God Created the Integers
God Created the Integers: The Mathematical Breakthroughs That Changed History
Author: Stephen Hawking
Amazon info
The premise of this book is wonderful - a tour through the mathematical breakthroughs over 2000 years. Up to a point, the book delivers. It would be very hard to pick and choose the right set of mathematicians and Hawking has done a good job here (although perhaps a little too much emphasis on Calculus). Each of the 31 chapters has two parts - an overview of the mathematican's accomplishements, followed by their original work.
The overviews can be understood by the "mathematically minded amateur", but the original work is for real mathematicians only. The original works account for much of the books 1300 pages (about 1000 pages is original work and 300 pages is the overviews).
My disappointment with the book comes from the errors that I think I found. Errors in a math book are deadly, as I spent a long time wondering if I was mistaken (and hey, I might be mistaken). The ones I found were:
1. Page 386 - "the probability of that chance would be exceedingly small (1/2)^29, about 1 in 100 billion. But (1/2)^29 is 1.86 * 10^-11 and "1 in 100 billion" is 1.0 * 10^-9, a difference of nearly 100x.
2. Page 567 - "if p and q are primes both congruent to 3 module 4 then exactly one of the equations
x^2 = p (mod q) is x^2 = q (mod p)
This isn't even parseable. I think it meant to say that
"if p and q are primes both congruent to 3 module 4 then exactly one of the equations
x^2 = p (mod q)
x^2 = q (mod p)
is solvable"
3. Page 673
"He represented the assertion that all X are Y as 'XY=Y'. Thus if all X are Y and all Y are X, then XY=YX"
But if all X are Y, we have XY=Y and if all Y are X, we have YX=X. It seems pretty clear that this implies that X and Y are the same and thus XY does in fact equal YX, but it seems like a something was left out. (OK, this is pretty minor and I probably shouldn't include it - but it puzzled me for a while.)
4. Page 890
"If for ever E, a D be found so that for all values of x such that x0-D < x < x0+E then |f(x)-f(x0)| < E."
I think that he means:
"If for ever E, a D be found so that for all values of x such that x0-D < x < x0+D then |f(x)-f(x0)| < E." Proably a simple typo, but it can be awfully confusing.
Bottom line, I enjoyed the overviews very much and others would get even more out of the original book - but the sloppy editing should be fixed. Recommend the second edition for all mathematically inclined readers.