In Praise of Small Numbers
by Marc Chamberland
The book is a caboodle carrying an eclectic assortment of short mathematical stories clustered by association with integers 1 through 9. Sometimes the association is loose as, for example, when in Chapter 1 the measure of the complement of the ternary Cantor set is shown to be 1, the quadratic rate of convergence of the Newton method is discussed in Chapter 2, the angle trisection and Morley's theorem in Chapter 3, the Ducci sequences in Chapter 4, the Platonic solids in Chapter 5, the six degrees of separation phenomenon in Chapter 6, the Seven Bridges of Köningsberg problem is mentioned in Chapter 7, the Game of Life in Chapter 8 (due to a cell having 8 neighbors), or Casting out 9s in Chapter 9. Thus the book does not exactly fit with, say, Lure of The Numbers by Joe Roberts (MAA, 1992) or The Kingdom of Infinite Number by Bryan Bunch (W.H. Freeman & Co, 2000) that deal exclusively with properties of integers. This in no way makes the book any less interesting. On the contrary, the scope of the book is broader, with the author highlighting many a fascinating fact from various fields of mathematics, not just number theory.
The preceding paragraph was my attempt to convey the feeling of the breadth of the book's selections. This is my estimate that the author compiled about 120 topics that throw light on very different facets of mathematics. Here's a few more:
- Art gallery theorem
- Brahmagupta-Fibonacci identity
- Beatty sequences
- Benford's law
- Brouwer's fixed point theorem
- Collatz conjecture
- Cutting a hole in half
- Equilateral Triangle on a Closed Curve
- Euler's polyhedron formula
- Four Travelers Problem
- Petersen graph
- Sierpinski gasket
- Tower of Hanoi
- Pythagorean theorem
- Realeaux triangle
The book can be read sequentially or browsed randomly. I began by leafing through but soon settled into reading out of fear of missing an exciting math nugget of which there are plenty in the book. I found myself ignorant of some topics and familiar with others but enjoyed reading without discrimination. The collection is outright delightful. It will agitate the minds of students and shake the sense of know-all off many a professional and most of the amateurs.
One aspect of the book is a little disappointing. The book ends with suggestions for further reading but has no topical references. It may not be a big deal nowadays as it is always possible to search the web for extra information on a topic of interest, still, the readers would have been better accommodated if the author pointed to his sources or listed specific recommendations. That's an omission in an otherwise well written and superbly organized book.
Single Digits. In Praise of Small Numbers by Marc Chamberland. Princeton University Press, 2015. Hardcover, 248 pp, $26.95. ISBN 978-0691-161143.