eíπ + 1 = 0. Often called the most beautiful equation in mathematics, Euler's identity links five of the most important constants — e, i, π, 1, and 0 — together with three fundamental operations (addition, multiplication, exponentiation) in a single, startlingly compact statement.
This single line brings together five of the most important numbers in all of mathematics:
The identity is a special case of a more general formula, Euler's formula: eⁱˣ = cos(x) + i·sin(x). Setting x = π gives eⁱπ = cos(π) + i·sin(π) = −1 + 0 = −1, which rearranges to eⁱπ + 1 = 0.
Named after Leonhard Euler, the most prolific mathematician in history, who worked extensively with the general formula in the 1740s, though it's uncertain whether Euler himself ever wrote down this specific special case in exactly this form — the compact identity as usually presented may be a later distillation of his broader results.
This formula connects exponential functions to trigonometric functions via complex numbers — one of the deepest and most useful bridges in mathematics.
The exponential, sine, and cosine functions all have infinite Taylor series expansions:
Substituting x = i·θ into the exponential series and using i² = −1, i³ = −i, i⁴ = 1 (and this pattern repeating), the series splits neatly into the cosine series (the real terms) plus i times the sine series (the imaginary terms) — yielding eⁱθ = cos(θ) + i·sin(θ) directly.
eⁱˣ traces out the unit circle in the complex plane as x varies — at angle x (measured in radians), the point eⁱˣ sits at coordinates (cos x, sin x). Setting x = π corresponds to going exactly halfway around the circle, landing at the point (−1, 0) — which is exactly −1 in complex number terms, giving eⁱπ = −1.
Roger Cotes (1714) discovered a closely related logarithmic identity (i·x = ln(cos x + i·sin x)) but did not develop it further or express it in exponential form. Euler independently derived the exponential formula around 1740 and published it in his 1748 work Introductio in analysin infinitorum, which also established much of the modern notation and framework for analysis using infinite series.
The formal justification for extending eˣ to complex arguments relies on analytic continuation: the complex exponential function is defined so that it agrees with the real exponential on the real axis and remains complex-differentiable (holomorphic) everywhere. This uniqueness of analytic continuation (a consequence of the identity theorem in complex analysis) guarantees there is only one sensible way to extend eˣ to complex numbers, and Euler's formula is what that extension yields.
The map θ ↦ eⁱθ is a group homomorphism from (ℝ, +) to the unit circle group (complex numbers of modulus 1, under multiplication). This map is 2π-periodic, meaning it identifies ℝ with the circle group ℝ/2πℤ — this periodicity is precisely why trigonometric functions are periodic, and it lies at the heart of Fourier analysis, where periodic functions are decomposed into sums of terms like eⁱⁿˣ.
The identity is sometimes cited as evidence for a kind of "unity" in mathematics — evidence that seemingly disconnected branches (algebra via i, geometry via π, analysis via e) are secretly deeply interconnected. This aesthetic argument has been used by mathematicians including Benjamin Peirce, who reportedly told his students that the formula "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth" (Crease, 2004) — though the precise wording and context of this widely repeated quote have some historical uncertainty.
In informal polls of mathematicians (Mathematical Intelligencer, 1988; Physics World, 2004), Euler's identity has repeatedly been voted the most beautiful theorem or equation in mathematics, ahead of results like the Pythagorean theorem and the Fundamental Theorem of Calculus — reflecting a broadly shared aesthetic sense among mathematicians that maximal simplicity combined with maximal depth constitutes mathematical beauty.