Eli Maor writes clearly, interestingly, and soberly. No dumb jokes to lighten up the subject matter!

Detailed Review:

Preface:
The Pythagorean Theorem, why is it interesting? Just a teaser, really.

Prologue: Cambridge, England, 1993
Wiles proves Fermat's Last Theorem. This happened soon after I moved to Madison and everybody I knew, since they were all math grad students, was talking about it. I was younger back then. Euler showed it true for n=3, and it had been shown for all n up to 100,000 by the time Wiles did his work. Now it is certainly proved.

Chapter 1: Mesompotamia, 1800 BCE
A Babylonian tablet found with the square root of 2 represented as a rational number to a very great accuracy. Maor hypothesis that the Babylonians were familiar with the formula

a = 2uv, b = u^2 - v^2, c = u^2 + v^2

where u, v are any two positive integers and u is greater than v. If u and v are of opposite parity and have no factors in common then each Pythagorean triple produced by the formula is primitive, i.e, it can not be produced by multiplying a, b, and c of some other triple by a constant.

1. Using algebra is it fairly easy to show that this formula must yield Pythagorean triples.

2uv^2 + (u^2 - v^2)^2 = (u^2 + v^2)^2

Let u^2 = p, v^2 = q.
2pq + (p - q)^2 = (p + q)^2
4pq + p^2 - 2pq +q^2 = p^2 + 2qp + q^2
It all cancels pretty easily, QED.

2. It is hard to show that restricting u, v to mutally prime numbers with opposite parity always gives primitive triples.
a. It is obvious why the opposite parity restriction exists. If they are the same parity, then u^2 - v^2 and u^2 + v^2 are both even. 2uv is also even, so each element in the triple can be divided by two, showing that the
triple is not primitive.
b. If they aren't mutually prime then each element in the triple is divisible by their shared factor.
So these criteria are obviously necessary, but are they sufficient?
NOTE: Some other book I read stated that the existence of this formula and the fact that its possible arguments are an infinite set proves that there are an infinite number of Pythagorean triples. This is obviously not true in itself, it is necessary to reason pretty carefully over the domain. It seemed rather hard to me to prove that there are an infinite number of primitive triples, based just on this fact.
But now it seems easy, because there are an infinite number of primes. Let u be a prime and let v = 1. That works. Then let u be some other prime and v = 1. That works to. But those are definitely different primitive triples. And there are an infinite number of primes. So there are, in fact, and infinite number of Pythagorean triples.

I feel I've concluded thinking about this formula for Pythagorean triples. It seems good to me.

Appendix A: How did the Babylonians approximate root 2?
Maor asserts that it was probably by something like the Newton-Raphson method. Discussion relies on the well-known fact that the arithmetic mean is always greater than or equal to the geometric mean of two numbers.

Proof of this fact:
It's obviously equal when the the two numbers are equal, both by intuition, and by algebra: (a + a)/2 = sqrt a * a. To demonstrate this fact when a != b choose b to be the larger, wlog. Then b = a + x, x > 0. So, (a + b)/2 = (a + a + x)/2 = a + x/2. We know that the geometric mean, squared, equals the product of a and b, by definition. g = sqrt a(a + x), so g^2 = a(a + x). Square the arithmetic mean, and we find that it is larger than g^2. This implies that the arithmetic mean is larger than g, so long as the arithmetic mean is larger than 1. (a + x/2)^2 = a^2 + ax + (x/2)^2 = a(a + x) + (x/2)^2. QED.

Appendix B: Pythagorean Triples
TODO

Sidebar 1: Did the Egyptians Know It?
Eli Maor thinks not. He claims there is no evidence although he also admits that the Egyptians wrote on papyrus, which really doesn't last like clay tablets do. He says that all the writers who talk about rope stretchers laying out 3/4/5 triangles to get right angles are just making it up.

Chapter 2: Pythagoras
Brief discussion of how little we know about the life of Pythagoras. Pythagoras and acoustics. Brief discussion of acoustics of which I know nothing, unfortunately. Pythagoreans and numerology. Nothing fun about this except there is a pretty diagram of the five Platonic solids. Pythagoreans and figurative numbers, triangular and square, i.e., sums of succeeding numbers and sums of succeeding odd numbers. This is the general idea of an arithmetic series. Also geometric proofs of algebraic formulae, as (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 -2ab + b^2. New formula for triples which is just a degenerate version of the Babylonian one where v = 1.

a = 2u, b = u^2 - 1, c = u^2 + 1

with a little manipulation:

a =u, b =( u^2 - 1)/2 c = (u^2 + 1) /2

Setting v to 1 only meant the Pythagoreans could only get a bunch of Pythagorean triples belonging to increasingly pointy triangles, since the hypotenuse was always just one longer than the longer side.

On page 25 there is a nice visual proof of the special case of the Pythagorean theorem for an isosceles triangle; the "Chinese" proof is shown on the following page.

Troublingly: if a = b = 1, then c = root 2 which is irrational. This really bothered the Pythagorans who convinced themselves that this was true. root 2 is obviously algebraic, and there is a nice diagram of its construction on the number line. Easy as pie!

There is a brief aside about modern theoretical physicists obsession with symmetry and beauty (and string theory).

Footnote 3: TODO

Sidebar 10: Four Pythagorean Brain Teasers

Appendix C: Sums of Two Squares
TODO

Appendix D: Proof that root 2 is irrational.
TODO

Appendix H: Solutions to Brainteasers
These are the solutions to the problems from Sidebar 10: Four Pythagorean Brainteasers.

Epilogue:
Description of a trip to Samos that the author made. Not sure what it's supposed to illustrate except that the people who visit are sometimes more informed and more interested than the people actually located at the site of interest.

Interesting selection of math topics surrounding "a squared plus b squared equals c squared." Not equation-phobic, nor even calculus-phobic, yet pretty undemanding. *Badly* marred by the

overleaf placement of figures.