A Most Elegant Equation

Introduction

Who can resist lists of the ten best this or that? Not me. So a few years ago when I ran across a ranking by math experts of the most beautiful theorems, I tuned in. I decided to treat it as a pop quiz: How many of them could I dredge up from my distant undergraduate days as a math major? Alas, I’m afraid I flunked. Still, I was consoled by being able to recall nine of the top 10 (of two dozen). But the No. 1 beauty, an equation known as Euler’s formula, bothered me—I’d seen it before, yet couldn’t remember going over it in school.

It’s probably overrated, I thought, a little defensively. Lacking arcane symbols or other bona fides of serious mathematical artistry, it features only numbers, and just five at that (as it’s usually written). True, three of them are designated by letters, showing they’re special. But the equation* itself looks hardly more scintillating than a confused first grader’s 2 + 1 = 0. See: e^iπ + 1 = 0.

I was certainly familiar with Leonhard Euler (pronounced “oiler”), the eighteenth-century mathematician it’s named after. He’s known as the Mozart of mathematics, and his fingerprints, so to speak, were all over the pages of my old math books. But that didn’t tell me much. And as the formula started playing through my head like a tune whose provenance maddeningly eluded me, I discovered via Google that a number of authorities on math have considered it not only beautiful, but also one of the most remarkable results in the history of mathematics. Among them was one of my heroes, Richard Feynman, a brilliant theoretical physicist who worked on the Manhattan Project, won a Nobel Prize, led the investigation of the 1986 Space Shuttle Challenger disaster, and, to top it off, radiated almost superhuman joie de vivre. Why did this simple-looking little formula light up his multi-gigawatt mind?

OK, I decided, it’s time for some investigative reporting. Although I’d moved on to science writing after college, I’d warded off total math-muscle atrophy by acting as my eye-rolling kids’ tragicomically enthused mathematics tutor, all the way through high school calculus in the case of my son. So I looked up the derivation of Euler’s formula—it’s straightforward if you know a little calculus—and reconnoitered its history and significance. And like many math lovers before me, I came away thinking, “wow,” or, more precisely, “WOW!”

For one thing, it effectively compresses about two millennia’s worth of big ideas in mathematics into a fantastically small package, among them the nature and uses of infinity (∞ is basically tucked away inside the formula), the weird ubiquity of the number π in math, the great utility of the misleadingly named imaginary numbers, and the wonderfulness of nothing, i.e., zero. What really grabbed me, though, was the fact that on his way to the formula, Euler uncovered a set of hidden connections among math concepts that many students go over in high school without ever realizing that they’re deeply linked in a way that could aptly be described as scary-cool. (That is, several levels of cool up from merely awesome.)

So I got the beauty thing. But I still wondered about the blank space in my memory where I should have had a beauty queen. Continuing the investigation, I pulled out my college calculus books, which I’d kept as trophies for all the hours I spent hunched over them. Euler’s formula wasn’t listed in their indexes. Paging through them, I finally found a single, fleeting mention of a general equation (also Euler’s) from which the most beautiful formula is derived as a special case. The closest thing to “e^iπ + 1 = 0” that I could find was an exercise whose answer happened to be a version of it.

So that’s it, I thought. I didn’t forget Euler’s formula. It’s just that, inexplicably, it got infinitesimal shrift during my student days. Come to think of it, the most beautiful equation didn’t come up in any of my son’s high school math courses either.

This last thought led to my recollecting how I knew all too much about those courses. It’s true, I inwardly sighed, as Quentin’s tutor I was more on top of his daily math assignments than he was. A budding artist, he regarded math classes as a boring waste of time. And by the time he left for college, I was seeing double when I looked at his math books. One of the two images I saw was the engaging one I knew as a math lover. But the other image was the one that I knew he saw. Novelist Nicholson Baker memorably described it in a 2013 piece in Harper’s about how mandatory high-school math classes often foster math hate. In particular, he wrote, Algebra II students “are forced, repeatedly, to stare at hairy, square-rooted, polynomialed horseradish clumps of mute symbology that irritate them, that stop them in their tracks, that they can’t understand. The homework is unrelenting, the algorithms get longer and trickier, the quizzes keep coming. Sooner or later, many of them hit the wall.”

You can probably see where I’m going with this. My next thought was: If only Quentin, as well as millions of other people out there who regard math as the supreme soporific, could experience the frisson I felt when reliving Euler’s great discovery. But let’s get real, I firmly told my inner soliloquist, who was already whispering, “Do a book!” in my ear. There’s no way that people who have forgotten most of their high school math could experience that epiphany. Preposterous. Forget it. The very idea could do incalculable harm to my reputation as a somewhat grounded thinker.

And then, of course, I sat down to write this book. Well, that’s a slight exaggeration. I mulled the idea over for about a year. Finally I remembered how American philosopher Oets Kolk “O.K.” Bouwsma had overcome his own hesitation to put together a book he’d had doubts about: “I have tossed a coin and it came down as I thought it would,” he explained. “It stood on its edge. And I knocked it down.”

So here we are. But before you delve into this book—or turn away from it—please finish the intro. I’ll be brief.

One reason I flattened the coin is that Euler’s formula offers a very rare combination of beauty, depth, surprise, and, most importantly for my purposes, understandability—few profound math results are as accessible as it is. (Although it does take some explaining—a short book’s worth, apparently.) And I knew that writing about it would give me license to roam across the history of mathematics as I went over the ideas that it so cunningly encapsulates.

Also, the formula isn’t just a math version of abstract art. Long after Euler’s era, scientists and engineers realized that the general equation mentioned above (the conceptual parent of e^iπ + 1 = 0) is immensely useful for mathematically modeling phenomena, such as the rhythmic flow of alternating current. Thus, Euler’s brilliant pure-math discovery is now effectively embedded in electrical devices all around us. I’d call that scary-cool, too, if I hadn’t already used up my weekly allotment of that epithet.

The formula’s timelessness also appealed to me. As electrical engineer Paul Nahin nicely put it in a book he wrote about the equation for people who’ve taken college math: “Unlike the physics or chemistry or engineering of today, which will almost surely appear archaic to technicians of the far future, Euler’s formula will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time.”

I hope this book is approximately equal to the sum of these pluses. But I feel obliged to mention a couple of things up front by way of truth in advertising. First, it isn’t intended to increase quantitative skills or impart a thorough grounding in the math that it covers—its sole mission is to bring home that great mathematics is as provocative, beautiful, and deep as great art or literature. Second, if you’re a long-time math lover, you’ll probably find most of it too elementary (with the possible exception of Appendix 1). I’ve tried to make the book’s math accessible to those who have forgotten, or perhaps repressed, most of the mathematics they learned after sixth grade. Thus, I assume that readers are acquainted with little more than the basics needed to cope in life: arithmetic, fractions, ratios, decimals, percentages—essentially what American kids are expected to know before being introduced to algebra in seventh or eighth grade.

Still, I have included a number of equations (along with lots of hand-holding as they rear their Medusa-like heads). You might conclude from this that I somehow missed physicist Steven Hawking’s famous admonition about the use of equations in nontechnical books: “Someone told me,” he remarked, “that each equation I included in the book [A Brief History of Time] would halve the sales. I therefore resolved not to have any equations at all.” (In the end he did include one: E = mc².) Actually, I’m so acutely aware of this comment that it’s virtually tattooed on my cerebral cortex. But I decided that giving an account of Euler’s formula without equations would be like describing van Gogh’s The Starry Night without showing a good-sized image of it. It would defeat my purpose, which is to enable readers to make an authentic personal approach to a high point in the history of mathematics, and, indeed, of human thought.

It has occurred to me, of course, that if the halving principle is correct this book is likely to attract less than one-millionth of a reader. (There are several dozen equations.) I’m happy to report, however, that more than a million times that many people are now known to have dipped into it (meaning you, dear whole number of a reader), which suggests that the principle’s originator may need remedial math. In any case, I hope that at least a few hungry-minded persons will read on and find themselves experiencing some unexpected moments of amazement and jubilation. What better reason to read a book? Or, for that matter, to write one?

* Euler’s equation is often called a formula or an identity. I refer to it as an equation or a formula, glossing over the fact that these terms aren’t synonymous in formal mathematics.

CHAPTER 1

God’s Equation

Leonhard Euler seemed as curious, cheerful, and acute as ever during the morning of September 18, 1783, the last day of his life. It was nearly fall in St. Petersburg, Russia, where the 76-year-old Swiss mathematician lived with his extended family and worked at the Russian Academy of Sciences. Despite having been nearly blind for more than a decade, he’d continued to produce math and science papers at an astounding rate—he’d actually become more productive after losing his sight. Euler did complex calculations in his head and dictated the results to assistants, who recorded them on two large writing slates he kept in his study.

That morning, as was his custom, he gave one of his grandsons a lesson in elementary science. Soon after, two colleagues arrived to discuss scientific matters, including the recently discovered planet Uranus and innovative hot-air balloons that had been launched in unpiloted experiments a few months earlier by two French brothers, Joseph-Michel and Jacques-Étienne Montgolfier—they would soon go down in history for achieving the first manned ascent.

After informing his visitors that he’d lost his remaining vestige of vision, Euler proceeded to work out a complex differential equation (an exercise in calculus) to model the rise of hot-air balloons and thereby determine how high they could go. He also made mental calculations related to Uranus’s orbit. After lunch, he said he felt faint and lay down for a nap.

A few hours later he rejoined his family and friends for four o’clock tea. Sitting on a couch, he jovially played with one of his grandsons, and then asked his wife for a second cup. Two minutes later, he suddenly dropped the pipe he was smoking, stood up, clasped both hands to his forehead, and exclaimed, in German, “I am dying.” They were his last words, and they represented one of his characteristic flashes of deep intuition—he was suffering a stroke that would prove fatal. Soon losing consciousness, he died that evening.

Later that day, Johann, Euler’s eldest son, came across the latest calculations on his father’s slates. Despite his shock and sadness (or perhaps because of them) he quickly set to work fleshing out the ones on hot-air balloons and published the results in a French journal as Euler’s first posthumous paper. Many followed—Euler left a huge collection of unpublished manuscripts filled with important findings that came out for decades after his death.

EULER’S LIFE SPANNED most of the Enlightenment, an explosive efflorescence of thought that followed the spring-like flowering of the Renaissance. During its high time in the eighteenth century, intellectuals gathered in coffeehouses and literary salons to thrash out world-changing ideas about science, individual liberty, religious tolerance, and free-market economics. America’s soul was forged during the Enlightenment—Jefferson was channeling its spirit when he penned the Declaration of Independence in 1776. That same year, Scottish philosopher Adam Smith published The Wealth of Nations, which spread the enormously influential idea that rational self-interest can foster economic prosperity. A few years later, England’s Mary Wollstonecraft wrote A Vindication of the Rights of Woman, one of the earliest feminist tracts.

The Enlightenment’s great minds included George Frideric Handel, Wolfgang Amadeus Mozart, Franz Joseph Haydn, Jonathan Swift, Alexander Pope, Samuel Johnson, Daniel Defoe, Voltaire, Thomas Paine, Benjamin Franklin, Montesquieu, David Hume, Immanuel Kant, Denis Diderot, William Herschel, Antoine Lavoisier, and Émilie du Châtelet, a mathematician and physicist who was the first woman to have a scientific paper published by the French Academy of Sciences. Many were part of an international republic of letters whose unofficial motto, famously enunciated by Kant, was Sapere aude! (Latin for “Dare to know!”). The era’s most influential writers, such as Swift, Paine, and Voltaire, attacked received wisdoms and the powers-that-be with unprecedented gusto and wit. A number of them regarded human reason, not divine revelation, as the ultimate arbiter of truth. That led many of the intellectuals of the time to embrace deism, which held that God set up the universe to run according to discoverable natural laws and then permanently retired from the scene; deists rejected miracles and other supernatural phenomena as superstitions.

Some, following England’s John Locke, believed that natural law applied to human society, and among its dictates were the rights to life, liberty, and property—a truly revolutionary idea. It was also the age of “enlightened despots,” such as Prussia’s King Frederick II and Russia’s Empress Catherine II, who shared the era’s belief in the primacy of reason, and who fostered education, religious tolerance, and property rights. Frederick and Catherine also took pride in their countries’ scientific academies, which vied to attract Europe’s most brilliant minds and served as leading research centers. Euler began his career at the Russian academy and then spent a quarter century at the Prussian academy before returning to the one in Russia for the rest of his life.

The leading thinkers of the time often disagreed, and the big questions they took up are still being debated. Euler, for example, was more religiously conservative than many of his peers, and he rejected the deists’ then-radical elevation of human reason over Christian revelation. But almost all of the era’s greats, including Euler, shared the Enlightenment’s defining inspiration: that the world is governed by natural laws whose regularities can be discovered by the methods of science. This idea was closely allied with the view that nature’s laws can be expressed in mathematical terms, a concept that seemed to be written in the very stars after Isaac Newton showed during the late 1600s that everything from the tides to the planets’ trajectories across the sky could be explained in terms of mathematically-formulated laws of motion and gravitation. The energetic pursuit of math-based explanations during the Enlightenment paved the way for the Industrial Revolution and the rise of modern technology. This pursuit also informed the period’s formal style of discourse, which sometimes had a distinctly mathematical flavor. Consider the Declaration of Independence’s famous second sentence: “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness”—it has the ring of a mathematical axiom from which the document’s subsequent assertions logically follow like theorems.

Such ornate, well-knit prose may seem quaint in the age of tweets and blogs. But it is difficult to overstate the importance of the passion for balance, order, and reason that informed Jefferson’s words and that Euler’s era bequeathed to posterity. Indeed, when we glance back through the corridors of history, the Enlightenment stands out like a brightly illuminated room whose glow is still helping us make our way forward in the dark. And as I write this in late 2016, it seems that today’s post-truth autocrats, religious fundamentalists, and science deniers have joined forces to pursue a single terrible end: to extinguish its guiding light and replace it with greed, lies, ignorance, and hate.

EULER WAS THE ENLIGHTENMENT’S greatest mathematician, as well as the era’s greatest physicist. He also made many important contributions to astronomy and engineering. Science historian Clifford Truesdell has estimated that Euler wrote a quarter of all the mathematical and scientific works published during the eighteenth century. During most of the Enlightenment he was, in effect, the leading carrier of the torch that had been lit during the Renaissance, and that Newton and others had raised high during the late seventeenth century.

Euler won the French science academy’s annual prize for innovative solutions to science, mathematics, and technology problems—the eighteenth century’s version of today’s Nobel Prize—no less than 12 times. He was history’s most prolific mathematical innovator; his collected works, which fill 80 thick volumes (so far), are math’s Mt. Everest. And he made major advances across virtually every subdiscipline of mathematics. As mathematician William Dunham has noted, you’d need a forklift to transport the roughly 25,000 pages of his Opera Omnia (Euler’s complete works, which the Swiss Academy of Sciences has been compiling for over a century).

Euler also made technology advances. In 1752, for instance, he originated then-futuristic designs for a paddle wheel and a rotating propeller for ships—devices that wouldn’t be practical until steam engines were developed in the nineteenth century to turn them. His ideas on how to construct achromatic lenses helped inspire an English inventor to put together one of the first such color-correcting lenses. He even proposed a design for a logic machine—a kind of early, mechanical computer—although it’s not clear whether the machine was constructed.

Although Euler isn’t considered a philosopher, his writings clearly influenced Kant’s metaphysics. His work on the mathematics of population growth helped inspire Darwin to come up with the theory of natural selection. He had a hand in solving the famous longitude problem—the devising of a practical way for determining a ship’s longitude at sea—and in 1765 the British government awarded him part of the prize money it had offered in 1714 to spur inventors to attack the problem. Today, the mathematics he pioneered has been applied in systems used for secure Internet communications, the analysis of social networks, electronic circuit design, and many other things. Even Hollywood has recognized his contributions—the 2016 movie Hidden Figures mentioned “Euler’s method,” an approximating technique one of the film’s stars used to calculate trajectories of 1960s spacecraft so they didn’t burn up when re-entering the atmosphere and then splashed down where Navy ships could quickly pick them up.

It’s clear that Euler had one of the most beautiful minds in history. And unlike many great innovators, he achieved international fame during his lifetime—perhaps only Voltaire, with his ebullient irreverence and memorably caustic wit, was more renowned in Europe than he was during the late 1700s. Today, however, while many people are familiar with the works of other Enlightenment geniuses—Handel’s music, Voltaire’s satires, Defoe’s novels, to name a few examples—Euler’s works are known to relatively few. That’s a shame. The intellectual gems he carved are as scintillating as any to be found in the history of thought. One, in particular, stands out as the epitome of math’s unsurpassed power to surprise and delight.

IT IS AS GALVANIZING, puzzling, and concise as a Zen koan—math’s one hand clapping:

e^iπ + 1 = 0.

Mathematics textbooks call it Euler’s formula. But some people feel that that name is too mundane for what is arguably math’s most magnetic truth, as well as one of its most startling, and so they call it God’s equation.*

Benjamin Peirce, considered America’s first world-class mathematician, once showed how to prove a variation of it during a class he taught at Harvard, where he was a professor from 1831 to 1880. Then, after contemplating what he’d written on the chalkboard for a few minutes, he turned to his students and observed that it “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth.”

Mathematician Keith Devlin, “The Math Guy” on National Public Radio, has been bowled over too. “Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence,” he commented.

Richard Feynman was briefer but no less enthusiastic: It’s “the most remarkable formula in math,” he stated in a notebook he compiled at age 14, along with a sketch of its proof. As mentioned in the introduction, math experts have ranked Euler’s formula as mathematics’ most beautiful equation. And in 2014 fifteen mathematicians were presented with 60 different equations, including Euler’s formula, while undergoing brain scans; the study showed that Euler’s formula had the greatest ability to activate brain areas associated with experiencing visual and musical beauty.

On its far left side are numbers of infinite complexity (two of them, e and π, are designated with letters because it would literally take forever to write them down with numerals), yet their endlessness collapses into a small, neat integer when they’re combined. That is, the equation shows that e^iπ—which seems the kind of numerical monstrosity that would turn a young math student to stone—is actually equal to a simple integer, −1.* Thanks to this startling fact, the equation’s five seemingly unrelated numbers (e, i, π, 1, and 0) fit neatly together in the formula like contiguous puzzle pieces. One might think that a cosmic carpenter had jig-sawed them one day and mischievously left them conjoined on Euler’s desk as a tantalizing hint of the unfathomable connectedness of things.

* The nickname “God’s equation” is meant to underscore its profundity, not to suggest that a deity handed it down from on high. However, it is true that mathematician Jules Henri Poincaré once called Euler the “god of mathematics.” And if God existed, my guess is that she would have Euler’s formula, carved on a stone tablet, sitting on her desk.

* To see this, it helps to remember that adding the same number to both sides of an equation (by “sides” I mean the stuff to the left and right of the equals sign) turns the two sides into other stuff that’s equal. If −1 is added to both sides of Euler’s formula, the left side becomes e^iπ + 1 + (−1), which equals e^iπ since 1 + (−1) equals 0, and adding 0 to a number leaves it just as it is. Meanwhile, the right side becomes 0 + (−1), which, since added 0s can be ignored, equals −1. Thus, this maneuver turns Euler’s formula into e^iπ = −1, which means that e^iπ is really just −1 in a fantastically good disguise.

CHAPTER 2

A Constant That’s All About Change

At first glance, the number e, known in mathematics as Euler’s number, doesn’t seem like much. It’s about 2.7, a quantity of such modest size that it invites contempt in our age of wretched excess and relentless hype. Puffy numbers like a bazillion clearly get a lot more ink in the media.* With e dollars in your pocket, you wouldn’t even have enough to buy a small latte at my local Starbucks:

But e is not to be trifled with. It’s one of math’s most versatile superheroes.

To begin with, it’s uniquely valuable for mathematically representing growth or shrinkage. That alone makes it a standout. In fact, e’s usefulness for dealing with problems related to the growth of savings via compound interest is what brought about its discovery in the 1600s.* Let’s revisit that moment in history with a modern twist.