Eulers identity

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For other meanings, see Euler function (disambiguation)

In complex analysis, Euler's identity is the equation

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where

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\,\!} is the imaginary unit, the complex number whose square is negative one, and
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi\,\! } is Pi, the ratio of the circumference of a circle to its diameter.

The identity is also sometimes expressed equivalently as

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

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

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for any real number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\,\!} . If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = \pi\,\!} , then

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and since, by definition

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and

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it follows that

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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:

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.

External links

ca:Identitat d'Euler de:Eulersche Identität es:Identidad de Euler fr:Identité d'Euler it:Identità di Eulero ko:오일러의 등식 nl:Formule van Euler he:זהות אוילר ja:オイラーの等式 pt:Identidade de Euler sl:Eulerjeva enačba th:เอกลักษณ์ของออยเลอร์ zh:歐拉恆等式