**Alex’s Adventures in Numberland By Alex Bellos (Bloomsbury, £18.99)**

When I was a teenager I devoured every book the Folkestone public library had on mathematics, which wasn’t many. They did a good job, I hasten to add, with classics like George Gamow’s One Two Three… Infinity and most of Eric Temple Bell’s output. But in those days, there weren’t a lot of books about maths that could sensibly grace the shelves of a public library.

Things have changed. During the 1980s and 1990s, scientists and mathematicians came to terms with the need to communicate their ideas to a general audience. The massive sales of Stephen Hawking’s A Brief History of Time briefly opened the floodgates for popular science to such an extent that airport bookshops had special shelves for the topic, but mostly this muddied the waters. Publishers and authors quickly discovered that we can’t all be Hawkings. When the mud had cleared, the market for popular science settled down again, but at a higher level than before. Nowadays, when someone cracks one of the big mathematical problems like Fermat’s last theorem or the Poincaré conjecture, you know that soon after there will be not just one book about it, but several.

Science writing is a competitive field, and most of the books that get into print are of reasonable quality. Today, my former self would have no difficulty satisfying his reading appetites, and would learn a lot of useful things in the process. But writing about maths remains a very particular subset of science writing, and it has its own special challenges. On the whole, mathematicians can’t parade gigantic apparatus like the Large Hadron Collider, or talk about sexy experiments involving Mars rovers, or make movies about cute meerkats.

Conventional wisdom also states that they can’t include formulas, which is a bit like expecting astronomers to avoid mentioning stars or planets. There is no shortage of good popular maths books. It’s only every so often, though, that one comes along that is outstanding—and Alex’s Adventures in Numberland is one of them. Science writers have to understand science and be able to write, but hardly anyone gets trained in both. So the profession splits into scientists who have acquired the knack of writing for the public, and journalists who have picked up enough of the science. Alex Bellos falls into the second category, but got into writing about maths by accident. In 1992, he tells us, he was a cub reporter at the Brighton Evening Argus.

Excessive use of traffic cones was big news, thanks in part to Prime Minister John Major promising to get rid of them. So the editors challenged the paper’s readers to guess how many cones were on the A23. A few hours later, the first entry came in, and it had the right answer. The editors, hoping to parody the prime minister, had made fools of themselves. Bellos reckons he was the only person in the building who worked out why. He alone understood that you don’t have to make wild guesses: you just divide the length of the motorway by the average distance between cones. The result will be an estimate, but a good one. This level of mathematical reasoning, he realised, was beyond the abilities of his journalistic colleagues. Bellos had a leg up: he had graduated in mathematics and philosophy. For a few years his career drifted further into news, and he ended up as a foreign correspondent in Brazil. Returning to Britain, and not knowing what to do next, he conceived the idea of writing a popular maths book.

When he revisited maths as an adult, he saw it in a very different light from his childhood experiences. No longer focused on passing the next exam or limited by the syllabus, he felt as though he was once more a foreign correspondent—in the abstract territory of Numberland. He tells us his motivation is a drive to “communicate the excitement and wonder of mathematical discovery.” To my mind this is by far the best reason for writing popular maths. Some people—especially television producers—seem to think that the way to convey excitement about science is to have everybody on the screen telling you how excited they are, preferably at high volume. This is like filming footballers enthusing about the excitement of the game, but never showing them playing a match. Bellos knows exactly how to convey excitement: show readers things that you find exciting, or fascinating, or thought-provoking, and leave it up to them to respond.

Alex’s Adventures, then, is not a book about advanced mathematics. Neither is it an attempt to rewrite the school textbooks. It is a guided tour through basic and important regions of mathematical terrain. The emphasis is on ideas, on the sheer joy of romping through the mathematical meadows, smelling the flowers, watching the squirrels and insects—so to speak. Numbers are the basis of mathematics, but mathematics deals with many other things: shapes, rules, probabilities. Numberland, à la Bellos, is populated with similarly diverse concepts. This is not an arithmetic book, but it can certainly help with your arithmetic. Not by teaching you to do sums, but by encouraging you to think of numbers as familiar friends, relax, and enjoy their company.

A refreshing sub-theme is the psychology of mathematical thinking. How is it that the human brain, which evolved on the savannahs to help our distant ancestors avoid being eaten by lions, can handle abstract symbols and calculate things that it has never experienced? How can we comprehend the fact that some numbers are not exact fractions, or that the digits of pi go on forever without repeating? Bellos also touches on the fascinating field of what the experts call ethnomathematics: how different human cultures deal with basic mathematical concepts (the shepherds of medieval Lincolnshire, for example, developed their own number system).

As I thumb through the book, I am struck by the realisation that most readers will be encountering things that they never in their wildest dreams associated with mathematics. Buddhist number names. Origami. The British 50-pence coin. Sudoku. Speaking of which: not long ago, on the internet, I found a site which stated that sudoku has nothing to do with maths because the digits 1-9 are used as symbols, not as numbers. This confuses mathematics with arithmetic. It’s probably true to say that sudoku has nothing much to do with arithmetic, but combinatorial conditions on symbols is an entirely sensible aspect of mathematics, and sudoku is in fact highly mathematical. No one who reads Alex’s Adventures will make this kind of mistake. Bellos’s last five chapters up the ante by heading into deeper regions of the mathematical realm: patterns in number sequences, probability, non-Euclidean geometry, and Georg Cantor’s theory of different sizes of infinity. The style is laced with humour, but at all times, the star of the show is the mathematics.

It’s a beautiful book and deserves to be a huge success. So let me end with a plea: as a matter of urgency, I’d like the author to get working on Through the Looking-Glass, and What Alex Found There. Though I suspect he’s several steps ahead of me.