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Euler's identity
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Euler's identity

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

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

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

for any real number . If we set , then

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

and therefore

Table of contents
1 Perceptions of the identity
2 References
3 External link

Perceptions of the identity

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: Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation. All the fundamental assumptions of complex analysis are present, and the integers 0 and 1 are related to the field of complex numbers.

In addition, the result is remarkable to most students learning it for the first time because it is so highly counterintuitive. Consider that


The simple addition of i where i2 = −1 changes the result dramatically.


External link