Euler's identity — the most beautiful formula

The most beautiful formula in mathematics

Euler’s identity $e^{i\pi} + 1 = 0$ is widely regarded as the most beautiful formula in mathematics — in one short equation it ties together the five most important constants: $e$ (the base of the natural logarithm), $i$ (the imaginary unit), $\pi$ (the circle constant) as well as $1$ and $0$ (the basic numbers), plus three fundamental operations: exponentiation, multiplication and addition.

It follows from Euler’s formula:

This formula (published by Leonhard Euler in “Introductio in analysin infinitorum”, 1748) links the exponential of an imaginary number with the trigonometric functions. Geometrically: $e^{i\varphi}$ is a point on the unit circle of the complex plane at angle $\varphi$, with coordinates $(\cos\varphi,\ \sin\varphi)$.

Where the identity comes from:

For $\varphi = \pi$ (a half turn, 180°) we land exactly on the point $-1$ on the real axis:

The invariant — the point is always on the circle:

— for every $\varphi$ the point $e^{i\varphi}$ lies at distance 1 from the center. The slider rotates the vector around the circle, but its length never changes.

Why this matters:

Euler’s formula is one of the pillars of mathematics and physics — it links analysis (the function $e^x$), complex numbers and trigonometry, and is indispensable for describing wave phenomena, alternating current, quantum mechanics and signal processing. It reveals a deep, unexpected unity among seemingly distant areas of mathematics.

Remember: $e^{i\varphi} = \cos\varphi + i\sin\varphi$ places a point on the unit circle; for $\varphi = \pi$ gives $-1$, hence $e^{i\pi} + 1 = 0$. (Extension topic — complex numbers, $e$ and $\pi$.)