Euler's Identity

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Euler's Identity, sometimes called Euler's Equation, 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.

Euler's identity is the equation e^iπ + 1 = 0.

The special numbers in Euler's Identity, are

• 0: zero, special because zero plus any number is still that same number
• 1: one, special because one times any number is still that same number
• π: 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.

  

• e, Euler's Number. Euler's Number appears in calculus and is related to the area between a curve that follows and the line y = 0.

  

• i, which is an imaginary number. The number and has the property .

[change] 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.

[change] Mathematical Proof using Euler's Formula

Euler's Formula is the equation e^ix = cos(x) + isin(x). Our variable x can be any real number, but for this proof x = π. Then e^iπ = cos(π) + isin(π). Since cos(π) = − 1 and sin(π) = 0, the equation can be changed to read e^iπ = − 1, which gives the identity e^iπ + 1 = 0.