# Single Digits: In Praise of Small Numbers

This is a caleidoscopic collection of mathematical short stories. Such a story is only a couple of pages long and can explain a concept, some historical fact or mathematician, a puzzle, an open problem, or a simple mathematical fact. These stories are somehow related to each of the digits 1-9. A short tenth chapter of only 2 pages gives solutions to 3 of the problems that were formulated in earlier chapters. The author has played with the idea to also include zero. However zero has such a particular position among the digits that it takes up long chapters in books on the history of mathematics and several books have been devoted to zero alone (e.g. *Zero: The Biography of a Dangerous Idea* (C. Seife, 2000), *The Nothing that Is: A Natural History of Zero* (R. Kaplan, 2000), *Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers* (A.D. Aczel, 2015),...).

The connection with the digit that forms the title of the chapters is rather loose. For example digit 1 stands for `unique', but the items covered in this chapter include origami, the golden ratio, number systems, factoring knots, countability, fractals, Sierpinski's gasket, Benford's law, Brouwer fixed point theorem, perfect squares, gamma function, Picard theorems and many others. Under the umbrella ot 2 we find Jordan curves, symmetry, the Pythagorian theorem, Euler's formula for polytopes, several conjectures related to prime numbers, the Towers of Hanoi pizzle, formulas for π, Apollonian circle packing, arithmetic and geometric means, Newton's method for root finding, etc. This illustrates the diversity of subjects that are involved. A minimal knowledge of mathematics suffices to understand and appreciate the items discussed. It may be already illustrated by the previous enumeration that the subjects become somewhat more advanced towards the end of each chapter, and there is also a graduate increase in complexity over the different chapters towards the end of the book.

The table of contents lists 118 of these short stories (some may cover more than a single item). Each one is a brief excursion in the world of mathematics, a gem that illustrates what keeps mathematicians from the past present and future fascinated about their subject. It is always summarized in an accessible way. Other books that have similar collections exist, and there is certainly some overlap with those books, but it is remarkable that many of the stories here contain facts that I have not seen in other books before. There are no proper mathematical proofs but the problem, puzzle, theorem, conjecture, open problem, result, or just a simple fact is made very clear without ever needing deep mathematics.

A similar idea of linking such a collection to numbers was used by Ian Stewart in his book *Professor Stewart's Incredible Numbers* (Profile Books, 2015). He however includes zero and negative integers, rationals, irrationals, and even infinity. Stewart's items are in general less advanced on a mathematical level. There are fewer topics but they are more elaborated.

**Submitted by Adhemar Bultheel |

**21 / Jul / 2015