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Euler's identity

Euler's identity, a special case of Euler's formula, is the following:

<math>e^{i \pi} + 1 = 0</math>

The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, <math>i</math> is the imaginary unit (an imaginary number with the property i² = -1), and <math>\pi</math> is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

It was called "the most remarkable formula in mathematics" by Richard Feynman. Feynman found this formula remarkable because it links some very fundamental mathematical constants:

The formula also involves the fundamental arithmetical operations of addition, multiplication and exponentiation.

The formula is a special case of Euler's formula from complex analysis, which states that

<math>e^{ix} = \cos x + i \sin x</math>

for any real number <math>x</math>. If we set <math>x = \pi</math>, then

<math>e^{i \pi} = \cos \pi + i \sin \pi,</math>

and since cos(π) = -1 and sin(π) = 0, we get

<math>e^{i \pi} = -1</math>

and

<math>e^{i \pi} + 1 = 0.</math>

References

wikipedia.org dumped 2003-03-17 with terodump