Eulers identity

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For other meanings, see Euler function (disambiguation)

In complex analysis, Euler's identity is the equation

$e^{i \pi} = -1 \,\!$ ,

where

$e\,\!$ is the base of the natural logarithm,

$i\,\!$ is the imaginary unit, the complex number whose square is negative one, and

$\pi\,\!$ is Pi, the ratio of the circumference of a circle to its diameter.

The identity is also sometimes expressed equivalently as

$e^{i \pi} + 1 = 0 \,\!$

in order to make explicit the relationship between these five fundamental mathematical constants.

1 Derivation
2 Perceptions of the identity
3 References
4 Notes
5 External links

Derivation

The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. The identity is a special case of Euler's formula from complex analysis, which states that

$e^{ix} = \cos x + i \sin x \,\!$

for any real number $x\,\!$ . If $x = \pi\,\!$ , then

$e^{i \pi} = \cos \pi + i \sin \pi \,\!$

and since, by definition

$\cos \pi = -1\,\!$

and

$\sin \pi = 0\,\!$

it follows that

$e^{i \pi} = -1 \,\!$

Perceptions of the identity

Benjamin Peirce, the noted 19th century mathematician and Harvard professor, after proving the identity in a lecture, said, "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth."¹

Richard Feynman called Euler's formula (from which the identity is derived) "the most remarkable formula in mathematics".² Feynman, as well as many others, found this formula remarkable because it links some very fundamental mathematical constants:

The number 0, the identity element for addition (for all a, a + 0 = 0 + a = a). See Group (mathematics).
The number 1, the identity element for multiplication (for all a, a × 1 = 1 × a = a).
The number π is a fundamental constant of trigonometry, Euclidean geometry, and mathematical analysis.
The number e is a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation dy / dx = y with initial condition y(0) = 1 is y = e^x).
The imaginary unit i (where i² = −1) is a unit in the complex numbers. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers (see fundamental theorem of algebra).

Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation.

References

Feynman, Richard P., The Feynman Lectures on Physics, vol. I Addison-Wesley (1977), ISBN 0201020106, ISBN 02010211161
Maor, Eli, e: The Story of a number, Princeton University Press (May 4, 1998), ISBN 0691058547

Notes

Template:Ent Maor, p. 160. Maor cites Edward Kasner and James Newman's, Mathematics and the Imagination, New York: Simon and Schuster (1940), pp. 103–104, as the source for this quote. Template:Ent Feynman p. 22-10.