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Euler’s formula
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A diagram illustrating Euler's formula on the complex plane, showing a unit circle with axes labeled real and imaginary. A red angle theta extends from the origin to a point on the circle, labeled as e^(i?) = cos ? + i sin ?, with the horizontal and vertical components labeled cos ? and sin ?.
Euler's Identity: The Most Beautiful of All Equations
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A diagram illustrating Euler's formula on the complex plane, showing a unit circle with axes labeled real and imaginary. A red angle theta extends from the origin to a point on the circle, labeled as e^(i?) = cos ? + i sin ?, with the horizontal and vertical components labeled cos ? and sin ?.
Euler's formula Representation of eiθ = cos θ + isin θ on the unit circle, where the real part corresponds to cos⁡ θ and the imaginary part to sin⁡ θ.

Euler’s formula

mathematics
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Top Questions

What are the two main theorems associated with Euler’s formula?

How can Euler’s exponential formula be visualized?

How does Euler’s polyhedron formula relate to topology?

Euler’s formula, either of two important mathematical theorems discovered by the Swiss mathematician Leonhard Euler.

Euler's Identity: The Most Beautiful of All Equations
Euler's Identity: The Most Beautiful of All EquationsBrian Greene shows how Euler's identity is considered the most beautiful of all mathematical equations, combining disparate fundamental quantities into a single mathematical formula. This video is an episode in his Daily Equation series.
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The more well-known of the two Euler formulas is eiθ = cos θ + i sin θ, where e is the exponential function; i is the imaginary unit, the square root of -1; cos and sin are the trigonometric functions cosine and sine, respectively, and θ is the value of an angle. This formula is sometimes called Euler’s identity.

When θ = π, the Euler formula becomeseiπ = cos π + i sin π = -1.This last result can be rewritten as eiπ + 1 = 0,which relates five important mathematical constants—0, 1, e, i, and π—in a single equation.

The second formula is a numerical relationship between the faces F, edges E, and vertices V of a polyhedron (a solid shape with polygons as its faces):V - E + F = 2.This result is also called Euler’s polyhedral formula.

For example, a cube has 6 faces, 12 edges, and 8 vertices; that is,8 - 12 +6 = 2.The left-hand side of the formula is more generally known as the Euler characteristic in topology.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.