Euler’s Identity (1748): From Rotation to the Most Beautiful Equation in Mathematics

Why does the equation e^(iπ) + 1 = 0 connect the five most important constants in mathematics? In this visual explanation, we build Euler’s identity step by step from the geometry of the complex plane. We start from the meaning of the imaginary unit: i² = -1 Then we move to complex numbers as points on a plane: z = a + ib From there, we introduce polar form: z = ρ(cos θ + i sin θ) and show how the real and imaginary parts are just projections: a = ρ cos θ b = ρ sin θ The key idea is rotation. A tiny change in angle, dθ, produces a tiny tangent change, dz. On the unit circle, this leads to the differential relation: dz/dθ = iz Solving this equation gives: z = e^(iθ) But the same point on the unit circle is also: z = cos θ + i sin θ So we obtain Euler’s formula: e^(iθ) = cos θ + i sin θ Finally, when θ = π, the rotating point reaches -1: e^(iπ) = -1 and therefore: e^(iπ) + 1 = 0 This is Euler’s identity: a single equation connecting exponentials, rotation, imaginary numbers, geometry, zero, and one. Euler’s formula was published in 1748 in Introductio in analysin infinitorum. References Leonhard Euler, Introductio in analysin infinitorum, 1748 https://en.wikipedia.org/wiki/Introdu... Euler’s formula https://en.wikipedia.org/wiki/Euler%2... Portrait of Leonhard Euler by Jakob Emanuel Handmann, 1753 https://commons.wikimedia.org/wiki/Fi...