This fine book gives what its title promises. It was written for students and teachers of mathematics and, of course, anyone else who would appreciate a well-written treatment of the subject. It is a popularization in that it is not burdened with footnotes or other scholarly apparatus but it is not a popularization because the author is not writing down to his readers. He is writing right at them.

The history follows the conventional course, from the ancient Egyptians and Greeks through the Arabs and medieval and renaissance Europe to Descartes, Newton, and on through the twentieth century. The author says that he left out most of Asia, Africa, and all of the Americas because there wasn’t space for everything.

One thing, among many, that the book does well is bring out the tremendous difficulty we have had in arriving at our present understanding of numbers. For the ancient Greeks, 1 was not a number. It was the unit, out of which numbers were made. It wasn’t too long, in historical terms, before it became a number, but I’m sure that struggle was involved, the dinosaurs against the revolutionaries. The struggle was even greater with negative numbers. They were first rejected, and then sort of accepted without being understood. Wallis (1616–1703), while doing things like writing an infinite product for \(\pi\), was unable to get his head around them, and even in the first half of the nineteenth century they had their opponents.

Another is to point out the very different ways our predecessors thought. For at least a millennium and a half, a ratio like 2:3 was not a number. It was a ratio, an entirely different thing. A proportion like 2:3 :: 4:6 had nothing to do with numbers. It was a proportion. Equalities were comparisons, not equations that could be operated on. Numbers were like labels. You can’t multiply labels. 2:3 = .666… made no sense at all.

“What was *wrong* with those people?” is what too many of us think, consciously or not. The author shows that there was nothing wrong with them. They were doing difficult work and making progress. Try doing algebra without symbols, using nothing but words, and you’ll see what they were up against.

The book contains much that will be new, even to experienced readers. For example, I never thought about how Euclid was careful to put geometry on what he thought was a firm foundation with his axioms and postulates, but when it came to proving that there are infinitely many primes he didn’t include any similar foundation for numbers.

In his last chapters the author doesn’t hesitate to lay out material that some mathematics undergraduates don’t encounter. There are algebraic numbers, with a proof of their countablilty, Kummer’s ideal numbers, and a development of the number system, starting with Peano’s postulates for the positive integers, to integers as pairs of positive numbers, rationals as pairs of integers, reals as Dedekind cuts, and complexes as pairs of reals. There are transfinite cardinals and ordinals. There is the axiom of choice and well-ordering. Good stuff.

The copy editing leaves something to be desired. Displayed lines have inconsistent punctuation, there are symbols not in italics that should be, and in the space of sixteen pages we find

p. 148: Bombelli writing “*R*.*q*.21” for the square root of ‒121

p. 152: congruences “module” 9

p. 153: a parenthesis closed that was never opened

p. 158: 6 times 216 is 393216

p. 163: Viète writing “it is not the reckoning’s fault bu the reckoner’s”.

The meaning is not obscured, though you may have to think a second or two about the fourth, but one expects better. If the OUP needs to save money I would rather it use cheaper paper instead of cheaper copy editors.

Woody Dudley was born so long ago that his days are numbered. They are now positive and rational, but may become irrational and then in any case, inevitably, imaginary.