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: eiπ + 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 “eiπ + 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 eiπ + 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.