# Euler's identity

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The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus e is the limit of (1 + iπ/N)N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)N. It can be seen that as N gets larger (1 + iπ/N)N approaches a limit of −1.
 Part of a series of articles on The mathematical constant, e Natural logarithm Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay People John Napier  · Leonhard Euler

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:

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.[1] Another reader poll conducted by Physics World in 2004 named Euler's identity the "greatest equation ever", together with Maxwell's equations.[2]

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."[3]

Constance Reid claimed that Euler's identity was "the most famous formula in all mathematics."[4]

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.[5]

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." [6]

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."[7]

## Derivation

Euler's formula for a general angle.

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,

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

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 nth 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.[8] (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.

## Notes

1. ^ Nahin, 2006, p.2–3 (poll published in summer 1990 issue).
2. ^ Crease, 2004.
3. ^ Cited in Crease, 2007.
4. ^ Reid.
5. ^ Derbyshire p.210.
6. ^ Maor p.160 and Kasner & Newman p.103–104.
7. ^ Nahin, 2006, p.1.
8. ^ Sandifer.

## References

• Crease, Robert P., "The greatest equations ever", PhysicsWeb, October 2004.
• Crease, Robert P. "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, "Euler's Greatest Hits", MAA Online, February 2007.