Here's the equation, in full:

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

Five symbols, three operations, and one statement of equality — and yet this line shows up on t-shirts, in physics textbooks, and in polls of mathematicians asked to name the most beautiful result in the field. The rest of this book explains why. This first stretch just introduces the five characters so you know who you're dealing with.

$e$ is a number, approximately $2.71828$, that shows up whenever something grows or shrinks at a rate proportional to its own size — compound interest, radioactive decay, population growth. It's called Euler's number, after the Swiss mathematician Leonhard Euler, though it was already lurking in earlier work on compound interest before he studied it closely. Like $\pi$, it's irrational (its decimal expansion never repeats) and in fact transcendental (it's not the root of any polynomial with rational coefficients). You'll see exactly where it comes from, series and all, in the next subsection.

$i$ is the imaginary unit, defined by the property $i^2=-1$. No real number squares to a negative number, so mathematicians invented a new kind of number to fill the gap — one you can't place on the ordinary number line, but can place on a second axis perpendicular to it, giving you the complex plane. Numbers of the form $a+bi$ live there, and $i$ itself sits one unit straight up from $0$.

$\pi$, approximately $3.14159$, is the ratio of a circle's circumference to its diameter. You've known it since grade school as a geometry fact. What might surprise you is that in Euler's identity, $\pi$ isn't measuring a circle's size at all — it's measuring an angle, specifically a half-turn, in radians. That shift from "circle ratio" to "angle of rotation" is the key that unlocks the whole identity, and it's the subject of subsection 4.