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The complex exponential is a time machine for waves. e^(iθ) = cosθ + i sinθ means that multiplying by e^(iθ) rotates a number by angle θ, combining rotation and growth.

How it works

Euler's formula underlies the analysis of AC circuits, where impedances become complex. In quantum mechanics, wave functions are written as e^(i(kx-ωt)). Any musical chord can be decomposed into pure tones thanks to Fourier series, based on Euler's formula.

💡 The most beautiful equation in mathematics: e^(iπ) + 1 = 0 brings together five important numbers — e, i, π, 1, 0 — in one equality.
e^{i\theta} = \cos\theta + i\sin\theta
e is the base of the natural logarithm (approximately 2.71828), i is the imaginary unit (i² = −1), θ is a real number (angle in radians), cosθ and sinθ are trigonometric functions.
e^{i\pi} + 1 = 0
π is the number pi (approximately 3.14159), 0 is zero, 1 is one.
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Discovered by
Leonhard Euler
Related concepts
eigenvalueHilbert spaceQuantum Fieldsuperpositionwave functionwave-particle duality
Related laws
de Broglie formulaSchrödinger equationspectral theorem

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