Customer Reviews

Gamma: Exploring Euler's Constant (Princeton Science Library)

5 star:		(21)
4 star:		(4)
3 star:		(1)
2 star:		(3)
1 star:		(0)

 

 

Who would have thought that there can be so much life in a constant? And one with a Greek name! If you have some math interests, I predict that you will get caught up in the thread of events: They are mathematical topics, but are presented like in a novel or a drama. A book that I couldn't put down. The main characters are the harmonic series, the sub-harmonic series, Riemann's Zeta function, its functional equation, its zeros, the Riemann hypothesis(it is worth a million dollars!), the prime number theorem, (..hard stuff! but it somehow seems easy in this book),Bernoulli numbers, Pell's equation, the distribution of prime numbers.... And if you forgot some of your math, you will have it reviewed in the appendices. They are attractive, well written, and to the point.

I agree wholeheartedly with all the positive comments and enthusiasm that other reviewers have shown. This is a remarkable book, and there should be more like it. I am astounded at how much and what range of mathematics there is in a book of this length and level of accessbility. Which raises a very good point: This would be a superb book for "Calc III". It's unfortunate that many students end their study of mathematics slugging through integration by parts, partial fractions, sequences and series, the logarithm as integral, etc., the traditional hodge-podge of topics called Calculus II. And the ones who progress end up going straight into multivariable calculus with its div, grad, curl, and all that. There is never really any reward for all the work in hacking through Calc II. This book, however, would tie so much of it together, it would all suddenly seem so mysteriously connected and beautiful, and the reader (I hope) would want to go on to Complex Analysis. Thank you, Prof. Havil! I hope you find the proof to the Riemann Hypothesis.

I debated for a while whether this book deserved four stars or five. There's a lot of very interesting material here: if there's one thing this book does--perhaps better than any book I've read in quite some time--is show just how interrelated far-flung mathematical concepts can be (how are the prime numbers related to pi, for example?).

My one complaint about the book--and the reason for giving it four stars instead of five--is that there are times when the formulae and notation get so dense that it's extremely difficult to follow the author's train of thought: I can think of a number of places where diagrams would have helped immensely. Likewise, since there's no list of symbols or formulae, it's not a book that you can simply browse through, in the sense that you can browse through, say, "A Brief History of Time."

Finally, let me reiterate that this book assumes that you already know a fair amount of math: if you don't know what a capital pi means, for example, you're probably going to have a hard time understanding this book. But if you *do* know what that symbol means, though, then by all means, give this book a try.

I recommend this book to anyone who has learned calculus. The book is full of mathematical gems set in historical context. For example, one of the Bernoullis showed that the integral of 1/x^x from x=0 to 1 is the sum of 1/n^n from n=1 to infinity. If you appreciate that result, then this book is for you. I have found only one error in the first 45 pages (which is as far as I have read). On page 13 the author is trying to compute the average distance of a planet from the sun, assuming an elliptical trajectory, and tacitly assuming a uniform distribution in the angle that astronomers call true anomaly. For this distribution, despite the assertion at the top of page 13, it is easy to show that if the ellipse is not a circle, then the average distance from the sun is less than the semimajor axis of the ellipse. So the author is human after all.

There is more mathematics in this book, more interestingly explained than any other of its type that I have read. What is its type? Well, I guess it comes under the category of 'Popular math' or in the jargon 'trade books' but don't be fooled by that; to get something out of it you need some math knowledge-and more than that, some math ability. There are symbols everywhere. The constant Gamma just had to be the next single number to appear as the subject of a book and with the books on e and i under their belt I guess that Princeton would be the obvious publisher. Havil, a first-time author, could easily have made the grave mistake of writing a gritty book on analysis with gamma as its focus but he didn't do that. He decided on a historical approach and to divide his attention between the harmonic series, logarithms and gamma itself. As a result the book really is a mathematical odyssey (publisher's blurb)which embraces a huge number of ideas, each of which has a natural place in the development. For the first time I really understand Napier's approach to logarithms and how it relates to what we now define them to be. For the first time I understand that wakky Benford's Law. For the first time I understand the Riemann Hypothesis. There is stacks more stuff too. The Continued Fraction chapter has really got me into that cool idea, which I had only vaguely heard about. Euler was for me the man who solved the 7 bridges of Konigsberg problem-I had no idea just how fantastic he was.

They tell us that Havil is a math teacher at Freeman Dyson's old high school. What a school that must be-I wish I had gone there and I would like to have been taught by a teacher who is so clearly knowledgeable and enthusiastic about his subject! Dyson wouldn't put his name to the book just becaause he is an alumni. He rates it and its real easy to see why.

Every so often you come across a winner...for me this is the first this year.

Per the foreword, this book is "aimed at students of mathematics, be they eager high school students or undergraduates". As a summa cum laude graduate math major (of some years ago) I expected to enjoy an romp thru some beautiful mathematical ideas. Well, the ideas are there, and Havil is to be commended for gathering some unusual and interesting topics. And much of the extensive mathematical notation is supported with nice numeric examples. However, much of it is not. All too often there are pages of integrals, sums, and products that go happily on without a clue to some of the beautiful things that are happening. The most frustrating example is the "proof" of Euler's zeta function formula, one of the prettiest pieces of mathematics. I still cannot understand Havil's presentation. (It was thrilling to read the same proof in "Prime Obsession" by Derbyshire so I know it can be explained with simple algebra.) Also, "Gamma" appears to be intended to be read in one sitting since it is rarely possible to begin at an advanced chapter. It is assumed that you remember definitions and notations which have appeared long before. To the author's credit, there are occasional backward references by page number, but then, about half of these are frustratingly wrong. Finally, it would be nice to see a copy of the errata for this book. I hope this book appears in a 2nd edition where the level of its presentation is made much more consistent.

An amazing amount of mathematics very clearly and enthusiastically explained. I had never heard of some of the stuff and some that I did know about has been explained in very interesting ways; for example, the original definition of logarithms. Gamma I had vaguely heard of and name of Euler I had come across, but I had no idea what the number really was and why it is important until I read the book. Also, I had not realised just how big Euler was; it's amazing that his name is not better known. The 3 applications of logs are really surprising and very well chosen. The PNT and RH stuff is awesome. Now I can really say that I understand the first of them and appreciate the second (although the maths there was a bit too advanced for me). To have Freeman Dyson write the preface says a lot about the quality of the book and, as he says, you are gonna have to work at some areas to understand them-but who says that maths is easy? What it should be is INTERESTING and well explained and this author has a great knack with both. As the earlier reviewer says, he has a contagious enthusiasm. Bring on his next book!

Gamma seemed like an odd topic for a book, which looked intriguing and attracted me to the book. Rather than being 'a biography' of a number, Havil uses it as a way to introduce lots of really interesting math. Some of the contents I was familiar with, but he brings a fresh approach to what could be dull. Much of it was new and really enjoyable. It will keep any keen math student busy for hours, and give many instructors ideas to enliven their teaching. The math isn't trivial - you need a good calculus course behind you - but it's worth the effort. Highly recommended. I'll be at the front of the queue for this author's next book.

The spectrum of 'popular math' books extends from those which describe the ideas, without enmeshing themselves in the gritty detail, to those which deal with that detail, but avoid being text books; Havil has written one of the latter. In the Introduction he says that the reader will judge the level of success of the project, which was to 'explain interesting mathematics interestingly'; that's an honest statement which naturally provokes a firm answer either way; for me he has done a fantasic job. I am trained in mathematics but sold my soul to commerce (some years ago) and miss the excitement and beauty of a subject that I found frustratingly addictive. A book such as this rejuvinates the wonder, challenges the intellect, informs and entertains. I reckon that it's a gamble on the part of PUP to continue to develop a series of books which ask more than most of the reader, but I admire their commitment-and I guess that these sort of books sell, otherwise the axe would fall. So, Gamma is well thought-out, beautifully structured and incredibly wide in its content; it's impossible not to learn from it and not to be fascinated by it, whether it be a historical or technical nugget that appeals. Havil's love of Euler is very evident-and very appealing (and justified). I have seen several reviews of the book in respected places (MAA online and New Scientist for example) and they have been unqualified in their admiration; I am too (for what it is worth!) I admire those who work hard enough to give us enlightening books to read, it must take years to produce the finished product, and here we have an example of its kind which is simply exemplary. If you love maths, buy it-but only if you love symbols too- he doesn't avoid the technical details, he revels in them!

Evidently some "reviewers" should be reviewing books involving simpler mathematics; they clearly didn't get what this book is about!!

Havil's book is not really for the person in the street despite his introductory comments. You must have sufficient background to stand up to some lengthy derivations and the willingness to work through them in detail. If you lack these, I don't think you'll get much out of the book.

All in all, a very nice piece of mathematical writing!

Though I've found only a few errors in the entire book, a complete errata list would be nice.