Customer Reviews

A Friendly Introduction to Number Theory (2nd Edition)

5 star:		(8)
4 star:		(2)
3 star:		(3)
2 star:		(1)
1 star:		(0)

 

 

I just wanted to make clear the point that each textbook or math book written is written for an INTENDED audience, and it's not fair to negatively criticize a book by using the reviewer's own personal background, rather than the INTENDED audience, as the guide for criticism.

This book was not written for math majors. So, I find it kind of distressing to hear that many math majors are saying this was textbook for a beginning number theory class for math majors. Silverman makes effort to point out that the book was written as the textbook for a general liberal arts math class, which is actually taken by non-science and non-math majors at the university where Silverman teaches. It requires nothing beyond basic calculus (if that), and I don't see anywhere where Silverman gives the impression that the book is meant to be used as a strong introduction to writing proofs or becoming fluent in rigorous mathematical arguments which math majors will later see.

So, of course, math majors will find fault...but the book wasn't written for them. It was written primarily to get people who have little interest in math or little exposure to math, some opportunity to see something more interesting beyond high school algebra and calculus. The emphasis on computation is warranted in any case, because although number theory is mathematics and has rigorous proofs, intuition and working familiarity with the concepts and constructions of number theory only come through hours and hours of simple computations with the positive integers. Computation is a legitimate and necessary part of number theory.

As for rational points on the circle (and Fermat's Last Theorem) being unusual or out of the ordinary material, this is farthest from the truth. The example of rational points on the circle is one of the oldest (2,000 years or so???) and most basic constructions of number theory, revealing how geometric number theory is, and the example directly leads to more general ideas and concepts which are central to current research (Diophantine equations, elliptic curves, projective geometry, for example) and pick up many of the standard graduate references on elliptic curves and the first 5-10 pages are a detailed examination of this very example.

I'm a graduate student studying number theory, so I'm pretty far away from the intended audience. But I can see that the book does a pretty good job at what it sets out to do, namely present an exposition of certain problems in mathematics, accessible to non-science and liberal arts majors, in a leisurely and engaging fashion, and to get the students to do their own basic pattern-searching, computation, data collection and conjecturing (ALL important facets of mathematics...proof is the polished product, but lots of time is spent by mathematicians before even GETTING to the point of proving things.)

This sounds like a fairly "friendly" introduction to me. If you want more, check out Niven, Hardy/Wright, Ireland/Rosen, Apostol, Gauss, etc.

I very much enjoyed this book. The book is indeed an excellent and "friendly" introduction to number theory. Dr. Silverman writes in a conversation style. I felt like I had a friendly tutor standing over my shoulder explaining not only how the mathematics worked, but, more importantly, why the topics he described or was about to describe are important and their relevancy in either the world of mathematics or in the "real" world. While he has very few "formal" proofs compared to most number theory texts, Dr. Silverman thoroughly works through numerous numerical examples to give the reader a "feel" for what is going on.

I was particularly pleased with Dr. Silverman's chapter explanation of why quadratic residues are important and how they are used.

Dr. Silverman presents introductory explanations of a number of frequently mentioned number theory topics such as Mersenne Primes, number sieves, RSA cryptography, elliptic curves. He ties together lucid explanations of Pythagorean triples, x2 + y2 = z2, x4 + y4 = z4, and elliptic curves to build to an explanation of Wiles proof of Fermat's Last Theorem.

Although the book is intended for non-math majors in college, it's ideal for advanced high school or even junior high students. Only knowledge of high school algebra is needed for the book. I recommend that the book be made mandatory reading in a advanced high school math class such as calculus or precalculus. My reasoning is that most advanced high school math classes such as calculus are too application-oriented and students often mistake manipulation of formulas for what mathematics is about. A book like Silverman's can spark the beginning of a brilliant career in mathematics.

I admit that this book might not be suitable especially for pure mathematicians. But I very much liked Silverman's way of writing: He cast questions and encourage readers to tackle them! Indeed, this is a unique number thoery book written in that way.

This book is popularly used for an introductory course in number theory, which also often serves as an introduction to proofs. This is how I came across this text and in that context it fails. To the author's credit, however, he never intended the book to be used in this manner and he states that in the beginning. This book was designed for non-math majors (and perhaps eager high-school students) and is clearly aimed towards them.

The book succeeds in giving a survey of number theory but is not rigorous enough or detailed enough for math majors. Many of the exercises focus on computations and discovery of patterns but not on proofs. While this is not necessarily a bad thing, especially for the intended audience, math majors need to be reading and writing rigorous proofs rather then merely spending their time performing computations. Attempting to provide this rigor, my professor required proofs of many of the exercises which were not intended to be proven and because of this many students were often unable to provide a proof, as they lacked other necessary tools.

For people interested in learning a little bit about number theory, or just math in general, this is an excellent text. For an undergraduate math course for math majors, however, look somewhere else, just as the author intended.

I'm a bit shocked by the bad reviews of this book. I guess if a book is clear, understandable, and interesting it doesn't qualify as a worthy math book. I've noticed this in other math book reviews, there seems to be a real element of machismo among alot of mathematicians, if the book is very formal usually theorem proof theorem it gets high marks. If the book is like this, one where the author is clear and engaging the book is discounted. I personally found this book to be one of the great intellectual adventures of my life, but here again in college I only minored in math not majored.

I am a working oceanographer with a physics background who is interested in browsing through various areas of mathematics, particularly ones like number theory which are not a common part of a physicist's background. I picked up and read Dr. Silverman's "Friendly Introduction to Number Theory" and was thoroughly charmed. The book presented many of the basic results of number theory in a clear, concise fashion, and also gave a bit of context and background to the results. Basis computations for "non-experts" are stressed, and the reader for whom this book was intended goes away with a nice feeling of having picked up a bit of knowledge of a new topic. I would also add my voice to those who chided the math majors for panning this book. There are plenty of high level "theorem-proof" books out there for mathematicians, and to criticise a book that popularizes mathematics is both snobbish and counterproductive. We should heartily applaud and value good popularizations of science and technology. This book is a first rate popularization.

I think its a brilliant introduction to someone like myself, who teaches high school maths to some very able students who may well go on to undergraduate maths courses.
My first degree was in engineering, so I havent had the priviledge of a second level number theory course.. Its ideal! I love the conversational style.. and there are recommendations of books with more rigour, but dont be fooled into thinking this is easy.. Its a demanding read, at least I think so.

I can understand much of the criticism that I read here from frustrated math majors. I just want to say that, as an engineer who took a number theory course for fun, this was a great introduction to the subject. I found it very readable and easy to understand. It got me interested in number theory - enough so that I would consider reading a bit of it on my own time as I pursue further education in science. It seems that engaging non-mathematicians is the intent, so I have to consider the book a success.

P.S. Can you really complain about the style of the book after reading the title? You certainly can't claim false advertisement.

I used this book for my Introduction to Number Theory class. I enjoy Silverman's writing style, but I wish there were some more examples and a little bit more theory involved.

It seemed to me as though there were a LOT of topics covered in a short about of time, but I would have liked to have seen some more of the actual meat behind it.

Not bad though!