Customer Reviews

An Adventurer's Guide to Number Theory

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

 

 

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.

I felt like I suited up for space travel and got grounded by equipment malfunction. Perhaps I took the title too literally. Since there are so many books on number theory, surely one with such a title should cover the outer reaches. This is nothing but a basic introduction. More is covered in Albert Beiler's "Recreations in the Theory of Numbers" and it's much more adventurous. Still worth 3 stars, and worth owning - but not worth keeping under your pillow.

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.

...but the dropdown allows you to pick either four or five; I felt generous.

This book is great in that I managed to become "number-theory literate" in a matter of days. Historical tidbits not only make the book flow smoothly, but make it fun to read. The actual mathematical content that is covered nails down the fundamental concepts of number theory pretty well. For clarity, the author is generous with examples.

My only complaint is that the writing, while clear and conversational, is almost too conversational. In the first part of the book, you have to question the author's mathematical background when he makes an embarassing claim and corrects himself in a footnote. Granted, we're all human, but this is a book for goodness sake, you can take your words back! Also, the examples occasionally skip steps, forcing you to stare at the problem longer if it's not clear to you what happened. This isn't always a bad thing, I suppose, but it can be distracting. Still, the book serves it's purpose well and is a good primer for anyone who at least understands high school algebra.

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.

A lot of us know that you can't double the square. You can't find two square whole numbers, one of which is twice the other. This, of course, is an ancient Greek problem.

If b squared were equal to two times a squared, the right side of the equation would contain an odd factor of two, which is obviously impossible by the fundamental theorem of arithmetic. This is the modern way of proving this assertion.

Richard Friedberg prefers the old way. He uses Fermat's method. On page 45 we read:

"At each point we can prove that the numbers we have reached are even. So we can go on dividing forever. But this is impossible. Eventually we must reach 1, or some other odd number. Since we have proved something that is impossible, we must have assumed something that isn't true. The only thing we assumed was that there are two numbers a and b, such that two times a squared equals b squared. So there can be no such numbers."

He continues in this way all the way to quadratic recipocity, and concludes with a Table of Theorems, all rigorously proved in his own quirky way.

I continue to be frustrated by Friedberg's approach to number theory. It is historically accurate but very difficult to assimilate or combine into present day orthodoxy. I'm not sure whether he is worth my time, but nevertheless I continue to study his book. I've read it on and off now for more than five years. There is no doubt in my mind that he is a genius . . . hence the five stars.

Whether one wants to embark on this slippery slope of classical geometry, historical number theory, the defects in Euler's reasoning and other incredibly obscure topics in number theory, the reader must decide for himself or herself. I don't think I'll ever know as much about the history of number theory as Richard Friedberg does, so I decided to put in my two cents mid-way through the course of my studies.

great light reading for anyone who is interested in number theory. basic understanding of mathematics required

You must have a medium understanding of mathematics and algebra.