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Unformatted text preview: ALEX’S ADVENTURES IN NUMBERLAND OceanofPDF.com ALEX’S ADVENTURES IN NUMBERLAND OceanofPDF.com ALEX BELLOS OceanofPDF.com For my mother and father OceanofPDF.com First published in Great Britain in 2010 Copyright © 2010 by Alex Bellos Illustrations © 2010 by Andy Riley The moral right of the author has been asserted Bloomsbury Publishing Plc, 36 Soho Square, London W1D 3QY A CIP catalogue record for this book is available from the British Library Mathematical diagrams by Oxford Designers and Illustrators All papers used by Bloomsbury Publishing are natural, recyclable products made from wood grown in well-managed forests. The manufacturing processes conform to the environmental regulations of the country of origin. Plate Section Picture Credits: p. 1 (top), p. 1 (bottom), p. 6 (top), p. 6 (bottom), p. 7 (top), p. 7 (bottom), p. 12 (top), p. 15 (top), p. 15 (bottom), p. 16 (top) © Alex Bellos; pp. 2–3 SR Euclid Collection, UCL Library Services, Special Collections; p. 4 (top), p. 4 (bottom) © Robert Lang; p. 5 (top) © Eva Madrazo, 2009. Used under license from Shutterstock.com; p. 5 (bottom) © Neil Mason; p. 8 Le Casse-tête en portraits, Gandais, Paris, 1818, from the Slocum Puzzle Collection, Lilly Library; p. 9 Thanks to Jerry Slocum; p. 10 (top left), p. 10 (top right), p. 10 (bottom left), p. 10 (bottom right), p. 11 © Christopher Lane; p. 12 (bottom), p. 13 Thanks to Eddy Levin; p. 14 © FLC/ADAGP, Paris and DACS, London 2009; p. 16 (bottom) © Daina Taimina. OceanofPDF.com Contents Introduction CHAPTER ZERO A Head for Numbers In which the author tries to find out where numbers come from, since they haven’t been around that long. He meets a man who has lived in the jungle and a chimpanzee who has always lived in the city. CHAPTER ONE The Counter Culture In which the author learns about the tyranny of ten, and the revolutionaries plotting its downfall. He goes to an after-school club in Tokyo, where the pupils learn to calculate by thinking about beads. CHAPTER TWO Behold! In which the author almost changes his name because the disciple of a Greek cult leader says he must. Instead, he follows the instructions of another Greek thinker, dusts off his compass and folds two business cards into a tetrahedron. CHAPTER THREE Something about Nothing In which the author travels to India for an audience with a Hindu seer. He discovers some very slow methods of arithmetic and some very fast ones. CHAPTER FOUR Life of Pi In which the author is in Germany to witness the world’s fastest mental multiplication. It is a roundabout way to begin telling the story of circles, a transcendental tale that leads him to New York and a new appreciation of the 50p piece. CHAPTER FIVE The x-factor In which the author explains why numbers are good but letters are better. He visits a man in Braintree who collects slide-rules and hears the tragic tale of their demise. Includes an exposition of logarithms, a dictionary of calculator words and how to make a superegg. CHAPTER SIX Playtime In which the author is on a mathematical puzzle quest. He investigates the legacy of two Chinese men – one was a dim-witted recluse and the other fell off the Earth – and then flies to Oklahoma to meet a magician. CHAPTER SEVEN Secrets of Succession In which the author is first confronted with the infinite. He encounters an unstoppable snail and a devilish family of numbers. CHAPTER EIGHT Gold Finger In which the author meets a Londoner with a claw who claims to have discovered the secret of beautiful teeth. CHAPTER NINE Chance is a Fine Thing In which the author remembers the dukes of hasard and goes gambling in Reno. He takes a walk through randomness and ends up in an office block in Newport Beach, California – where, if he looked across the ocean, he might be able to spot a lottery winner on a desert island in the South Pacific. CHAPTER TEN Situation Normal In which the author’s farinaceous overindulgence is an attempt to savour the birth of statistics. CHAPTER ELEVEN The End of the Line In which the author terminates his journey with crisps and crochet. He’s looking at Euclid, again, and then at a hotel with an infinite number of rooms that cannot cope with a sudden influx of guests. GLOSSARY APPENDICES NOTES ON CHAPTERS ACKNOWLEDGEMENTS PICTURE CREDITS OceanofPDF.com 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 OceanofPDF.com OceanofPDF.com CHAPTER ZERO OceanofPDF.com A Head for Numbers When I walked into Pierre Pica’s cramped Paris apartment I was overwhelmed by the stench of mosquito repellent. Pica had just returned from spending five months with a community of Indians in the Amazon rainforest, and he was disinfecting the gifts he had brought back. The walls of his study were decorated with tribal masks, feathered headdresses and woven baskets. Academic books overloaded the shelves. A lone Rubik’s Cube lay unsolved on a ledge. I asked Pica how the trip had been. ‘Difficult,’ he replied. Pica is a linguist and, perhaps because of this, speaks slowly and carefully, with painstaking attention to individual words. He is in his fifties, but looks boyish – with bright blue eyes, a reddish complexion and soft, dishevelled silvery hair. His voice is quiet; his manner intense. Pica was a student of the eminent American linguist Noam Chomsky and is now employed by France’s National Centre for Scientific Research. For the last ten years the focus of his work has been the Munduruku, an indigenous group of about 7000 people in the Brazilian Amazon. The Munduruku are hunter-gatherers who live in small villages spread across an area of rainforest twice the size of Wales. Pica’s interest is the Munduruku language: it has no tenses, no plurals and no words for numbers beyond five. To undertake his fieldwork, Pica embarks on a journey worthy of the great adventurers. The nearest large airport to the Indians is Santarém, a town 500 miles up the Amazon from the Atlantic Ocean. From there, a 15-hour ferry ride takes him almost 200 miles along the Tapajós River to Itaituba, a former gold-rush town and the last stop to stock up on food and fuel. On his most recent trip Pica hired a jeep in Itaituba and loaded it up with his equipment, which included computers, solar panels, batteries, books and 120 gallons of petrol. Then he set off down the Trans-Amazon Highway, a 1970s folly of nationalistic infrastructure that has deteriorated into a precarious and often impassable muddy track. Pica’s destination was Jacareacanga, a small settlement a further 200 miles southwest of Itaituba. I asked him how long it took to drive there. ‘Depends,’ he shrugged. ‘It can take a lifetime. It can take two days.’ How long did it take this time, I repeated. ‘You know, you never know how long it will take because it never takes the same time. It takes between ten and twelve hours during the rainy season. If everything goes well.’ Jacareacanga is on the edge of the Munduruku’s demarcated territory. To get inside the area, Pica had to wait for some Indians to arrive so he could negotiate with them to take him there by canoe. ‘How long did you wait?’ I enquired. ‘I waited quite a lot. But, again, don’t ask me how many days.’ ‘So, it was a couple of days?’ I suggested tentatively. A few seconds passed as he furrowed his brow. ‘It was about two weeks.’ More than a month after he left Paris, Pica was finally approaching his destination. Inevitably, I wanted to know how long it took to get from Jacareacanga to the villages. But by now Pica was demonstrably impatient with my line of questioning: ‘Same answer to everything – it depends!’ I stood my ground. How long did it take this time? He stuttered: ‘I don’t know. I think…perhaps…two days…a day and a night…’ The more I pushed Pica for facts and figures, the more reluctant he was to provide them. I became exasperated. It was unclear if underlying his responses was French intransigence, academic pedantry or simply a general contrariness. I stopped my line of questioning and we moved on to other subjects. It was only when, a few hours later, we talked about what it was like to come home after so long in the middle of nowhere that he opened up. ‘When I come back from Amazonia I lose sense of time and sense of number, and perhaps sense of space,’ he said. He forgets appointments. He is disoriented by simple directions. ‘I have extreme difficulty adjusting to Paris again, with its angles and straight lines.’ Pica’s inability to give me quantitative data was part of his culture shock. He had spent so long with people who can barely count that he had lost the ability to describe the world in terms of numbers. No one knows for certain, but numbers are probably no more than about 10,000 years old. By this I mean a working system of words and symbols for numbers. One theory is that such a practice emerged together with agriculture and trade, as numbers were an indispensable tool for taking stock and making sure you were not ripped off. The Munduruku are only subsistence farmers and money has only recently begun to circulate in their villages, so they never evolved counting skills. In the case of the indigenous tribes of Papua New Guinea, it has been argued that the appearance of numbers was triggered by elaborate customs of gift exchange. The Amazon, by contrast, has no such traditions. Tens of thousands of years ago, well before the arrival of numbers, however, our ancestors must have had certain sensibilities about amounts. They would have been able to distinguish one mammoth from two mammoths, and to recognize that one night is different from two nights. The intellectual leap from the concrete idea of two things to the invention of a symbol or word for the abstract idea of ‘two’, however, will have taken many ages to come about. This occurrence, in fact, is as far as some communities in the Amazon have come. There are tribes whose only number words are ‘one’, ‘two’ and ‘many’. The Munduruku, who go all the way up to five, are a relatively sophisticated bunch. Numbers are so prevalent in our lives that it is hard to imagine how people survive without them. Yet while Pierre Pica stayed with the Munduruku he easily slipped into a numberless existence. He slept in a hammock. He went hunting and ate tapir, armadillo and wild boar. He told the time from the position of the sun. If it rained, he stayed in; if it was sunny, he went out. There was never any need to count. Still, I thought it odd that numbers larger than five did not crop up at all in Amazonian daily life. I asked Pica how an Indian would say ‘six fish’. For example, just say that he or she was preparing a meal for six people and he wanted to make sure everyone had a fish each. ‘It is impossible,’ he said. ‘The sentence “I want fish for six people” does not exist.’ What if you asked a Munduruku who had six children: ‘How many kids do you have?’ Pica gave the same response: ‘He will say “I don’t know”. It is impossible to express.’ However, added Pica, the issue was a cultural one. It was not the case that the Munduruku counted his first child, his second, his third, his fourth, his fifth and then scratched his head because he could go no further. For the Munduruku, the whole idea of counting children was ludicrous. The whole idea, in fact, of counting anything was ludicrous. Why would a Munduruku adult want to count his children, asked Pica? The children are looked after by all the adults in the commun...
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