Euler's Identity

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Euler's Identity is a simple equation that links several important numbers in mathematics in an unexpected way. Euler's identity is named after the Swiss mathematician Leonhard Euler.

The special numbers in Euler's Identity, are

  • zero, special because zero plus any number is still that same number
  • one, special because one times any number is still that same number
  • pi, special because it is one of the most common numbers in mathematics, and the distance around the outside of a circle divided by the distance across the circle (pi ≈ 3.14159).
  • e, also known as Euler's Number, (e ≈ 2.71828). Euler's number appears in calculus and is related to the area between a curve that follows y = 1/x and the line y=0.
  • the number "i", which is a "imaginary" number. The number "i" has the property that i x i = -1.


Euler's identity is the equation
eiπ + 1 = 0

where

e is the base of natural logarithms, also known as Euler's Number, (e ≈ 2.71828 18284 59045 23536)
i is the Imaginary Unit, one of two complex numbers, with a square equal to -1,
π is Pi, the distance around the outside of a circle divided by the distance across the circle.

Euler's Identity is sometimes called "Euler's Equation".

[edit] Reputation

A reader poll done by Physics World in 2004 called Euler's identity the "greatest equation ever", together with Maxwell's equations. Richard Feynman called Euler's identity "the most beautiful equation". The Identity is well-known for its mathematical beauty; that is, the equation is very pretty and pleasing to the eye. Some say this is because it is simple, and others because it uses many basic mathematical elements.

[edit] Mathematical Proof using Euler's Formula

Euler's Formula is the equation eix = cosx + isinx. X can be any real number, but for this proof x = π. Then eiπ = cosπ + isinπ. Since cosπ = − 1 and sinπ = 0, the equation can be changed to read eiπ = − 1, which gives the identity eiπ + 1 = 0.