The Most Beautiful Equation: Euler's Identity and the Mystique of $e^{i\pi}+1=0$

Awe at a Single Line

On a Tuesday afternoon in 1943, a sixteen-year-old high school student named Richard Feynman sat in a classroom in Far Rockaway, New York, leafing through a library book he had not intended to borrow. The book presented a derivation, and at the climax of that derivation there was a single line:

$$e^{i\pi} + 1 = 0$$

Feynman later recalled — in talks, in interviews, and in his autobiography that spanned his entire career — that he had stared at the line, closed the book, walked outside into the autumn air, and felt something he could not easily describe. He called it, repeatedly, "a sense of awe." Decades later, when asked what he meant, he said: "The equation stood alone, it said what it said, and yet it seemed to mean more than it said. It connected. It connected things that I had thought had no reason to be connected."

This book begins with Feynman's reaction because the reaction is the puzzle. Why should any person, much less a person whose profession is to interrogate the universe with equations, look at a line of symbols and feel awe? What is it about $e^{i\pi} + 1 = 0$ that has prompted professional mathematicians to write essays about it, that has prompted *The Mathematical Intelligencer* to name it the most beautiful equation in history in a poll of its readers, and that has prompted popular writers to call it "God's equation" in headline after headline? What is mathematical beauty, that it can move us?

The answer to these questions is not a matter of taste. As we will see, the identity is mathematically deep in ways that have nothing to do with elegance. It is also historically meaningful in ways that have everything to do with elegance. By the end of this chapter, you will know why every term in the identity is what it is, why the operations binding the terms are the operations they are, how to derive the identity yourself from first principles, and why mathematicians throughout history have treated aesthetic reactions to identities like this one as a reliable, even indispensable, guide to mathematical truth. This last claim is the boldest of the four, and it is the thesis that runs through every page of this book.

The equation is short. Five symbols of constant, three operations of arithmetic, two signs. The five constants are $0$, $1$, $e$, $i$, and $\pi$. They were not invented together. They were not named by the same person. They were not even studied in the same century, in many cases. And yet here they are, riding the same line of typesetting, bound by addition, multiplication, and exponentiation. This is the mystery we must unpack. Let us begin with the constants themselves.

The Five Constants and Their Independent Origins

The identity is short, but each symbol in it carries the weight of human civilization. To understand why the identity is striking, we have to appreciate that the five constants it contains are usually studied as if they had nothing to do with each other.

Zero: The Last Number to Be Invented

Of all the constants in the identity, zero is the youngest as a named concept, but it carries perhaps the deepest philosophical freight. The ancient Babylonians had placeholders in their cuneiform number system as early as 400 BCE, and they used a slanted mark to keep track of them. The ancient Greeks, despite their extraordinary achievements in geometry, declined to admit zero as a number in its own right. To Aristotle, the void was an absurdity, and a "nothing number" violated the intuition that numbers count things. To Euclid, the multiplication of nothing gave "nothing," trivially, and so he saw no need to elevate the empty set to the dignity of arithmetic.

The status of zero changed, dramatically and irreversibly, in India between the fifth and ninth centuries. Around 628 CE, the astronomer and mathematician Brahmagupta wrote the *Brāhmasphuṭasiddhānta*, in which he laid out rules for arithmetic that treated zero as a number with its own behaviors. "When zero is added to a number or subtracted from a number, the number is unchanged," Brahmagupta wrote. He defined zero plus zero, zero minus zero, and zero times zero. He did not quite manage the rule that division by zero is undefined, but he came close. Crucially, he established zero as an integer on the same footing as one, two, and three. From India, the concept of zero traveled along the Silk Road to Baghdad, where al-Khwārizmī used it in the ninth century in his treatise on algebra. From Baghdad it traveled to Moorish Spain, and from there into Christian Europe, where Fibonacci's *Liber Abaci* of 1202 introduced it to a continent that had not seen it before.

We should not underestimate what this required. Calling zero a number is not a small notational convenience. It is the recognition that nothing can be a thing — can be operated on, can be added, can be transformed. Two thousand years after Brahmagupta, we still occasionally forget this and write sentences like "the answer is zero," as if zero were the absence of an answer rather than an answer in its own right. Zero is the only number on the number line that, when multiplied by any other number, gives itself rather than the other number. It is the absorbing element of multiplication, and we will see in later chapters that this property — being "absorbing" — turns out to be significant in unexpected places.

One: The Multiplicative Identity

One is so elementary that we rarely pause over it, but on closer look it is also mysterious. It is the multiplicative identity: any number multiplied by one is unchanged. There is exactly one such number among the integers, exactly one among the rationals, exactly one among the reals, exactly one among the complex numbers. This uniqueness is not a theorem we prove; it is a definition. We declare that the number we call one has the property that $1 \cdot n = n$, and any other number with this property is, by definition, the same number. The element "one" is fundamental to the structure of any multiplicative system — it is the pivot point that makes multiplication invertible in a sense that we will make precise later.

One's appearance in the identity is structural: it is what shifts the value $-1$ to the value $0$. Without it, the identity would read $e^{i\pi} = -1$, which is already a striking relation, but it would lack the cancellation punch that ties the equation to the empty set, to Brahmagupta's invention, to nothingness. The "one" is a way of saying: "Take the result, subtract it from itself, and the answer is nothing — find zero."

e: The Base of Natural Growth

The constant $e \approx 2.71828$ entered mathematics through the problem of compound interest. In 1683, Jacob Bernoulli posed a question: if you lend a sum of money at 100% annual interest and compound continuously, halving the compounding interval each time, what is the limiting value of the principal at the end of the year?

To compute the answer, suppose you compound once a year. Your multiplier at the end of the year is $1 + 1 = 2$. Compound twice a year instead, with the interest rate halved each period: your end-of-year multiplier is $\left(1 + \frac{1}{2}\right)^2 = 2.25$. Compound four times: $\left(1 + \frac{1}{4}\right)^4 \approx 2.4414$. Compound $n$ times in a year: the multiplier is $\left(1 + \frac{1}{n}\right)^n$.

For $n = 100$, this gives $2.7048$. For $n = 1000$, $2.7169$. For $n = 10000$, $2.7181$. The numbers are increasing and converging. The limit is:

$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.71828$$

Several important things are happening in this limit. The first is that compounding more often yields more money, which makes intuitive sense — every additional compounding period adds a small amount of "interest on interest." The second is that the limit exists and converges to a single number, $e$. The third, which is the profound part, is that $e$ shows up everywhere else in mathematics once you start looking. It app