Euler's formula

Euler's formula
1. Math. the theorem that ei x = cos(x) + i sin(x).
2. Mech. a formula for determining the maximum load that can be applied to a given column without causing it to buckle.
[1945-50; named after L. EULER]

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Either of two important mathematical theorems of Leonhard Euler.

The first is a topological invariance (see topology) relating the number of faces, vertices, and edges of any polyhedron. It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges, and satisfies this formula. The second formula, used in trigonometry, says eix = cos x + isin x where e is the base of the natural logarithm and i is the square root of -1 (see irrational number). When x is equal to π or 2π, the formula yields two elegant expressions relating π, e, and i: eiπ = -1 and e2iπ = 1.

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Universalium. 2010.

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