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The Personality of Numbers

By Philip J. Davis

Single Digits: In Praise of Small Numbers. By Marc Chamberland. Princeton University Press, Princeton, NJ, 2015, 240 pages, $26.95.

Almost sixty years ago, the New Mathematical Library was set up to publish a series of monographs that would appeal to a wide audience. Guided by an editorial panel of a dozen distinguished mathematicians, the series had as its technical editor Anneli Lax at New York University.

Lax tapped me to be one of the first to contribute a volume to the series. I agreed, provided that my text could contain some material that was exceedingly elementary. She said OK to that, and so The Lore of Large Numbers (Yale University Press,1961) was born.

I titled a section of this book “The Personality of Numbers,” and by that I meant simply such arithmetic features of numbers as odd, even, square, triangular,  prime, perfect; after discussing various tests for divisibility, with trumpets blaring, I wound up with Wilson’s theorem: A number \(N\) is prime if and only if it divides \((N – 1)! + 1.\)

Marc Chamberland, a professor of mathematics at Grinnell College, has produced a book that I would sub-subtitle “The Personality of the Numbers One to Nine.” Devoting a separate chapter to each of these numbers, he extends my notion of “personality” far beyond mere divisibility. Here are examples from some of the chapters (each of which contains much, much more).

The Miquel five-circles theorem.
In chapter three Chamberland discusses, among other things, Morley’s theorem; Poincaré and the three-body problem; the Lorenz attractor and chaos (“Period Three Implies Chaos”); Ramanujan sequences; and Monge’s three-circle theorem.

In chapter five we meet the platonic solids, the difficult quintic equation, and the Miquel five-circles theorem.

In chapter eight we find the pizza theorem, the game of life, and Lie groups.

In chapter nine, among much else, we learn about two circle packings and casting out nines. But I had hoped to see the famous nine-point circle theorem, of which Dame Mary L. Cartwright once told me she needed to know two proofs in order to advance her career.

But where is the number zero – about which a whole book could, should, and perhaps has been written?* Some authorities have claimed that zero is not a number at all.

Chamberland’s writing is lucid, interesting, often amusing, and informative. The reader will find numerous conjuctures and unsolved problems. I came away from the book knowing of many things I had never encountered.

One of the pleasures I find in reviewing comes when something  an author mentions—however remotely—connects up with something I had thought about or carried out. An example in Chamberland’s book in chapter two shows the Apollonian packings of a circle by smaller circles. This reminded me of a proof I once gave, using interpolation theory, of the visually obvious result that a circle cannot be filled out with a finite number of non-overlapping smaller circles. Now, if something is visually obvious, what is the need for proof? Call up Mr. Euclid and ask him about the role intuition should play in the math biz. 

*SIAM News published reviews of two books about zero in the September 2000 issue: "The Nothing That Is: A Natural History of Zero" by Robert Kaplan and "Zero: The Biography of a Dangerous Idea" by Charles Seife.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island and can be reached at philip_davis@ brown.edu.

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