Introduction

In the summer of 1992 I was working as a cub reporter at the Evening Argus in Brighton. My days were spent watching recidivist teenagers appear at the local magistrates court, interviewing shopkeepers about the recession and, twice a week, updating the opening hours of the Bluebell Railway for the paper’s listings page. It wasn’t a great time if you were a petty thief, or a shopkeeper, but for me it was a happy period in my life.

John Major had recently been re-elected as prime minister and, flush from victory, he delivered one of his most remembered (and ridiculed) policy initiatives. With presidential seriousness, he announced the creation of a telephone hotline for information about traffic cones – a banal proposal dressed up as if the future of the world depended on it.

In Brighton, however, cones were big news. You couldn’t drive into town without getting stuck in roadworksThe main route from London – the A23 (M) – was a corridor of striped orange cones all the way from Crawley to Preston Park. With its tongue firmly in its cheek, the Argus challenged its readers to guess the number of cones that lined the many miles of the A23 (M). Senior staff congratulated themselves on such a brilliant idea. The village fête-style challenge explained the story while also poking fun at central government: perfect local-paper stuff.

Yet only a few hours after the competition was launched, the first entry was received, and in it the reader had guessed the correct number of cones. I remember the senior editors sitting in dejected silence in the newsroom, as if an important local councillor had just died. They had aimed to parody the prime minister, but it was they who had been made to look like fools.

The editors had assumed that guessing how many cones there were on 20 or so miles of motorway was an impossible task. It self-evidently wasn’t and I think I was the only person in the building who could see why. Assuming that cones are positioned at identical intervals, all you need to do is make one calculation:

Number of cones = length of road ÷ distance between cones

The length of road can be measured by driving down it or by reading a map. To calculate the distance between cones you just need a tape measure. Even though the space between cones may vary a little, and the estimated length of road may also be subject to error, over large distances the accuracy of this calculation is good enough for the purposes of winning competitions in local papers (and was presumably exactly how the traffic police had counted the cones in the first place when they supplied the Argus with the right answer).

I remember this incident very clearly because it was the first moment in my career as a journalist that I realized the value of having a mathematical mind. It was also disquieting to realize just how innumerate most journalists are. There was nothing very complicated about finding out how many cones were lined alongside a road, yet for my colleagues the calculation was a step too far.

Two years previously I had graduated in mathematics and philosophy, a degree with one foot in science and the other in the liberal arts. Entering journalism was a decision, at least superficially, to abandon the former and embrace the latter. I left the Argus shortly after the cones fiasco, moving to work on papers in London. Eventually, I became a foreign correspondent in Rio de Janeiro. Occasionally my heightened aptitude for numbers was helpful, such as when finding the European country whose area was closest to the most recently deforested swathe of Amazon jungle, or when calculating exchange rates during various currency crises. But essentially, it felt very much as if I had left maths behind.

Then, a few years ago, I came back to the UK not knowing what I wanted to do next. I sold T-shirts of Brazilian footballers, I started a blog, I toyed with the idea of importing tropical fruit. Nothing worked out. During this process of reassessment, I looked again at the subject that had consumed me for so much of my youth, and it was there that I found the spark of inspiration that led me to write this book.

Entering the world of maths as an adult was very different from entering it as a child, where the requirement work onass exams means that often the really engrossing stuff is passed over. Now I was free to wander down avenues just because they sounded curious and interesting. I learned about ‘ethnomathematics’, the study of how different cultures approach maths, and about how maths was shaped by religion. I became intrigued by recent work in behavioural psychology and neuroscience that is piecing together exactly why and how the brain thinks of numbers.

I realized that I was behaving just like a foreign correspondent on assignment, except the country I was visiting was an abstract one – ‘Numberland’.

My journey soon became geographical, since I wanted to experience mathematics in the real world. So, I flew to India to learn how the country invented ‘zero’, one of the greatest intellectual breakthroughs in human history. I booked myself into a mega-casino in Reno to see probability in action. And in Japan, I met the world’s most numerate chimpanzee.

As my research progressed, I found myself being in the strange position of being both an expert and a non-specialist at the same time. Relearning school maths was like reacquainting myself with old friends, but there were many friends of friends I had never met back then and there are also a lot of new kids on the block. Before I wrote this book, for example, I was unaware that for hundreds of years there have been campaigns to introduce two new numbers to our ten-number system. I didn’t know why Britain was the first nation to mint a heptagonal coin. And I had no idea of the maths behind Sudoku (because it hadn’t been invented).

I was led to unexpected places, such as Braintree, Essex, and Scottsdale, Arizona, and to unexpected shelves on the library. I spent a memorable day reading a book on the history of rituals surrounding plants to understand why Pythagoras was a notoriously fussy eater.

The book starts at Chapter Zero, since I wanted to emphasize that the subject discussed here is pre-mathematics. This chapter is about how numbers emerged. At the beginning of Chapter One numbers have indeed emerged and we can get down to business. Between that point and the end of Chapter Eleven the book covers arithmetic, algebra, geometry, statistics and as many other fields as I could squeeze into 400-ish pages. I have tried to keep the technical material to a minimum, although sometimes there was no way out and I had to spell out equations and proofs. If you feel your brain hurting, skip to the beginning of the next section and it will get easier again. Each chapter is self-contained, meaning that to understand it one does not have to have read the previous chapters. You can read the chapters in any order, although I hope you read them from the first to the last since they follow a rough chronology of ideas and I occasionally refer back to points made earlier. I have aimed the book at the reader with no mathematical knowledge, and it covers material from primary school level to concepts that are taught only at the end of an undergraduate degree.

I have included a fair bit of historical material, since maths is the history of maths. Unlike the humanities, which are in a permanent state of reinvention, as new ideas or fashions replace old ones, and unlike applied science, where theories are undergoing continual refinement, mathematics does not age. The theorems of Pythagoras and Euclid are as valid now as they always were – which is why Pythagoras and Euclid are the oldest names we study at school. The GCSE syllabus contains almost no maths beyond what was already known in the mid seventeenth century, and likewise A-level with the mid eighteenth century. (In my degree the most modern maths I studied was from the 1920s.)

When writing this book, my motivation was at all times to communicate the excitement and wonder of mathematical discovery. (And to show that mathematicians are funny. We are the kings of logic, which gives us an extremely discriminating sense of the illogical.) Maths suffers from a reputation that it is dry and difficult. Often it is. Yet maths can also be inspiring, accessible and, above all, brilliantly creative. Abstract mathematical thought is one of the great achievements of the human race, and arguably the foundation of all human progress.

Numberland is a remarkable place. I would recommend a visit.

Alex Bellos
January 2010