If, like Euler, you were given to plumbing depths that no one even realized were there before you revealed them, where better to look for hidden passages to the altogether elsewhere than e and π, which harbor a form of infinity and, even more intriguing, were cropping up in mathematics with striking regularity during the eighteenth century? The number i must have exerted the same pull on a mind like Euler’s. Indeed, eiπ seems to me exactly like the kind of bizarre, but interesting, expression that Euler would have mused about as he expanded the universe of mathematical ideas—it’s positively Euler-esque.
But what about the other two numbers in eiπ + 1 = 0? At first glance, 1 and 0 seem to lack the endless charisma of the equation’s other three numbers. But, again, looks can be deceiving in mathematics—these two are also Very Important Numbers. In fact, they’re arguably the only numbers that outrank e, i, and π in the VIN pantheon.
One, in a nutshell, is just that—the first quantity that came on the scene when people starting counting things. It’s the mother of all the positive integers: you can generate them from one with the help of addition. It also has a marvelously light touch—one is the one and only number that when multiplied times other numbers leaves them just the way they were. When author Alex Bellos surveyed people’s favorite numbers and adjectives associated with them, he was informed that one is independent, strong, honest, brave, straightforward, pioneering, and lonely.
Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line—at least as the line is usually drawn—making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one. (The trick for doing that is to use the binary number system, in which numbers are expressed in terms of 1s and 0s.)
However, in contrast to one, which is singularly straightforward, zero is secretly peculiar. If you pierce the obscuring haze of familiarity around it, you’ll see that it is a quantitative entity that, curiously, is really the absence of quantity. It took people a long time to get their minds around that. In fact, zero was just a big nothing, conceptually speaking, until Indian mathematicians accepted it as a legitimate number somewhere between the fifth and ninth centuries—thousands of years after numbers greater than zero were as common as dirt.*
So here’s the main reason that Euler’s formula is flabbergasting: the top five celebrity numbers of all time appear together in it with no other numbers. (In addition, it includes three primordial peers from arithmetic: +, =, and exponentiation.) This conjunction of important numbers, which sprang up in different contexts in math and thus would seem to be completely unrelated, is staggering, and it accounts for much of the hullabaloo about the equation.
Here’s an analogy: Say that future astronomers identify scores of distant solar systems with planets that are almost exactly like Earth (right down to the level of oxygen in the atmosphere), and that in every such case the Earth-lookalike turns out to be the third planet out from its local star, and, furthermore, that the five planets nearest to the lookalike are nearly identical in all respects to Mercury, Venus, Mars, Jupiter, and Saturn—the five planets closest to Earth. This would be nothing short of astounding, and it would suggest the existence of a completely unanticipated, deep regularity in the structure of the universe. The tight-knit pattern of seemingly unrelated, supremely important numbers in Euler’s formula is similarly provocative.†
But that’s not all that’s surprising about it. Try another little thought experiment: Imagine that you’d never heard of Euler’s formula but were familiar with the basics sketched above on e, i, and π. Now be honest—wouldn’t you have expected eiπ to be (a) gibberish along the lines of “elephant ink pie,” or, if it were mathematically meaningful, to be (b) an infinitely complicated irrational imaginary number? Indeed, eiπ is a transcendental number raised to an imaginary transcendental power.* And if (b) were the case, surely eiπ would not compute no matter how much computer power were available to try to pin down its value.
As you know, neither (a) nor (b) is true, because eiπ = −1. (I suspect the fact that both (a) and (b) are provably false is the reason that Benjamin Peirce, the nineteenth-century mathematician, found Euler’s formula (or a closely related formula) “absolutely paradoxical.”) In other words, when the three enigmatic numbers are combined in this form, eiπ, they 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. It’s as if greenish-pink androids rocketing toward Alpha Centauri in 2370 had hit a space-time anomaly and suddenly found themselves sitting in a burger joint in Topeka, Kansas, in 1956. Elvis, of course, was playing on the jukebox.
* A version of zero had been used as a placeholder in number systems (a kind of spacer, that is, which is a role that zero plays in our decimal number system—for instance, to distinguish between 606 and 66) as far back as the ancient Babylonians. But it was later Indian mathematicians who are credited with making the conceptual leap of accepting nothing as something, number-wise. The fact that their Hindu religion included the concept of nothingness—the void—may account for the fact that this major conceptual leap took place in India.
† I feel obliged to mention here that some people feel that the equation isn’t all that interesting. They have their reasons, which I disagree with and will address in the last chapter.
* If you were given to thinking of numbers as having human-like qualities, you might picture eiπ as a guru into transcendental meditation who’d achieved infinite enlightenment. But there’s a problem with that—Euler’s formula shows that eiπ can never free itself from worldly concerns. Recall, eiπ is really −1 in disguise, and −1 is just a mathism for owing a dollar to your friend, Steve. One hand clapping.