The authors don’t say for whom (or why) this book was written, but its contents show that it is meant for readers without much mathematical training. There is some algebra, but only a little. They say that (p. 383) “A definition of the cardinality of infinite sets is beyond the scope of this book.” A few easy things about Pythagorean triples are proved, but most results are only stated.

The authors start (p. 18) with the five principles of counting (can you name them without reading further?), which they capitalize: Bijection, Ordinality, Cardinality, Invariance, and Abstraction. After discussing them, they move on to such things as number names and symbols, place value, and number bases. There follow even and odd numbers, rectangular and square numbers, triangular numbers, polygonal numbers, and tetrahedral numbers.

Then comes a chapter on counting in which Pascal’s triangle appears, then Fibonacci numbers, magic squares, Napier’s bones, and so on: topics that appear in much elementary popular exposition. Continued fractions, which don’t often appear, pop up also, along with the more usual suspects: Pythagorean triples, Mersenne primes, amicable and perfect numbers, tests for divisibility, and so on.

Of course, the golden ratio is here, and the authors can’t resist bringing up the Great Pyramid of Egypt. Though they don’t come out and assert that the ancient pyramid builders put the ratio into the pyramid, they ask (p. 319) “Could it be that the architects of the pyramid indeed chose to encode the golden ratio in the proportions of this pyramid?” The answer, as the authors should know, is “no”. Because the pyramid builders knew nothing about the golden ratio, they could not choose to encode it. To be fair, the authors sort of — not explicitly — say no, but not before stating three “Hypotheses” connecting the pyramid with the golden ratio and \(\pi\), which isn’t built into the pyramid either. Numbers are sufficiently rich and fascinating that they don’t need the pyramid for support. And the pyramid didn’t need numbers, except for arithmetic.

There is little in the book that is new, but some things are less familiar than others. For example, I can’t remember encountering (p. 211) \[(12345654321)(1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1) = 666666666666\] before.

The book is pleasing to the eye. The publishers are to be congratulated on their editing: not a single typographical error! (That I noticed, anyway.)

General readers may find the book entertaining and informative, but its best use, I think, is as a gift for eighth- or ninth-graders who are good at and interested in mathematics. Some of it may be a bit over their heads, but that’s all right — it encourages stretching. They should be told that they don’t have to memorize the five principles of counting.

Woody Dudley is the author of *Numerology* and *Elementary Number Theory*, which are largely about numbers.