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For other meanings, see Euler function (disambiguation)

In mathematical analysis, Euler's identity is the equation

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

where

${\displaystyle e\,\!}$ is Euler's number, the base of the natural logarithm,

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

${\displaystyle \pi \,\!}$ is Pi, the ratio of the circumference of a circle to its diameter.

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

Derivation

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

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

for any real number x. In particular, if ${\displaystyle x=\pi \,\!}$, then

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

Since

${\displaystyle \cos \pi =-1\,\!}$

and

${\displaystyle \sin \pi =0\,\!}$,

it follows that

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

which gives the identity.

Perceptions of the identity

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 is ubiquitous in trigonometry, Euclidean geometry, and mathematical analysis.
• The number e occurs widely in mathematical analysis.
• The number i generates the complex numbers, which contain the roots of all nonconstant polynomials and lead to deeper insights into many operators, such as integration.

Equations with zero on one side are generally useful in mathematical analysis.

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."(refactored from Maor)

Notes

Template:Ent Maor p. 160 and Kasner and Newman p.103

References

• E. Kasner and J. Newman, Mathematics and the imagination (Bell and Sons, 1949) pp. 103–104
• Maor, Eli, e: The Story of a number (Princeton University Press, 1998), ISBN 0691058547

See also

• Exponential function

External links

• Euler-like equations (Equations that are similar to Euler's Identity)

• Complex analysis
• Exponentials
• Mathematical identities
• Mathematical theorems