David Stipp’s short book, A Most Elegant Equation, aims to persuade the “math-averse” that “great mathematics is as provocative, beautiful, and deep as great art or literature.” His exemplar is Euler’s identity, which can be written as the gnomic formula eiπ + 1 = 0. Stipp offers to explain what it means, why it’s true, and why it is significant as science and as art. The discussion, he says, will take pains to assume no mathematical prerequisites beyond checkbook arithmetic, and he isn’t kidding. For example, every algebraic manipulation that crops up is accompanied by a verbal paraphrase (often lengthy). A 101-word footnote on page 15 is devoted to explaining why “x = −1” means the same thing as “x + 1 = 0.”
“Euler” is Leonhard Euler, the master mathematician of the 18th century and one of the greatest of all time—also the most prolific. Publication of his collected works, begun in 1911 and ongoing, will total more than 80 large volumes. He is by all accounts an appealing character—a pious family man who, according to one contemporary, could work happily with “a child on his knees, a cat on his back.” Euler was generous in his dealings with other scholars, a good teacher, and something of a polymath who, in addition to his native German, knew Latin, Russian, French, and English and published works on mathematics, science, philosophy, and music. He could recite the entire Aeneid from memory. Euler began to lose his sight at an early age but blindness seemed if anything to increase his productivity: He worked things out in his head and dictated the results.
The exotic ingredient in Euler’s identity is eiπ: π is what you think, the ratio of a circle’s circumference to its diameter; e and i need considerable explaining, as does the use of iπ as an exponent (“raising e to the power iπ”). Without rehearsing those lengthy explanations it’s possible to scan the terrain in which that intellectual adventure takes place.
To begin concretely, but not too helpfully, e is a number a bit greater than 2.7. Like π, it cannot be expressed as a decimal that stops or settles into a repetitive pattern, and it crops up everywhere in mathematics, physics, and engineering (among other places). To go further we must expand our minds to accept the idea of carrying out operations, such as addition, infinitely often. Here’s a very simple example: Imagine a stool that stands on a single post that’s one foot long. Chop off the post’s bottom half (leaving 1/2 foot); chop off half of what’s left (so you’ve now chopped off 1/2 + 1/4, leaving 1/4); chop off half of what’s left (you’ve now chopped off 1/2 + 1/4 + 1/8, leaving 1/8); carry on as long as you like. Any point on the post can be chopped off by carrying on long enough. If one could somehow finish performing all of the infinitely many chops the entire post would be consumed; the seat of the stool would lie on the floor. So it’s tempting to say that we can meaningfully add together all the infinitely many lengths that were chopped off and that the resulting sum must total the one-foot length that was consumed:
1/2 + 1/4 + 1/8 + 1/16 + . . . = 1
(The “. . .” means “You get the idea; go on like this forever.”) One might say that, if only to seem mysterious and clever, but why bother? Because deploying infinite operations and manipulating them by something like the ordinary rules of algebra is a powerful way to solve old problems and discover new truths—including truths, like Euler’s identity, that don’t explicitly refer to infinities. Euler displayed his virtuosity with these methods in Introduction to the Analysis of the Infinite, perhaps the most influential mathematics textbook since Euclid’s. Another century of work was needed to put them on sound logical footing and avoid lurking fallacies and errors. Meanwhile, Euler’s imagination (or chutzpah)—guided by a deep, if not quite infallible, intuition—expanded the boundaries of mathematics.
Coming to grips with i requires overcoming a regrettable piece of terminology too old to change: “imaginary number.” i is called “imaginary” because it is assumed, by fiat, to satisfy the equation i×i = −1, even though none of the numbers we’re used to—now to be called, by contrast, “real” numbers—can possibly fill that bill: The result of multiplying a negative number by itself, or a positive number by itself, is always positive, so can’t equal −1. Thus, i is neither positive nor negative, but is still somehow something. As early as the 16th century, procedures for solving equations gave rise to expressions that, if they meant anything at all, could only denote such “imaginary” entities. They were embarrassments but could not simply be shunned: Faith-based persistence, applying the usual algebraic rules (and replacing i×i, when convenient, with −1), sometimes caused the unwanted expressions to drop out, leaving the “real” answers originally sought. Far from avoiding imaginaries, Euler exploited them with glee, opening up whole new mathematical vistas. Subsequent work has developed a logical foundation for the entire domain of complex numbers, those—such as 2, 3i, e+πi—that result from applying the operations of arithmetic to real and imaginary numbers.
What about eiπ? Those who remember high school algebra will recall that e2 means e×e (multiply two copies of e), that e3 means e×e×e, etc. But this hardly helps make sense of eiπ. What could “iπ copies of e” possibly mean? Euler proceeds by first finding an infinite sum that gives a formula for computing ex. That is, it computes e2 if we replace the infinitely many xs in the formula by 2, e3 if we replace them by 3, etc. He then declares that the meaning of eiπ is the result of replacing them all with iπ. Which is obvious if you’re the sort who can pioneer “analysis of the infinite” and the theory of complex numbers—and can, along the way, extend infinite analysis to complex numbers. Stipp quotes the 20th-century mathematician Mark Kac: “An ordinary genius is a fellow you and I would be just as good as, if we were only many times better. . . . It is different with the magicians . . . the working of their minds is for all intents and purposes incomprehensible.” Euler was a magician.
It unsettles ordinary mortals to follow rules without an account of what the rules are about, to accept without proof their internal consistency, and to trust that results about the real numbers reached by calculations that detour through the complex domain are true. Stipp’s next-to-last chapter sketches a modern representation of complex numbers as points in a two-dimensional plane. The “real” numbers lie along one straight line in that plane; another, perpendicular to it, contains the purely “imaginary” numbers. Arithmetic operations have a simple geometric meaning, as does Euler’s identity. Stipp’s account of all this seems pitched just right, a few worked examples that give a satisfying sense of how everything hangs together.
The final chapter, called “The Meaning of It All,” asks what makes it beautiful. Stipp begins by noting qualities that mathematicians have attributed to beautiful results. From G. H. Hardy, for example, he gets this famous list: seriousness, generality, depth, unexpectedness, inevitability, and economy. Such reflections will help those who already sense beauty in mathematics to articulate their experience; they won’t persuade others that beauty is there to be found. But persuasion is not Stipp’s aim. His book is not a work of philosophy. What he offers amounts not to an argument but to an experience, especially “the feeling of exaltation that we get from an encounter with an example of our species outdoing itself.” He trusts that someone who manages to “get” a beautiful result will recognize a kinship between that experience and the rewards provided by works of other kinds of art. Stipp’s prose can be overripe—in Euler’s identity, he writes, e, i, and π “react together to carve out a wormhole that spirals through the infinite depths of number space to emerge smack dab in the heartland of integers”—but he gives his reader a good shot at getting hold of something beautiful.
