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# Euler's identity

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Euler's identity, sometimes called Euler's equation, is a simple equation. It links several important numbers (mathematical constants) in mathematics in an unexpected way. Euler's identity is named after the Swiss mathematician Leonhard Euler, though it is not clear that he did invent it.[1]

Euler's identity is the equation $\displaystyle e^{i\pi} + 1 = 0$ .

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
• $\displaystyle \pi$ : 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.
$\displaystyle \pi \approx 3.14159$
• $\displaystyle e$ , Euler's Number. Euler's Number appears in calculus and is related to the area between a curve that follows $\displaystyle y = {1 \over x}$ and the line $\displaystyle y = 0$ .
$\displaystyle e \approx 2.71828$
• $\displaystyle i$ , which is an imaginary number. The number $\displaystyle i = \sqrt{-1}$ and has the property $\displaystyle i \times i = i^2 = -1$ .

## 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: It combines the fields of geometry and algebra, and yet does so using only 7 of the most common and important mathematical symbols.

## Mathematical proof using Euler's formula

Euler's Formula is the equation $\displaystyle e^{ix} = \cos(x) + i \sin(x)$ . Our variable $\displaystyle x$ can be any real number, but for this proof $\displaystyle x = \pi$ . Then $\displaystyle e^{i\pi} = \cos(\pi) + i \sin(\pi)$ . Since $\displaystyle \cos(\pi) = -1$ and $\displaystyle \sin(\pi) = 0$ , the equation can be changed to read $\displaystyle e^{i\pi} = -1$ , which gives the identity $\displaystyle e^{i\pi} + 1 = 0$ .

## References

1. Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
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