# Single Digits: In Praise of Small Numbers

###### Marc Chamberland
Publisher:
Princeton University Press
Publication Date:
2015
Number of Pages:
248
Format:
Hardcover
Price:
26.95
ISBN:
9780691161143
Category:
General
[Reviewed by
William J. Satzer
, on
06/6/2015
]

The exotics like $\pi$ and $e$ have gotten their share of attention in the world of popular mathematical writing. Now it’s time to give proper attention to the integers 1 through 9. Well, sort of. Chamberland uses the small numbers to create a series of vignettes that address over a hundred different mathematical topics. Each chapter typically begins with an easier concept and then progresses to more sophisticated ideas. All topics are linked in some fashion to a single integer.

Chapter 8 is representative. It begins with the Pizza Theorem; this describes a fair way to divide a pizza among two people when it is first cut conventionally, but well off center. A proof by picture shows a division of such a pizza into eight pairs of congruent regions. This is followed by segments on the Game of Life (each cell has eight neighbors), repetition in Pascal’s triangle (the number 3003 appears eight times), the Sierpinski Carpet (eight squares remain after one is deleted at each stage of its construction), the octonions and then the exceptional Lie group $E_8$.

Exceptional Lie groups are the extreme upper end of abstraction here. Each chapter has at least some geometry and some number theory. Several use a bit of calculus. The range of topics includes enough material to appeal to anyone from a high school student to a practicing mathematician. Finding enough subjects for each integer must have been challenging. It does occasionally force a bit of a stretch; sometimes the connection between the vignette and the associated integer is pretty tangential. Yet the book is consistently entertaining and well-written. It also has a good collection of illustrative figures and photographs.

This would be an ideal gift for a high school student who might have an interest in mathematics. It’s also fun just to pick up and browse.

Bill Satzer ([email protected]) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.