Euler formula

[ˈɔɪlər ˈfɔrmjələ]
nounpl: Euler formulas
fórmula de Euler
1. A mathematical formula discovered by Leonhard Euler that establishes a fundamental relationship between trigonometric functions and complex exponentials, expressed as e^(ix) = cos(x) + i·sin(x)
Euler formula is one of the most elegant equations in mathematics, connecting exponential and trigonometric functions.
A fórmula de Euler é uma das equações mais elegantes da matemática, conectando funções exponenciais e trigonométricas.
2. In graph theory, the formula V - E + F = 2 relating the number of vertices, edges, and faces in a planar graph
Using Euler formula, we can verify that a cube with 8 vertices, 12 edges, and 6 faces satisfies the relationship.
Usando a fórmula de Euler, podemos verificar que um cubo com 8 vértices, 12 arestas e 6 faces satisfaz a relação.
3. In topology, a formula calculating the Euler characteristic of a surface
The Euler formula for polyhedra helps determine the topological properties of three-dimensional shapes.
A fórmula de Euler para poliedros ajuda a determinar as propriedades topológicas de formas tridimensionais.
Euler formula is a cornerstone concept in advanced mathematics education in both Brazil and the United States, typically introduced in university-level courses in complex analysis, discrete mathematics, or topology. Named after Swiss mathematician Leonhard Euler (1707-1783), it is considered one of the most important discoveries in mathematics and is widely used in physics, engineering, and computer science. The formula e^(iπ) + 1 = 0, derived from Euler formula, is often cited as the most beautiful equation in mathematics.
Synonyms / Sinônimos
Euler's formulaEuler's identity (when x = π)

Regional Variations

General Brazilian
fórmula de Euler
Standard academic terminology used in mathematics education and research
Portugal
fórmula de Euler
Same as Brazilian Portuguese, used in academic mathematics contexts
São Paulo
fórmula de Euler
Standard usage in universities and mathematical institutions

Related Words

complex exponentialtrigonometric functiongraph theoryEuler characteristicLeonhard Eulerpolyhedraplanar graph
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