things that have little relevance to pi, like the evolution of the calendar. These sections did not bother me as they were generally interesting and did not make the book that much longer (it is a fairly short book as it is, less than 200 pages); however, I can see some people getting irritated by this. Beckmann has a fairly conversational style of writing and this has much of the same result as the above. Like I said earlier, he makes no attempt to hide his opinions of various historical events and people, nor does he claim to be doing so: in the Preface he clearly states that he has never hesitated to "vent" his opinions. While this does mean that one must be critical of basing one's opinions off of his, I find it fairly amusing myself. For example, Beckmann spends basically the entire fifth chapter ranting about how the Romans are completely overrated and are basically just a bunch of thugs. I started wondering just what they had done to him until the last paragraph gave me the answer: the Romans killed Archimedes. It is also fairly clear that he absolutely detests Communism, fascism, and, for some reason, the UN. While I didn't always agree with his opinion, I found them amusing and they did not detract from the book.

The highest level of math in the book is basic calculus. It is not, however, necessary to completely understand the actually mathematics as long as one can follow the basics of what Beckmann is saying. Obviously, understanding what the mathematics he does are adds to the pleasure of the book but it is not necessary to remember more than basic algebra to really keep track of what is going on. What I'm trying to say is that it's very easy to skip all the math and still understand what is going on (more or less).

The very last section did bother me because it was all too prophetic and trying very hard to be significant. Beckmann envisions a time when "intelligent computers will make a better job of keeping peace among men and nations than men have ever been able to" (pg 189). I really don't think this is possible. A History of Pi is a very good little book but it is not for everyone. Besides being interested in the subject matter, it is also necessary to be able to find amusement in a well-stated opinion - even if it doesn't agree with one's own.

A page turner.......2007-03-03

I found this book very entertaining and enlightening. It's a history of mathematics, rather than of Pi alone. It reads better than most novels and presents mathematics in a much more intriguing and appetizing way than what one learns in school. It should ignite interests in young readers and entice them to explore mathematics-related career fields. Good Christmas presents for your teenage nieces and nephews.

Entertaining and irksome.......2006-11-10

AHoP is a quick read, light on mathematical explanation, heavy (at times) on mathematical formulae, scant in detailing the lives and work of Pi-minded mathematicians, and annoying for its frequent negative pronouncements on historical personnages and entire civilizations (though I share many of the author's views, I tired of reading them in a work on piece of mathematical history). Let me note one more substantive annoyance: PB rarely explains how an original proof was arrived at, preferring instead to illustrate the relative accuracy of older methods by recourse to modern mathematics. Though I did find it interesting to learn how effective ancient and more recent methods have been at yielding approxmimations of pi, I would far rather have read detailed discussions of the proofs of Archimedes et al, in their own terms. I realize, of course, that such discussions would have necessitated a much longer book, and an increasingly abstruse one at that. Under the circumstances, PB packed quite a lot into AHoP. If nothing else, AHoP has greatly whetted my appetite to read other treatments of pi, squared circles, and the like.

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