# Eulers identity

*For other meanings, see Euler function (disambiguation)*

In complex analysis, **Euler's identity** is the equation

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

where

**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 e\,\!}**is the base of the natural logarithm,**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

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

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

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

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

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

and since, by definition

**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 \cos \pi = -1\,\!}**

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 \sin \pi = 0\,\!}**

it follows that

**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 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*i*^{2}= −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.

## External links

*Proof of Euler's Identity*by Julius O. Smith III*Proof of Euler's Identity for a Layman*by Ian Henderson*Proof of Euler's Relation*by Craig Lewis

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