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In this delightful guide, a noted mathematician and teacher offers a witty, historically oriented introduction to number theory, dealing with properties of numbers and with numbers as abstract concepts. Written for readers with an understanding of arithmetic and beginning algebra, the book presents the classical discoveries of number theory, including the work of Pythagoras, Euclid, Diophantus, Fermat, Euler, Lagrange and Gauss.
Unlike many authors, however, Mr. Friedberg encourages students to think about the imaginative, playful qualities of numbers as they consider such subjects as primes and divisibility, quadratic forms and residue arithmetic and quadratic reciprocity and related theorems. Moreover, the author has included a number of unusual features to challenge and stimulate students: some of the original problems in Diophantus' Arithmetica, proofs of Fermat's Last Theorem for the exponents 3and 4, and two proofs of Wilson's Theorem.
Readers with a mathematical bent will enjoy and benefit from these entertaining and thought-provoking adventures in the fascinating realm of number theory. Mr. Friedberg is currently Professor of Physics at Barnard College, where he is Chairman of the Department of Physics and Astronomy.
- Sales Rank: #613222 in Books
- Published on: 1995-01-09
- Released on: 1995-01-09
- Original language: English
- Number of items: 1
- Dimensions: 8.45" h x .49" w x 5.42" l, .53 pounds
- Binding: Paperback
- 240 pages
Most helpful customer reviews
15 of 15 people found the following review helpful.
What a carefully written exploration!
By Jeffrey L. Cooper
I think this book is a masterpiece in mathematical exposition. All you need to know is how to add, subtract, multiply, and divide and maybe a vague memory of algebra. Mr. Friedberg will walk you through a lot of number theory after which (or maybe even during which) you may find a number theory textbook more approachable. If you read carefully you will really internalize what a proof by contradiction is and what an infinite descent is. You'll get a real appreciation for the logic of a proof and you'll see some ingenius tricks used by some great mathematicians ... and you'll understand them!
This book is approachable and doable by anyone with a motivation for what can be understood about numbers. And I can't stress how carefully, thoughtfully, and articulately it is written.
19 of 21 people found the following review helpful.
An impishly, well-written introduction to number theory.
By A Customer
The author's enthusiasm shines through as he explains primes, perfect numbers, quadratic forms, and more. The explanations are clear: not too easy, but not too hard; Mr. Friedberg does a remarkable job of gauging the reader's level (at least MY level!).
I didn't realize number theory was so much fun until I started reading this book.
Note: I noticed a small typo on p.95: the equation to generate Pythagorean triplets is missing a 'square' on the left hand side.
14 of 15 people found the following review helpful.
An informal introduction to number theory with a historical perspective.
By N. F. Taussig
Friedberg's text, which is written in an inviting conversational tone, is an idiosyncratic introduction to number theory which stresses the subject's historical development. The material is introduced through problems that motivate the results that Friedberg discusses. These results include Euclid's theorem that there are infinitely many prime numbers, the use of the sieve of Eratosthenes to find prime numbers less than the square root of a positive integer n, Gauss' Fundamental Theorem of Arithmetic, perfect and amicable numbers, Pythagorean triples, modular arithmetic, factoring numbers of the form x^2 + ny^2, and the Law of Quadratic Reciprocity. Friedberg ably links these topics together and places them in historical perspective. However, there are better introductions to number theory. This text has no formal exercises, so you do not have an opportunity to reinforce what you are learning. It is also a poor reference because definitions, theorems, and proofs are stated within paragraphs, the whimsical chapter titles do not convey what topics are covered, and there is no subject index to help you find the definitions and theorems that are buried within the paragraphs. Also, the scope of coverage is less than that of other introductions to number theory.
In his introduction, Friedberg, a physicist, distinguishes between the common and scientific meanings of the word theory. He also discusses the difference between a scientific theory and a mathematical theorem.
Friedberg uses sequences to introduce proofs by mathematical induction. Friedberg shows how proofs of mathematical induction work and discusses why they are valid. In the text, however, he tends to use Fermat's method of infinite descent to prove assertions indirectly rather than using direct induction proofs.
While discussing these sequences, Friedberg refers to 1 as a prime number, contrary to the usual definition that a prime number is a positive integer larger than 1 whose only factors are 1 and itself. Defining 1 to be prime would violate the assertion of the Fundamental Theorem of Arithmetic that each positive integer has a unique prime factorization. For instance, if you allow 1 to be prime,
6 = 2 x 3 = 1 x 2 x 3 = 1 x 1 x 2 x 3 = ...
This is problematic, so Friedberg disregards his own assertion that 1 is prime when discussing the Fundamental Theorem of Arithmetic.
Friedberg then discusses some results from ancient Greece, including the fact that the square root of 2 is irrational, Euclid's theorem that there are infinitely many primes, the sieve of Eratosthenes, perfect numbers, and amicable numbers. He also proves the Fundamental Theorem of Arithmetic while covering these topics.
During a brief discussion of Diophantine equations, Friedberg discusses how to find and factor Pythagorean triples, that is, triples (x, y, z) of positive integers that satisfy the equation x^2 + y^2 = z^2. A better explanation of how to find Pythagorean triples is given by John Stillwell in his texts Mathematics and its History and Elements of Number Theory. After Friedberg's discussion of the problem, he tackles the more general problem of how to factor numbers of the form x^2 + ny^2, where n is a positive integer. The mathematics used to solve this problem, including modular arithmetic, is quite powerful, which is conveyed by the simple proofs Friedberg provides of results proved with more difficulty earlier in the book and by his proofs of Fermat's Last Theorem for the cases n = 3 and n = 4.
Friedberg concludes the book with a proof of Gauss' proof of the Law of Quadratic Reciprocity. The material on quadratic residues calls upon many of the previous results. However, while there is a table classifying the theorems in the text (albeit without their actual formal statements), the lack of a subject index makes finding the necessary definitions and theorems difficult. Consequently, Friedberg's arguments are more difficult to follow than they need to be.
If you are seeking a basic introduction to the subject, try working through Oyestein Ore's an Invitation to Number Theory (New Mathematical Library), which is accessible to a bright high school student. Ore is also the author of a slightly more advanced text, Number Theory and Its History (Dover Classics of Science and Mathematics), which, like Friedberg's text, introduces number theory through its historical development. There are numerous more advanced treatments of the subject, which serve as good introductions. They include, among others, The Higher Arithmetic: An Introduction to the Theory of Numbers by H. Davenport, Elementary Number Theory by Gareth A. Jones and J. Mary Jones, and An Introduction to the Theory of Numbers by Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery.
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