# Nice Numbers

For readers who are used to read books that promote and popularize mathematics, John Barnes may be a familiar name since he published before *Gems of Geometry*(Springer, 2009, 2012). What he did for geometry there, he is now doing for numbers, another topic commonly used to reach a broad audience. And audience is to be taken in a literal sense because both books are the result of a series of lectures given by the author.

There are 10 chapters in this book, corresponding to 10 lectures, each one discussing a general aspect related to numbers. Of course many authors have been excavating this topic with the same ambition before and much of what Barnes tells us has been told by others on several occasions. Yet there is always a fresh angle to look at them and there is always some sparkling gem, something unknown before, or an unexpected link or connection to be discovered.

A quick survey of the chapters.**1. Measures.** Defining prime numbers and factors leads to an extensive discussion of many non-decimal systems of units for currency, length, volume, time, weight,... In fact 10 has too few factors and therefore other numbers can be more suitable as a base.**2. Amicable numbers.** Perfect numbers equal the sum of their factors and amicable numbers is a couple where each one equals the sum of the factors of the other. This can be generalized to n-tuples or sociable cycles. Mersenne primes and Fermat numbers are introduced here. **3. Probability.** Of course this involves coin flipping and dice throwing and normal and other distributions. But as an aside, it is surprising to learn that there are 16 different possibilities to produce a classical die. Furthermore we recognize the Buffon needle problem and the Monty Hall problem. And there is the practical problem of false positives in tests and more playfully several game strategies and gambling theoretical problems.**4. Fractions.** This is explaining the Egyptian number system, continued fractions, repeating decimal expansions of rational numbers, and a long division-like algorithm to compute a square root (a forgotten skill now that pocket calculators are generally available). **5. Time.** This discusses the Roman and the Gregorian calendar, but also the astronomical origins of year, season, month, week, and night and day.**6. Notations.** Thinking of the notation of numbers, this must obviously include our familiar decimal positional system, but also predecessors like the Roman and the Babylonian system. This is also a good place to look at modular arithmetic (also used in other chapters). The repeating expansion of rational numbers is reconsidered in a different base system and connected with Fermat's little theorem.**7. Bells.** Bell ringing (or tintinnalogia) is a combinatorial problem in which one has to ring a set of bells with certain patterns of repetition. This will involve decomposition of permutations and even some group theory, although the latter is not elaborated here.**8. Primes.** This involves classical ideas like the Euclidean algorithm and the sieve of Eratosthenes, but also Gaussian (complex) primes, and prime polynomials with coefficients in modular arithmetic.**9. Music.** This is a relatively long chapter about the different music scales that can be used.**10. Finale.** This is a roundup of three remaining topics. The most important is an explanation of the working of the public key RSA encryption algorithm. Furthermore possible animal gaits, and finally the towers of Hanoi and the topologically equivalent Chinese ring puzzle.

In his lectures, Barnes did involve his public actively, which is reflected by the exercises that are given at the end of the chapters. Easy exercises that do not need extra knowledge than what was explained in that chapter. For those who are willing to read more, some references are provided with some advise about their content and their difficulty.

So, if you are not familiar with popularizing books using `numbers' as a master key to introduce mathematics, this is an excellent start. It is light, entertaining, richly illustrated, and still Barnes has disregarded the advise of many publishes to avoid all formulas since each formula allegedly would cut the sales in half. I tend to disagree with those publishers and I am happy that Barnes did too. I am sure that those who are willing to read this kind of books are not scarred away by a formula. The problem with formulas is that typos easily slip in, like a blatant one on page 70 claiming that $\sqrt{2}$ is a root of the equation $x^2−1=0$. Even if formulas are allowed, the content should still be digestible for anyone, and it should be easy for some of the readers to skip the more technical parts, while these are still available for those who are hungry for more. In this case there are 100 more pages with nine appendices of such marvelous items. Some are indeed more mathematical yet still entertaining (Ackermann function, Pascal's triangle and 3D generalization of triangular numbers, stochastics in game and queuing theory, the Chinese remainder theorem, group theory and an extensive one on the solution of the Rubik cube, etc.). This book is a most enjoyable read. Those who read *Gems of Geometry* need not be convinced of Barnes' entertaining style and will love to read this book too. Others who read this book without knowing his geometry book will be teased to also check that one out.

**Submitted by Adhemar Bultheel |

**19 / Feb / 2017