A Friendly Introduction to Number Theory (2nd Edition) [Hardcover]

Joseph H Silverman (Author)

Book Description

ISBN-10: 0130309540 | ISBN-13: 978-0130309549 | Publication Date: February 15, 2001 | Edition: 2

This introductory text is designed to entice non-math focused individuals into learning some mathematics, while teaching them to think mathematically. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. Pythagorean Triples, Linear Equations and the Greatest Common Divisor, Factorization and the Fundamental Theorem of Arithmetic, Congruences, Mersenne Primes, Squares Modulo p, Quadratic Reciprocity, Pell's Equation, Diophantine Approximation, Irrational Numbers and Transcendental Numbers, Sums of Powers, Binomial Coefficients and Pascal's Triangle, Elliptic Curves and Fermat's Last Theorem. For individuals with limited math experience who are interested in number theory.

Editorial Reviews

From the Publisher

This is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradu ally led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples which are analyzed for patterns and us ed to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. --This text refers to an out of print or unavailable edition of this title.

From the Back Cover

This introductory text is designed to entice non-math focused individuals into learning some mathematics, while teaching them to think mathematically. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. Pythagorean Triples, Linear Equations and the Greatest Common Divisor, Factorization and the Fundamental Theorem of Arithmetic, Congruences, Mersenne Primes, Squares Modulo p, Quadratic Reciprocity, Pell's Equation, Diophantine Approximation, Irrational Numbers and Transcendental Numbers, Sums of Powers, Binomial Coefficients and Pascal's Triangle, Elliptic Curves and Fermat's Last Theorem. For individuals with limited math experience who are interested in number theory.

Product Details

• Hardcover: 383 pages
• Publisher: Prentice Hall; 2 edition (February 15, 2001)
• Language: English
• ISBN-10: 0130309540
• ISBN-13: 978-0130309549
• Product Dimensions: 9.3 x 6 x 1 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.2 out of 5 stars  See all reviews (14 customer reviews)
• Amazon Best Sellers Rank: #1,125,914 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

72 of 74 people found the following review helpful:

4.0 out of 5 stars This book was NOT written for math majors, January 18, 2001

By 

Darin Brown "revolver13" (Goleta, CA United States) - See all my reviews
(REAL NAME)   

This review is from: A Friendly Introduction to Number Theory (Hardcover)

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.

16 of 16 people found the following review helpful:

5.0 out of 5 stars A really friendly, enjoyable introduction to number theory, February 1, 2001

By A Customer

This review is from: A Friendly Introduction to Number Theory (Hardcover)

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.

16 of 16 people found the following review helpful:

5.0 out of 5 stars A truly friendly introduction to number theory, November 3, 1998

By A Customer

This review is from: A Friendly Introduction to Number Theory (Hardcover)

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.