# The Mathematics Lover’s Companion

This book consists of a collection of mathematical vignettes that illustrate the fun, the beauty, and the applicability of mathematics to neophytes. This is not the first, and certainly not the last book of this type that will be published. There is necessarily some overlap between these books, and yet every author has his or her specific preferences for selecting the topics, the level of the mathematics, and the style adopted in telling their story.

The overall level of the book is really basic mathematics. For example the concept of a proof is illustrated in the introduction by proving that the sum of two odd numbers is even. Later the principle of a proof by contradiction (there are infinitely many primes) and a proof by induction (for the Fibonacci numbers it holds that $F_0+F_1+\cdots+F_n=F_{n+2}−1$) is illustrated. For the latter identity also an alternative combinatorial proof is given. That is by showing that the left-hand side and the right-hand side correspond to two different ways of counting the same number of events. Some chapters are slightly more advanced, but only after new concepts are introduced like the exponential or the logarithmic function, complex numbers,... but everything stays at a really modest level. The reader is supposed to work a bit to assimilate the material. Therefore in every chapter on one or two occasions it happens that after some example is explained, you, as a reader are asked to work out the next step or do an analogous example for yourself. Solutions are then provided at the end of the chapter. There are thus no long lists of "exercises" to be solved, and actually they do not feel like exercises, but more like an interesting puzzle you are challenged to solve. It remains entertainment all the way.

The subjects discussed involve the usual suspects at this level of introductory mathematics: numbers, geometry, and probability. The 23 relatively short chapters on these respective subjects are brought together in three separate parts. The first part, simply called *Number*, is the largest. It fills about the first half of the book. The second half is approximately equally divided over the other two topics. Although the subjects are entertaining, the book is building up the material in steps like in a textbook, proving the results (at least the simple ones) and sequentially introducing new concepts as the reader progresses. To illustrate this concept, we shall give a quick (and incomplete) survey of the topics that are discussed.

In the first part we obviously meet the natural numbers, prime numbers, and binary representation, but in subsequent chapters the integers are followed by rationals, irrationals (even transcendentals) and complex numbers. So we find a proof that 0.9999999... is equal to 1 and that $\sqrt{2}$ is irrational. We are introduced to $i=\sqrt{-1}$ and complex numbers, the transcendent number $\pi$, Euler's number e and the exponential function, infinity (and the transfinite numbers). And then there are some applications: Fibonacci numbers, factorials, Benford's law, and eventually algorithms for sorting, and for computing the GCD and LCM. Some famous problems and applications are only mentioned briefly (sometimes in one of the other parts): the Goldbach conjecture, Fermat's last theorem, the twin prime conjecture, the Collatz or 3n+1 conjecture, Kepler's sphere packing problem, cryptography, music theory,...

The part about geometrical topics is called *Shape* and proves theorems about triangles (of course several proofs of the Pythagoras theorems, but other properties too like formulas for area and perimeter, and the four possible centres that can be associated with a triangle), circles (kissing circles, Kepler's problem, Pascal's hexagon theorem), platonic solids (with Euler's characteristic linking the number of vertices, edges and faces), fractals (defining fractal dimension by box counting), and hyperbolic geometry (including several ways of tiling the hyperbolic plane).

The title of the third part is *Uncertainty*. Probability is employed to show that calculating the chances that some event will occur can give rather counter intuitive results. For example how in a game involving rolling dice it is possible that player A beats player B and player B beats player C, and yet player C beats player A. Another chapter investigates how you should interpret the result of a medical test predicting that you have (or have not) a disease if the test in only correct 95% of the time. There is an illustration that for the slightest change in the initial conditions of some dynamical system that originally converged to a stable steady state, can result in a chaotic, hence unpredictable, behaviour. Arrow's theorem shows that designing a fair democratic voting system is far from trivial. Finally the classic Newcomb paradox is discussed. It refers to a thought experiment where some player can either choose to take both boxes presented to him or only box 2. Box 1 contains a small amount, but the content of box 2 is unknown and depends on an oracle. The oracle is almost always correct and it has predicted what choice the player will make but does not reveal this. If the oracle predicted both boxes are chosen, then box 2 contains nothing, otherwise, box 2 contains a large amount. What choice should the player make? If you think this problem through, this touches upon the philosophical problem of free will.

So there is really a great diversity of entertaining topics that can catch the interest of any reader that has the mathematical level of a regular young secondary school (junior high) student from the age of about 12-14. Not only the content, sprinkled with some sparks of Scheinerman's dry sense of humour, also the layout of the page helps to make reading agreeable. The main text is in one column that takes about two thirds of the width of a page and to the right of it a wide margin is left mostly blank, but it is used to add in smaller font some extra explanation or a side remark or a graph. It is truly a mathematics lover's companion, containing masterpieces for everyone.

**Submitted by Adhemar Bultheel |

**16 / Nov / 2017