Euler's identity

[
300px|thumb|right|The_exponential function "e"^"z" can be defined as the limit of nowrap|(1 + "z"/"N")^"N", as "N" approaches infinity, and thus "e"^"iπ" is the limit of nowrap|(1 + "iπ/N")^"N". In this animation "N" takes various increasing values from 1 to 100. The computation of nowrap|(1 + "iπ/N")^"N" is displayed as the combined effect of "N" repeated multiplications in the (1 + "iπ/N")^"N". It can be seen that as "N" gets larger nowrap|(1 + "iπ/N")^"N" approaches a limit of −1.In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation

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

where:$e,!$ is Euler's number, the base of the natural logarithm,:$i,!$ is the imaginary unit, one of the two complex numbers whose square is negative one (the other is $-i,!$), and :$pi,!$ is pi, the ratio of the circumference of a circle to its diameter.

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

Nature of the identity

Euler's identity is considered by many to be remarkable for its mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
* The number 0.
* The number 1.
* The number "π", which is ubiquitous in trigonometry, geometry of Euclidean space, and mathematical analysis.
* The number "e", the base of natural logarithms, which also occurs widely in mathematical analysis (e ≈ 2.71828).
* 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.

Perceptions of the identity

A reader poll conducted by "Mathematical Intelligencer" named the identity as the most beautiful theorem in mathematics. [Nahin, 2006, p.2–3 (poll published in summer 1990 issue).] Another reader poll conducted by "Physics World" in 2004 named Euler's identity the "greatest equation ever", together with Maxwell's equations. [Crease, 2004.]

The book "Dr. Euler's Fabulous Formula" [2006] , by Paul Nahin (Professor Emeritus at the University of New Hampshire), is devoted to Euler's identity; it is 400 pages long. The book states that the identity sets "the gold standard for mathematical beauty." [Cited in Crease, 2007.]

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

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.]

Stanford mathematics professor Keith Devlin says, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." [Nahin, 2006, p.1.]

Derivation

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". (Note that the arguments to the trigonometric functions "sin" and "cos" are taken to be in radians.) In particular, if

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

then

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

Since

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

and

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

it follows that

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

which gives the identity

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

Generalization

Euler's identity is a special case of the more general identity that the "n"th roots of unity, for "n" > 1, add up to 0::$sum\_\{k=0\}^\{n-1\}\; e^\{2\; pi\; i\; k/n\}\; =\; 0\; .$Euler's identity is the case where "n" = 2.

Attribution

While Euler wrote about his formula relating "e" to "cos" and "sin" terms, there is no known record of Euler actually stating or deriving the simplified identity equation itself; moreover, the formula was likely known before Euler. [Sandifer.] (If so, then this would be an example of Stigler's law of eponymy.) Thus, the question of whether or not the identity should be attributed to Euler is unanswered.

ee also

* Exponential function
* Gelfond's constant

Notes

References

* Crease, Robert P., " [http://physicsweb.org/articles/world/17/10/2 The greatest equations ever] ", PhysicsWeb, October 2004.
* Crease, Robert P. " [http://physicsweb.org/articles/world/20/3/3/1 Equations as icons] ," PhysicsWeb, March 2007.
* 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 0-691-05854-7
* Nahin, Paul J., "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills" (Princeton University Press, 2006), ISBN 978-0691118222
* Reid, Constance, "From Zero to Infinity" (Mathematical Association of America, various editions).
* Sandifer, Ed, " [http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2040%20Greatest%20Hits.pdf Euler's Greatest Hits] ", MAA Online, February 2007.