# Professor Stewart's Incredible Numbers

Numbers are at the origin of mathematical activity. As soon as men started scratching notches on a tally bone, they started counting. It probably took a while before the abstract concept of a number arose, which is not just the bijection between a set of trading objects and the set of notches. Children can not really count when the are able to recite the poem one-two-three-four-... But anyway, numbers certainly form the backbone of mathematics, and fascination for numbers will have been the stepping stone for many to eventually become a mathematician. It certainly did for Ian Stewart. His fascination, is written out in this book, which is another exponent of his talent to make mathematics accessible for a broad readership.

What are the numbers interesting enough to tell about them. Certainly the numbers 1 to 10. So that is where the book starts with. The 1 is obviously a starting point because it generates all other numbers. With 2 we can distinguish between even and odd, we can talk about squares and the sum of squares, i.e., the Pythagorean triples, the binary system (and other number systems), the parity of a permutation (used to prove the existence of the 15-puzzle), quadratic equations and square roots. And Stewart continues with 3, 4, 5,... to just touch upon the most diverse issues in mathematics. Of course number theoretic problems like prime numbers, magic squares and Fermat's last theorem, but also tilings of the plane, the 4 color problem, the 4th dimension, quasi crystals, and many more. Even more interesting is the history of 0, which had to wait a long time to earn the status of being a number. This is followed by negative numbers (if you have $-1$, you have all the others) and the complex numbers.

The next group are the rationals (among them the popular approximations of $\pi$ like 22/7. Probably not many will recognize 466/885 as special. This is related to the Sierpinski gasket and the number of moves to solve the towers of Hanoi puzzle. Of course the rationals are followed by the irrationals. The most prominently being $\sqrt{2}$, $\pi$, the golden ratio $\varphi=(1+\sqrt{5})/2$, and e, the base of the natural logarithms, and a surprising one: log3/log2 which is related to fractal dimension. Another one that only few will recognize: $\pi/\sqrt{18}$, which is related to sphere packing. Perhaps some will connect $\sqrt[12]{2}$ in connection with musical scales, $\zeta(3)$ is well known as Apéry's constant and $\gamma\approx0.577215$ as Euler's constant. Of course much can be told about each of these and Stewart does this in his very familiar entertaining way.

You might think that must be it, what other numbers might be interesting enough to mention? Well there is the number of dimensions conjectured by string theory (11), the number of pentominoes (12), the number of wall paper patterns (17), the size of the group for two people having the same birthday to be larger than 0.5 (23), the number of letters in the English alphabet and hence related to coding theory (26). The number 56 is related to another sphere packing problem in the form of a sausage (hence the sausage conjecture) and 168 is the number of symmetries in a simple finite geometry.

And then there are the really big numbers which I will not explicitly mention here. The number of ways to arrange the alphabet, the number of possibilities offered by a Rubik cube, or a Sudoku, and the largest known prime. The superlatives are the infinities like $\aleph_0$ and the continuum. For all of these you can smell the creamy stories that Stewart can attach to each of these items.

And then there is *The hitchhiker's guide to the galaxy* by Douglas Adams where the answer to all questions one ever could ask was 42. This number was chosen because Adams asked around for the most boring number that came to mind. So Stewart's point being that there are no uninteresting numbers closes the book by summarizing what is special about 42.

Perhaps many of the ideas that Stewart includes in this journey will already be known to many readers. Yet, Stewart is always entertaining and there is always a surprise hidden somewhere. So even if many things are familiar, I would still advise you to read the book. At least I did enjoy it very much and I am sure you will do too.

**Submitted by Adhemar Bultheel |

**20 / Jun / 2015