The Joy of X E-Book

Steven Strogatz

• Herausgeber: Atlantic Books
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2012

Beschreibung

Award-winning Steven Strogatz, one of the foremost popularisers of maths, has written a witty and fascinating account of maths' most compelling ideas and how, so often, they are an integral part of everyday life. Maths is everywhere, often where we don't even realise. Award-winning professor Steven Strogatz acts as our guide as he takes us on a tour of numbers that - unbeknownst to the unitiated - connect pop culture, literature, art, philosophy, current affairs, business and even every day life. In The Joy of X, Strogatz explains the great ideas of maths - from negative numbers to calculus, fat tails to infinity - with clarity, wit and insight. He is the maths teacher you never had and this book is perfect for the smart and curious, the expert and the beginner.

Leseprobe

Also by Steven Strogatz

Sync: The Emerging Science of Spontaneous Order

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Maths

First published in the United States in 2012 by An Eamon Dolan Book, an imprint of Houghton Mifflin Harcourt Publishing Company, New York.

Published in trade paperback in 2012 by Atlantic Books, an imprint of Atlantic Books Ltd.

Published in hardback in Great Britain in 2013 by Atlantic Books, an imprint of Atlantic Books Ltd.

Copyright © Steven Strogatz, 2012

The moral right of Steven Strogatz to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

Every effort has been made to trace or contact all copyright holders. The publishers will be pleased to make good any omissions or rectify any mistakes brought to their attention at the earliest opportunity.

Chapters 1–3, 5, 7, 8, 11, 12, 16–18, 20, 23, 26, 28, and 30 are adapted, with permission, from pieces originally published in The New York Times. Grateful acknowledgment is made for permission to reprint an excerpt from the following copyrighted work: pp. 201–202, from p. 111 of The Solitude of Prime Numbers: A Novel by Paolo Giordano, translated by Shaun Whiteside, copyright © 2008 by Arnoldo Mondadori Editore S.p.A., translation copyright © 2009 by Shaun Whiteside. Used by permission of Pamela Dorman Books, an imprint of Viking Penguin, a division of Penguin Group (USA) Inc.

Further credits can be found on page 307

10 9 8 7 6 5 4 3 2 1

A CIP catalogue record for this book is available from the British Library.

Hardback ISBN: 978 84887 843 3 Trade paperback ISBN: 978 1 84887 844 0 Ebook ISBN: 978 0 85789 845 6

Printed in Great Britain by

Atlantic Books An Imprint of Atlantic Books Ltd Ormond House 26–27 Boswell Street London WC1N 3JZ

www.atlantic-books.co.uk

Contents

Preface

Part OneNUMBERS

1. From Fish to Infinity

An introduction to numbers, pointing out their upsides (they’re efficient) as well as their downsides (they’re ethereal)

2. Rock Groups

Treating numbers concretely — think rocks — can make calculations less baffling.

3. The Enemy of My Enemy

The disturbing concept of subtraction, and how we deal with the fact that negative numbers seem so . . . negative

4. Commuting

When you buy jeans on sale, do you save more money if the clerk applies the discount after the tax, or before?

5. Division and Its Discontents

Helping Verizon grasp the difference between .002 dollars and .002 cents

6. Location, Location, Location

How the place-value system for writing numbers brought arithmetic to the masses

Part TwoRELATIONSHIPS

7. The Joy of x

Arithmetic becomes algebra when we begin working with unknowns and formulas.

8. Finding Your Roots

Complex numbers, a hybrid of the imaginary and the real, are the pinnacle of number systems.

9. My Tub Runneth Over

Turning peril to pleasure in word problems

10. Working Your Quads

The quadratic formula may never win any beauty contests, but the ideas behind it are ravishing.

11. Power Tools

In maths, the function of functions is to transform.

Part ThreeSHAPES

12. Square Dancing

Geometry, intuition, and the long road from Pythagoras to Einstein

13. Something from Nothing

Like any other creative act, constructing a proof begins with inspiration.

14. The Conic Conspiracy

The uncanny similarities between parabolas and ellipses suggest hidden forces at work.

15. Sine Qua Non

Sine waves everywhere, from Ferris wheels to zebra stripes

16. Take It to the Limit

Archimedes recognized the power of the infinite and in the process laid the groundwork for calculus.

Part FourCHANGE

17. Change We Can Believe In

Differential calculus can show you the best path from A to B, and Michael Jordan’s dunks help explain why.

18. It Slices, It Dices

The lasting legacy of integral calculus is a Veg-O-Matic view of the universe.

19. All about e

How many people should you date before settling down? Your grandmother knows — and so does the number e.

20. Loves Me, Loves Me Not

Differential equations made sense of planetary motion. But the course of true love? Now that’s confusing.

21. Step Into the Light

A light beam is a pas de deux of electric and magnetic fields, and vector calculus is its choreographer.

Part FiveDATA

22. The New Normal

Bell curves are out. Fat tails are in.

23. Chances Are

The improbable thrills of probability theory

24. Untangling the Web

How Google solved the Zen riddle of Internet search using linear algebra

Part SixFRONTIERS

25. The Loneliest Numbers

Prime numbers, solitary and inscrutable, space themselves apart in mysterious ways.

26. Group Think

Group theory, one of the most versatile parts of maths, bridges art and science.

27. Twist and Shout

Playing with Möbius strips and music boxes, and a better way to cut a bagel

28. Think Globally

Differential geometry reveals the shortest route between two points on a globe or any other curved surface.

29. Analyze This!

Why calculus, once so smug and cocky, had to put itself on the couch

30. The Hilbert Hotel

An exploration of infinity as this book, not being infinite, comes to an end

Acknowledgments

Notes

Credits

Index

Note on the Author

Preface

I have a friend who gets a tremendous kick out of science, even though he’s an artist. Whenever we get together all he wants to do is chat about the latest thing in psychology or quantum mechanics. But when it comes to maths, he feels at sea, and it saddens him. The strange symbols keep him out. He says he doesn’t even know how to pronounce them.

Crazy as it sounds, that’s what I’ll be trying to do in this book. It’s a guided tour through the elements of maths, from preschool to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what maths is all about and why it’s so enthralling to those who get it.

We’ll discover how Michael Jordan’s dunks can help explain the fundamentals of calculus. I’ll show you a simple — and mind-blowing — way to understand that staple of geometry, the Pythagorean theorem. We’ll try to get to the bottom of some of life’s mysteries, big and small: Did O.J. do it? How should you flip your mattress to get the maximum wear out of it? How many people should you date before settling down? And we’ll see why some infinities are bigger than others.

Maths is everywhere, if you know where to look. We’ll spot sine waves in zebra stripes, hear echoes of Euclid in the Declaration of Independence, and recognize signs of negative numbers in the run-up to World War I. And we’ll see how our lives today are being touched by new kinds of maths, as we search for restaurants online and try to understand — not to mention survive — the frightening swings in the stock market.

By a coincidence that seems only fitting for a book about numbers, this one was born on the day I turned fifty. David Shipley, who was then the editor of the op-ed page for the New York Times, had invited me to lunch on the big day (unaware of its semicentennial significance) and asked if I would ever consider writing a series about maths for his readers. I loved the thought of sharing the pleasures of maths with an audience beyond my inquisitive artist friend.

“The Elements of Math” appeared online in late January 2010 and ran for fifteen weeks. In response, letters and comments poured in from readers of all ages. Many who wrote were students and teachers. Others were curious people who, for whatever reason, had fallen off the track somewhere in their maths education but sensed they were missing something worthwhile and wanted to try again. Especially gratifying were the notes I received from parents thanking me for helping them explain maths to their kids and, in the process, to themselves. Even my colleagues and fellow maths aficionados seemed to enjoy the pieces — when they weren’t suggesting improvements (or perhaps especially then!).

All in all, the experience convinced me that there’s a profound but little-recognized hunger for maths among the general public. Despite everything we hear about maths phobia, many people want to understand the subject a little better. And once they do, they find it addictive.

The Joy of x is an introduction to maths’s most compelling and far-reaching ideas. The chapters — some from the original Times series — are bite-size and largely independent, so feel free to snack wherever you like. If you want to wade deeper into anything, the notes at the end of the book provide additional details and suggestions for further reading.

For the benefit of readers who prefer a step-by-step approach, I’ve arranged the material into six main parts, following the lines of the traditional curriculum.

Part 1, “Numbers,” begins our journey with kindergarten and grade-school arithmetic, stressing how helpful numbers can be and how uncannily effective they are at describing the world.

Part 2, “Relationships,” generalizes from working with numbers to working with relationships between numbers. These are the ideas at the heart of algebra. What makes them so crucial is that they provide the first tools for describing how one thing affects another, through cause and effect, supply and demand, dose and response, and so on — the kinds of relationships that make the world complicated and rich.

Part 3, “Shapes,” turns from numbers and symbols to shapes and space — the province of geometry and trigonometry. Along with characterizing all things visual, these subjects raise maths to new levels of rigor through logic and proof.

In part 4, “Change,” we come to calculus, the most penetrating and fruitful branch of maths. Calculus made it possible to predict the motions of the planets, the rhythm of the tides, and virtually every other form of continuous change in the universe and ourselves. A supporting theme in this part is the role of infinity. The domestication of infinity was the breakthrough that made calculus work. By harnessing the awesome power of the infinite, calculus could finally solve many long-standing problems that had defied the ancients, and that ultimately led to the scientific revolution and the modern world.

Part 5, “Data,” deals with probability, statistics, networks, and data mining, all relatively young subjects inspired by the messy side of life: chance and luck, uncertainty, risk, volatility, randomness, interconnectivity. With the right kinds of maths, and the right kinds of data, we’ll see how to pull meaning from the maelstrom.

As we near the end of our journey in part 6, “Frontiers,” we approach the edge of mathematical knowledge, the borderland between what’s known and what remains elusive. The sequence of chapters follows the familiar structure we’ve used throughout — numbers, relationships, shapes, change, and infinity — but each of these topics is now revisited more deeply, and in its modern incarnation.

I hope that all of the ideas ahead will provide joy — and a good number of Aha! moments. But any journey needs to begin at the beginning, so let’s start with the simple, magical act of counting.

Part OneNUMBERS

1

From Fish to Infinity

THE BEST INTRODUCTION to numbers I’ve ever seen — the clearest and funniest explanation of what they are and why we need them — appears in a Sesame Street video called 123 Count with Me. Humphrey, an amiable but dimwitted fellow with pink fur and a green nose, is working the lunch shift at the Furry Arms Hotel when he takes a call from a roomful of penguins. Humphrey listens carefully and then calls out their order to the kitchen: “Fish, fish, fish, fish, fish, fish.” This prompts Ernie to enlighten him about the virtues of the number six.

Children learn from this that numbers are wonderful shortcuts. Instead of saying the word “fish” exactly as many times as there are penguins, Humphrey could use the more powerful concept of six.

As adults, however, we might notice a potential downside to numbers. Sure, they are great timesavers, but at a serious cost in abstraction. Six is more ethereal than six fish, precisely because it’s more general. It applies to six of anything: six plates, six penguins, six utterances of the word “fish.” It’s the ineffable thing they all have in common.

Viewed in this light, numbers start to seem a bit mysterious. They apparently exist in some sort of Platonic realm, a level above reality. In that respect they are more like other lofty concepts (e.g., truth and justice), and less like the ordinary objects of daily life. Their philosophical status becomes even murkier upon further reflection. Where exactly do numbers come from? Did humanity invent them? Or discover them?

An additional subtlety is that numbers (and all mathematical ideas, for that matter) have lives of their own. We can’t control them. Even though they exist in our minds, once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there’s nothing we can do about it except watch and try to understand. In that sense they are eerily reminiscent of atoms and stars, the things of this world, which are likewise subject to laws beyond our control . . . except that those things exist outside our heads.

This dual aspect of numbers — as part heaven, part earth — is perhaps their most paradoxical feature, and the feature that makes them so useful. It is what the physicist Eugene Wigner had in mind when he wrote of “the unreasonable effectiveness of mathematics in the natural sciences.”

In case it’s not clear what I mean about the lives of numbers and their uncontrollable behavior, let’s go back to the Furry Arms. Suppose that before Humphrey puts in the penguins’ order, he suddenly gets a call on another line from a room occupied by the same number of penguins, all of them also clamoring for fish. After taking both calls, what should Humphrey yell out to the kitchen? If he hasn’t learned anything, he could shout “fish” once for each penguin. Or, using his numbers, he could tell the cook he needs six orders of fish for the first room and six more for the second room. But what he really needs is a new concept: addition. Once he’s mastered it, he’ll proudly say he needs six plus six (or, if he’s a showoff, twelve) fish.

The creative process here is the same as the one that gave us numbers in the first place. Just as numbers are a shortcut for counting by ones, addition is a shortcut for counting by any amount. This is how mathematics grows. The right abstraction leads to new insight, and new power.

Before long, even Humphrey might realize he can keep counting forever.

Yet despite this infinite vista, there are always constraints on our creativity. We can decide what we mean by things like 6 and +, but once we do, the results of expressions like 6 + 6 are beyond our control. Logic leaves us no choice. In that sense, maths always involves both invention and discovery: we invent the concepts but discover their consequences. As we’ll see in the coming chapters, in mathematics our freedom lies in the questions we ask — and in how we pursue them — but not in the answers awaiting us.

2

Rock Groups

LIKE ANYTHING ELSE, arithmetic has its serious side and its playful side.

The serious side is what we all learned in school: how to work with columns of numbers, adding them, subtracting them, grinding them through the spreadsheet calculations needed for tax returns and year-end reports. This side of arithmetic is important, practical, and for many people joyless.

The playful side of arithmetic is a lot less familiar, unless you were trained in the ways of advanced mathematics. Yet theres nothing inherently advanced about it. Its as natural as a childs curiosity.

In his book A Mathematicians Lament, Paul Lockhart advocates an educational approach in which numbers are treated more concretely than usual: he asks us to imagine them as groups of rocks. For example, 6 corresponds to a group of rocks like this:

You probably dont see anything striking here, and thats right unless we make further demands on numbers, they all look pretty much the same. Our chance to be creative comes in what we ask of them.