the $27.00 joy · pdf file“a delightful exploration of the beauty and fun of...

xThe

Joyof

A Guided

Tour of Math,

from One to

Infinity

sTe ve n sTrogaTz

A world-class mathematician and regular contributor to the New York Times hosts a delightful tour of the greatest ideas of math, revealing how it connects to literature, phi-losophy, law, medicine, art, business, even pop culture in ways we never imagined.

Did o.j. do it? How should you flip your mattress to get the maximum wear out of it? How does Google

search the Internet? How many people should you date before settling down? Believe it or not, math plays a crucial role in answering all of these questions and more. Math underpins everything in the cosmos, including us, yet too few of us understand this universal language well enough to revel in its wisdom, its beauty — and its joy. This deeply enlightening, vastly entertaining volume trans-lates math in a way that is at once intelligible and thrilling. Each trenchant chapter of The Joy of x offers an “Aha!” moment, starting with why numbers are so helpful and progressing through the wondrous truths implicit in π, the Pythagorean theorem, irrational numbers, fat tails, even the rigors and surprising charms of calculus. Showing why he has won awards as a professor at Cornell and garnered exten-sive praise for his articles about math for the New York Times, Strogatz presumes of his read-ers only curiosity and common sense. And he rewards them with clear, ingenious, and often funny explanations of the most vital and excit-ing principles of his discipline. Whether you aced integral calculus or aren’t sure what an integer is, you’ll find profound wisdom and persistent delight in The Joy of x.

“A delightful exploration of the beauty and fun of mathematics, in the best traditions of Lewis Carroll, George Gamow, and Martin Gardner. The Joy of x will entertain you, amaze you, and make you smarter.” — Steven Pinker, Harvard College Professor of Psychology, Harvard University, and author of How the Mind Works and The Language Instinct

“Steven Strogatz should do for math what Julia Child did for cookery. He shows that this stuff really matters, and he shows that it can nourish us.” — James Gleick, author of The Information: A History, a Theory, a Flood and Chaos

“This joyous book will remind you just how beautiful and mesmerizing math can be. Steve Strogatz is the teacher we all wish we had.” — Joshua Foer, author of Moonwalking with Einstein

“I loved this beautiful book from the first page. With his unique ingenuity and affable charm, Strogatz disassembles mathematics as a subject, both feared and revered, and reassembles it as a world, both accessible and magical. The Joy of x is, well, a joy.” — Janna Levin, Professor of Physics and Astronomy, Barnard College, Columbia University, and author of How the Universe Got Its Spots and A Madman Dreams of Turing Machines

“Amazingly, mathematicians can see patterns in the universe that the rest of us are usually blind to. With clarity and dry wit, The Joy of x opens a window onto this hidden world with its landscapes of beauty and wonder.” — Alan Alda

“This book is, simply put, fantastic. It introduces the reader to the underlying concepts of mathematics — presenting reasons for its unfamiliar language and explaining conceptual frameworks that do in fact make understanding complex problems easier. In a world where mathematics is essential but, largely, poorly understood, Steve Strogatz’s teaching skills and deft writing style are an important contribution.” — Lisa Randall, Frank B. Baird, Jr., Professor of Science, Harvard University, and author of Warped Passages and Knocking on Heaven’s Door

“Strogatz has discovered a magical function that transforms ‘math’ into ‘joy,’ page after wonderful page. He takes everything that ever mystified you about math and makes it better than clear — he makes it wondrous, delicious, and amazing.” — Daniel Gilbert, Professor of Psychology, Harvard University, and author of Stumbling on Happiness

sTeve n sTr o g aTz is the Schur-man Professor of Applied Mathematics at Cor-nell University. A renowned teacher and one of the world’s most highly cited mathematicians, he has been a frequent guest on National Pub-lic Radio’s Radiolab. He is the author of Sync and The Calculus of Friendship, and the recipi-ent of a lifetime achievement award for math communication. He also wrote a popular New York Times online column, “The Elements of Math,” which formed the basis for this book.

isbn 978-0-547-51765-0

$27.00 Higher in Canada

sTe ve n

sTrogaTz

The

Joyofx

$27.00 /higher in canada

10121444894

Jacket design by Martha KennedyAuthor photograph © Jennifer S. Altman

an eaM o n D o Lan B oo K

houghton Mifflin harcourtwww.hmhbooks.com ©

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“Strogatz may be the only person alive with the skill to pied piper me into the murky abyss of set theory. I literally learned something on ev-ery page, despite my innumerate brain. This is a fantastic book, conveyed with clarity, techni-cal mastery, and infectious joy.” — Jad Abumrad, host of Radiolab

9

Uncle irv was my dad’s brother as well as his partner in a shoe store they owned in our town. He handled the business end of things and mostly stayed in the office upstairs, because he was good with numbers and not so good with the customers. When I was about ten or eleven, Uncle Irv gave me my first word problem. It sticks with me to this day, probably because I got it wrong and felt embarrassed. It had to do with filling a bathtub. If the cold-water faucet can fill the tub in a half-hour, and the hot-water faucet can fill it in an hour, how long will it take to fill the tub when they’re running together?

H

C

my Tub runneth over

Strogatz_JOY_X_int_F.indd 59 7/19/12 11:57 AM

60 r e l a t i o N S H i P S

I’m pretty sure I guessed forty-five minutes, as many people would. Uncle Irv shook his head and grinned. Then, in his high-pitched nasal voice, he proceeded to school me. “Steven,” he said, “figure out how much water pours into the tub in a minute.” The cold water fills the tub in thirty min-utes, so in one minute it fills 130 of the tub. But the hot wa-ter runs slower — it takes sixty minutes, which means it fills only 160 of the tub per minute. So when they’re both running, they fill

130 + 160

of the tub in a minute. To add those fractions, observe that 60 is their lowest com-mon denominator. Then, rewriting 130 as 260, we get

130 + 160 = 260 + 160 = 360 = 120,

which means that the two faucets working together fill 120 of the tub per minute. So they fill the whole tub in twenty minutes. Over the years since then, I’ve thought about this bathtub problem many times, always with affection for both Uncle Irv and the question itself. There are broader lessons to be learned here — lessons about how to solve problems approximately when you can’t solve them exactly, and how to solve them in-tuitively, for the pleasure of the Aha! moment. Consider my initial guess of forty-five minutes. By look-ing at an extreme, or limiting, case, we can see that that an-


61m Y T u b r u N N e T h o v e r

swer can’t possibly be right. In fact, it’s absurd. To understand why, suppose the hot water wasn’t turned on. Then the cold water — on its own — would fill the tub in thirty minutes. So whatever the answer to Uncle Irv’s question is, it has to be less than this. After all, running the hot water along with the cold can only help. Admittedly, this conclusion is not as informative as the ex-act answer of twenty minutes we found by Uncle Irv’s method, but it has the advantage of not requiring any calculation. A different way to simplify the problem is to pretend the two faucets run at the same rate. Say each can fill the tub in thirty minutes (meaning that the hot water runs just as fast as the cold). Then the answer would be obvious. Because of the symmetry of the new situation, the two perfectly matched fau-cets would together fill the tub in fifteen minutes, since each does half the work. This instantly tells us that Uncle Irv’s scenario must take longer than fifteen minutes. Why? Because fast plus fast beats slow plus fast. Our make-believe symmetrical problem has two fast faucets, whereas Uncle Irv’s has one slow, one fast. And since fifteen minutes is the answer when they’re both fast, Un-cle Irv’s tub can only take longer. The upshot is that by considering two hypothetical cases — one with the hot water off, and another with it matched to the cold water — we learned that the answer lies somewhere between fifteen and thirty minutes. In much harder problems where it may be impossible to find an exact answer — not just in math but in other domains as well — this sort of partial in-formation can be very valuable. Even if we’re lucky enough to come up with an exact an-swer, that’s still no cause for complacency. There may be easier



or clearer ways to find the solution. This is one place where math allows for creativity. For example, instead of Uncle Irv’s textbook method, with its fractions and common denominators, here’s a more playful route to the same result. It dawned on me some years later, when I tried to pinpoint why the problem is so confusing in the first place and realized it’s because of the faucets’ different speeds. That makes it a headache to keep track of what each faucet contributes, especially if you picture the hot and cold water sloshing together and mixing in the tub. So let’s keep the two types of water apart, at least in our minds. Instead of a single bathtub, imagine two assembly lines of them, two separate conveyor belts shuttling bathtubs past a hot-water faucet on one side and a cold-water faucet on the other.

C

H



Each faucet stands in place and fills its own tubs — no mixing allowed. And as soon as a tub fills up, it moves on down the line, making way for the next one. Now everything becomes easy. In one hour, the hot-water faucet fills one tub, while the cold-water faucet fills two (since each takes a half-hour). That amounts to three tubs per hour, or one tub every twenty minutes. Eureka! So why do so many people, including my childhood self, blunder into guessing forty-five minutes? Why is it so tempting to split the difference of thirty and sixty minutes? I’m not sure, but it seems to be a case of faulty pattern recognition. Maybe the bathtub problem is being conflated with others where split-ting the difference would make sense. My wife explained it to me by analogy. Imagine you’re helping a little old lady cross the street. Without your help, it would take her sixty seconds, while you’d zip across in thirty seconds. How long, then, would it take the two of you, walking arm in arm? A compromise around forty-five seconds seems plausible because when granny is clinging to your elbow, she slows you down and you speed her up. The difference here is that you and granny affect each other’s speeds, but the faucets don’t. They’re independent. Ap-parently our subconscious minds don’t spot this distinction, at least not when they’re leaping to the wrong conclusion. The silver lining is that even wrong answers can be edu-cational . . . as long as you realize they’re wrong. They expose misguided analogies and other woolly thinking, and bring the crux of the problem into sharper relief. Other classic word problems are expressly designed to trick their victims by misdirection, like a magician’s sleight of hand. The phrasing of the question sets a trap. If you answer by in-stinct, you’ll probably fall for it.



Try this one. Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence? It’s tempting to blurt out “one hour.” The words themselves nudge you that way. The drumbeat in the first sentence — three men, three fences, three hours — catches your attention by es-tablishing a rhythm, so when the next sentence repeats the pat-tern with one man, one fence, ____ hours, it’s hard to resist filling in the blank with “one.” The parallel construction sug-gests an answer that’s linguistically right but mathematically wrong. The correct answer is three hours. If you visualize the problem — mentally picture three men painting three fences and all finishing after three hours, just as the problem states — the right answer becomes clear. For all three fences to be done after three hours, each man must have spent three hours on his.

The undistracted reasoning that this problem requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.



Perhaps even more important, word problems give us prac-tice in thinking not just about numbers, but about relationships between numbers — how the flow rates of the faucets affect the time required to fill the tub, for example. And that is the essen-tial next step in anyone’s math education. Understandably, a lot of us have trouble with it; relationships are much more abstract than numbers. But they’re also much more powerful. They ex-press the inner logic of the world around us. Cause and effect, supply and demand, input and output, dose and response — all involve pairs of numbers and the relationships between them. Word problems initiate us into this way of thinking. However, Keith Devlin raises an interesting criticism in his essay “The problem with word problems.” His point is that these problems typically assume you understand the rules of the game and agree to play by them, even though they’re often artificial, sometimes absurdly so. For example, in our problem about three men painting three fences in three hours, it was implicit that (1) all three men paint at the same rate and (2) they all paint steadily, never slowing down or speed-ing up. Both assumptions are unrealistic. You’re supposed to know not to worry about that and go along with the gag, be-cause otherwise the problem would be too complicated and you wouldn’t have enough information to solve it. You’d need to know exactly how much each painter slows down as he gets tired in the third hour, how often each one stops for a snack, and so on. Those of us who teach math should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill — it’s called mathematical modeling. Scientists do it all the time when they apply math to the real world. But


r e l a t i o N S H i P S66

they, unlike the authors of most word problems, are usually careful to state their assumptions explicitly. So thanks, Uncle Irv, for that first lesson. Humiliating? Yes. Unforgettable? Yes, that too . . . but in a good way.


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