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Formal Sciences
Mathematics
1748
Intermediate

Euler's Identity

eiπ+1=0e^{i\pi} + 1 = 0

Connects five fundamental constants—e, i, π, 1, and 0—in one breathtaking equation.

By Leonhard Euler

Formal Sciences
Euler's Identity
1748 · Leonhard Euler
Human Reviewed
84%

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Story PortalWho discovered this?Discover how Leonhard Euler and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — signal processing.Math PortalWhat does it derive from?Trace the mathematical lineage from fourier transform.Future PortalWhere is it going?Quantum computing
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Why it matters: Unified exponential, trigonometric, and complex analysis—foundation of modern engineering and physics.

Discoverers: Leonhard Euler (1748)

What does it mean?

Connects five fundamental constants—e, i, π, 1, and 0—in one breathtaking equation.

Why should I care?

Unified exponential, trigonometric, and complex analysis—foundation of modern engineering and physics.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
eeEuler's number—Base of natural logarithm ≈ 2.718
iiImaginary unit—√(-1)
ππPi—Ratio of circumference to diameter

Worked Example

e^(iπ) = -1, so e^(iπ) + 1 = 0 exactly.

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Pictures & video

Portrait of Leonhard Euler by Jakob Emanuel Handmann, 1753
Leonhard Euler, who introduced e^{iθ} = cos θ + i sin θ.Jakob Emanuel Handmann / Wikimedia Commons · Public domain

e^(iπ) in 3.14 minutes, using dynamics

3Blue1Brown

Euler's formula with introductory group theory

3Blue1Brown

Sources & further reading

Euler's identity — WikipediaEuler Formula — Wolfram MathWorldSearch WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Euler's Identity

eiπ+1=0e^{i\pi} + 1 = 0

Real-world impact

Quantum technology

Wave mechanics enables next-generation devices.

Photo: Unsplash — quantum hardware

Connects five fundamental constants—e, i, π, 1, and 0—in one breathtaking equation.

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