An adventurer's guide to number theory (The History of science)

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.


Really four-and-a-half stars...   January 1, 2000
Brian Y. (Pittsburgh, Pa.)
12 out of 13 found this review helpful

...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.


"Quirky" is exactly the right word.   June 25, 2001
7 out of 9 found this review helpful

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.


Be carefull   August 24, 2001
Olinto Rodriguez (Maracaibo, ZU VEN)
You must have a medium understanding of mathematics and algebra.