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Geometric Growth

Did you fumble with fractions? Cry over calculus? Mathematician Steve Strogatz says there’s still hope for you to love his field.

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Math is everywhere,” says Steve Strogatz. “It’s an intoxicating way to look at the world. It’s always close to the surface for me, as if I’m wearing different glasses and can see colors that other people can’t. Mathematics permeates the whole world—including personal life, my wardrobe, everything.”

Professor Steve Strogatz

Professor Steve Strogatz

Photo: Robert Barker/UP

His wardrobe?

The professor of applied mathematics isn’t being glib. Some colleagues, he explains, once calculated that there are exactly eighty-five ways to tie a man’s tie. “It turns out you could ask the question, ‘What are all the possible, sensible ways of tying a tie?’ and people have answered it; some are new ways that weren’t known until mathematicians looked at this,” he says. “The point is that these people have the same bug that I have. It’s not that it’s useful to know that there are eighty-five ways; it’s the pleasure of being creative, letting your mind ask a new question. But a spooky aspect of math is that invariably, it is useful. You start out asking this frivolous question about a tie, and the things you learn are relevant to medicine—because DNA is a double helix, so it’s got a lot of geometry. This theme plays itself out over and over—that these curiosity questions that mathematicians have are mirrored in the structure of the world inside us and around us.”

In the spring of 2010, Strogatz had the chance to share his lifelong love of math with an audience far beyond campus, when the New York Times ran his weekly online column on the subject’s myriad pleasures. Dubbed “Steven Strogatz on the Elements of Math,” the fifteen-part series took readers from the kindergarten level (a “Sesame Street” clip in which Ernie, faced with a cadre of hungry penguins each clamoring for a fish, extols the value of counting) to advanced concepts like probability theory and differential geometry. “What I want above all is for people to share the joy of the subject,” Strogatz says. “People who love math love it for a reason—it gives them a lot of pleasure. It’s truly beautiful.”

The Joy of X: releasing in October.

In early October, Houghton Mifflin Harcourt will publish the columns, along with fifteen new ones, as The Joy of X: A Guided Tour of Math, from One to Infinity. The thirty essays contemplate topics familiar to students worldwide: algebra, complex numbers, word problems, geometric proofs, integral and differential calculus, prime numbers, and more. “This is not remedial; it’s not a course,” Strogatz says. “It’s not going to teach you math. It’s why math is enthralling.”

The essays explore complex concepts using cultural references both high and low. There’s talk of Pythagoras, Einstein, and Archimedes—but also musings on the topology of a bagel, and the tale of a Verizon customer service rep who couldn’t grasp the difference between .002 dollars and .002 cents. Strogatz even compares his field to Tony Soprano. “Math swaggers with an intimidating air of certainty,” he writes. “Like a Mafia capo, it comes across as decisive, unyielding, and strong. It’ll make you an argument you can’t refuse. But in private, math is occasionally insecure. It has doubts. It questions itself and isn’t always sure it’s right. Especially where infinity is concerned. Infinity can keep math up at night, worrying, fidgeting, feeling existential dread. For there have been times in the history of math when unleashing infinity wrought such mayhem, there were fears it might blow up the whole enterprise. And that would be bad for business.”

Strogatz’s Times series proved wildly popular. Each column made it to the list of the top-ten most e-mailed articles (some at number one) and garnered hundreds of comments. “Normally, the academic experience is that you write a paper and nobody reacts—it goes into the vacuum, or more like a black hole,” he says. “Except for the peer review, it’s a big thud. But here, people were writing to me, asking questions. Some were parents asking about things they could do to help their kids, or just curious grownups who always wanted to understand math better.”

As the weeks went by, he says, the column turned into a bit of a sociology experiment, as tussles emerged in the comments section between self-proclaimed experts and the interested laypeople for whom the column was intended. “Math people, especially schoolteachers, started weighing in, saying stuff like, ‘Let me tell you a little more about what Dr. Strogatz means…’ and they would write long expositions,” he recalls. “But other people said, ‘This is our column; this is not for people who are already math aficionados. Stop criticizing; you’re showing off; leave us alone.'” Strogatz’s wife, who took on the task of surveying the comments to spare his sanity, termed the former group “the pontificators.”

The column, Strogatz notes, clearly struck a nerve. It revealed that there’s a vast array of people who once loved math, but stumbled and never recovered. Years or decades after high school, it still eats at them—and they were enormously grateful that Strogatz was giving them another chance to appreciate the subject. “When people at a party hear that you teach math, it gets an emotional response,” he says. “It’s always regret—‘I liked math, until. . . .’ For some, division was the problem, or algebra, or geometry. Some say, ‘I was great until calculus.’ There’s also a feeling of shame. Sometimes they blame the teacher, but there is often a sense of failure—that because it’s sequential, you’re done and there’s no second chance. Often, for very smart people, this is the only time they’ve felt that they couldn’t do something in school. So it becomes personal.”

The Joy of X is Strogatz’s third general-audience book; he previously published Sync: How Order Emerges from the Universe, Nature, and Daily Life and The Calculus of Friendship: What a Teacher and a Student Learned About Life While Corresponding About Math. This fall, he’ll write more online essays for the Times, this time an eight-week series using multimedia to explore mathematical concepts. “My neighbor down the street told me, ‘Your columns make me want to like math,'” Strogatz muses. “I thought that was a beautiful formulation, because she still doesn’t like it—but now she wants to like it, which is the first step.”


Magic Numbers

In an excerpt from The Joy of X, the math professor explains—with a little help from Ezra—how arithmetic came to the masses.

By Steven Strogatz

I’d walked past Ezra Cornell’s statue [on the Arts Quad] hundreds of times without even glan­cing at his greenish likeness. But then one day I stopped for a closer look.

Ezra appears outdoorsy and ruggedly dignified in his long coat, vest, and boots, his right hand resting on a walking stick and holding a rumpled, wide-brimmed hat. The monument comes across as unpretentious and disarmingly direct—much like the man himself, by all accounts.

Which is why it seems so discordant that Ezra’s dates are inscribed on the pedestal in pompous Roman numerals:

EZRA CORNELL
MDCCCVII–MDCCCLXXIV

Why not write simply 1807–1874? Roman numerals may look impressive, but they’re hard to read and cumbersome to use. Ezra would have had little patience for that.

Finding a good way to represent numbers has always been a challenge. Since the dawn of civilization, people have tried various systems for writing numbers and reckoning with them, whether for trading, measuring land, or keeping track of the herd.

What nearly all these systems have in common is that our biology is deeply embedded in them. Through the vagaries of evolution, we happen to have five fingers on each of two hands. That peculiar anatomical fact is reflected in the primitive system of tallying; for example, the number 17 is written as:

Here, each of the vertical strokes in each group must have originally meant a finger. Maybe the diagonal slash was a thumb, folded across the other four fingers to make a fist?

Roman numerals are only slightly more sophisticated than tallies. You can spot the vestige of tallies in the way Romans wrote 2 and 3, as II and III. Likewise, the diagonal slash is echoed in the shape of the Roman symbol for 5, V. But 4 is an ambiguous case. Sometimes it’s written as IIII, tally style (you’ll often see this on fancy clocks), though more commonly it’s written as IV. The positioning of a smaller number (I) to the left of a larger number (V) indicates that you’re supposed to subtract I, rather than add it, as you would if it were stationed on the right. Thus IV means 4, whereas VI means 6.

The Babylonians were not nearly as attached to their fingers. Their numeral system was based on 60—a clear sign of their impeccable taste, for 60 is an exceptionally pleasant number. Its beauty is intrinsic and has nothing to do with human appendages. Sixty is the smallest number that can be divided evenly by 1, 2, 3, 4, 5, and 6. And that’s just for starters (there’s also 10, 12, 15, 20, and 30). Because of its promiscuous divisibility, 60 is much more congenial than 10 for any sort of calculation or measurement that involves cutting things into equal parts. When we divide an hour into 60 minutes, or a minute into 60 seconds, or a full circle into 360 degrees, we’re channeling the sages of ancient Babylon.

But the greatest legacy of the Babylonians is an idea that’s so commonplace today that few of us appreciate how subtle and ingenious it is.

To illustrate it, let’s consider our own Hindu-Arabic system, which incorporates the same idea in its modern form. Instead of 60, this system is based on ten symbols: 1, 2, 3, 4, 5, 6, 7, 8, 9, and, most brilliant, 0. These are called digits, naturally, from the Latin word for a finger or a toe.

The great innovation here is that even though this system is based on the number 10, there is no single symbol reserved for 10. Ten is marked by a position—the tens place—instead of a symbol. The same is true for 100, or 1,000, or any other power of 10. Their distinguished status is signified not by a symbol but by a parking spot, a reserved piece of real estate. Location, location, location.

Contrast the elegance of this place-value system with the much cruder approach used in Roman numerals. You want 10? We’ve got 10. It’s X. We’ve also got 100 (C) and 1,000 (M), and we’ll even throw in special symbols for the 5 family: V, L, and D, for 5, 50, and 500.

The Roman approach was to elevate a few favored numbers, give them their own symbols, and express all the other, second-class numbers as combinations of those.

Unfortunately, Roman numerals creaked and groaned when faced with anything larger than a few thousand. In a work-around solution that would nowadays be called a kludge, the scholars who were still using Roman numerals in the Middle Ages resorted to piling bars on top of the existing symbols to indicate multiplication by a thousand. For instance, X meant ten thousand, and M meant a thousand thousands or, in other words, a million. Multiplying by a billion (a thousand million) was rarely necessary, but if you ever had to, you could always put a second bar on top of the M. As you can see, the fun never stopped.

But in the Hindu-Arabic system, it’s a snap to write any number, no matter how big. All numbers can be expressed with the same ten digits, merely by slotting them into the right places. Furthermore, the notation is inherently concise. Every number less than a million, for example, can be expressed in six symbols or fewer. Try doing that with words, tallies, or Roman numerals.

Best of all, with a place-value system, ordinary people can learn to do arithmetic. You just have to master a few facts—the multiplication table and its counterpart for addition. Once you get those down, that’s all you’ll ever need. Any calculation involving any pair of numbers, no matter how big, can be performed by applying the same sets of facts, over and over again, recursively.

If it all sounds pretty mechanical, that’s precisely the point. With place-value systems, you can program a machine to do arithmetic. From the early days of mechanical calculators to the supercomputers of today, the automation of arithmetic was made possible by the beautiful idea of place value.

But the unsung hero in this story is 0. Without 0, the whole approach would collapse. It’s the placeholder that allows us to tell 1, 10, and 100 apart.

All place-value systems are based on some number called, appropriately enough, the base. Our system is base 10, or decimal (from the Latin root decem, meaning “ten”). After the ones place, the subsequent consecutive places represent tens, hundreds, thousands, and so on, each of which is a power of 10:

10 = 101
100 = 10 × 10 = 102
1,000 = 10 × 10 × 10 = 103.

Given what I said earlier about the biological, as opposed to the logical, origin of our preference for base 10, it’s natural to ask: Would some other base be more efficient, or easier to manipulate?

A strong case can be made for base 2, the famous and now ubiquitous binary system used in computers and all things digital, from cell phones to cameras. Of all the possible bases, it requires the fewest symbols—just two of them, 0 and 1. As such, it meshes perfectly with the logic of electronic switches or anything else that can toggle between two states—on or off, open or closed.

Binary takes some getting used to. Instead of powers of 10, it uses powers of 2. It still has a ones place like the decimal system, but the subsequent places now stand for twos, fours, and eights, because:

2 = 21
4 = 2 × 2 = 22
8 = 2 × 2 × 2 = 23.

Of course, we wouldn’t write the symbol 2, because it doesn’t exist in binary, just as there’s no single numeral for 10 in decimal. In binary, 2 is written as 10, meaning one 2 and zero 1s. Similarly, 4 would be written as 100 (one 4, zero 2s, and zero 1s), and 8 would be 1000.

The implications reach far beyond math. Our world has been changed by the power of 2. In the past few decades we’ve come to realize that all information—not just numbers, but also language, images, and sound—can be encoded in streams of zeros and ones.

Which brings us back to Ezra Cornell.

Tucked at the rear of his monument, and almost completely obscured, is a telegraph machine—a modest reminder of his role in the creation of Western Union and the tying together of the North American continent.

As a carpenter turned entrepreneur, Cornell worked for Samuel Morse, whose name lives on in the code of dots and dashes through which the English language was reduced to the clicks of a telegraph key. Those two little symbols were technological forerunners of today’s zeros and ones.

Morse entrusted Cornell to build the nation’s first telegraph line, a link from Baltimore to the U.S. Capitol, in Washington, D.C. From the very start it seems that he had an inkling of what his dots and dashes would bring. When the line was officially opened, on May 24, 1844, Morse sent the first message down the wire: “What hath God wrought.”

Excerpted from The Joy of X by Steven Strogatz, to be published October 2012 by Houghton Mifflin Harcourt Publishing Company. Copyright © 2012 by Steven Strogatz. Reprinted by permission of Houghton Mifflin Harcourt Publishing Company. All rights reserved.

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