For other meanings, see List of topics named after Leonhard Euler

In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation

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

where

$e\backslash ,\backslash !$ is Euler's number, the base of the natural logarithm,

$i\backslash ,\backslash !$ is the imaginary unit, one of the two complex numbers whose square is negative one (the other is $-i\backslash ,\backslash !$), and

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

Euler's identity is also sometimes called "Euler's equation".

Perceptions of the identity

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Euler's identity is remarkable for its mathematical beauty. Three basic arithmetic functions are present exactly once: addition, multiplication, and exponentiation. As well, the identity links five fundamental mathematical constants:

• The number 0.
• The number 1.
• The number π, which is ubiquitous in trigonometry, Euclidean geometry, and mathematical analysis.
• The number e, the base of natural logarithms, which occurs widely in mathematical analysis.
• The number i, imaginary unit of the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insight into many operators, such as integration.

Furthermore, in mathematical analysis, equations are commonly written with zero on one side.

A reader poll conducted by Physics World in 2004 named Euler's identity the "greatest equation ever", together with Maxwell's equations.

Constance Reid even claimed that Euler's identity was "the most famous formula in all mathematics".

Gauss is reported to have commented that if this formula was not immediately apparent to a student on being told it, the student would never be a first-class mathematician.Derbyshire p.210.

After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "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." Maor p.160 and Kasner & Newman p.103–104.

Derivation

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

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

for any real number x. In particular, if

$x\; =\; \backslash pi,\backslash ,\backslash !$

then

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

Since

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

and

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

it follows that

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

which gives the identity

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

Notes

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References

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• Crease, Robert P., "The greatest equations ever", PhysicsWeb, October 2004.
• Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (New York: Penguin, 2004).
• Kasner, E., and Newman, J., Mathematics and the Imagination (Bell and Sons, 1949).
• Maor, Eli, e: The Story of a number (Princeton University Press, 1998), ISBN 0691058547
• Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions).

See also

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• Exponential function
• Gelfond's constant

Complex analysis | Exponentials | Mathematical identities | Mathematical theorems

Identitat d'Euler | Eulersche Identität | Identidad de Euler | Identité d'Euler | זהות אוילר | Identità di Eulero | オイラーの等式 | 오일러의 등식 | Formule van Euler | Identidade de Euler | Eulerjeva enačba | Ојлеров идентитет | เอกลักษณ์ของออยเลอร์ | 歐拉恆等式