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:<math>i\,\!</math> is the ], one of the two ]s whose square is negative one (the other is <math>-i\,\!</math>), and :<math>i\,\!</math> is the ], one of the two ]s whose square is negative one (the other is <math>-i\,\!</math>), and
:<math>\pi\,\!</math> is ], the ] of the circumference of a circle to its diameter. :<math>\pi\,\!</math> is ], the ] of the circumference of a circle to its diameter.

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


==Derivation== ==Derivation==
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* The number &pi; is a fundamental constant of ], ], and ]. * The number &pi; is a fundamental constant of ], ], and ].
* The number ''e'' is fundamental in the study of ]s and occurs widely in ]. * The number ''e'' is fundamental in the study of ]s and occurs widely in ].
* The ] ''i'' (where ''i''<sup>2</sup> = &minus;1) is a unit in the ]s. (Introducing this unit yields all non-constant ] equations soluble in the ] of complex numbers: see ].) * The number ''i'', which together with the real numbers and arithmetic, generates the ]s (this leads to solutions for all nonconstant polynomials and deeper insights into many operators such as ]).


Furthermore, the most fundamental functions of ] are also present exactly once: ], ], and ]. As well, an equation with zero on one side is the most fundamental relation in mathematics. Furthermore, the most fundamental functions of ] are also present exactly once: ], ], and ]. As well, an equation with zero on one side is the most fundamental relation in mathematics.

Revision as of 11:41, 31 January 2006

For other meanings, see Euler function (disambiguation)

In mathematical analysis, Euler's identity is the equation

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

where

e {\displaystyle e\,\!} is Euler's number, the base of the natural logarithm,
i {\displaystyle i\,\!} is the imaginary unit, one of the two complex numbers whose square is negative one (the other is − i {\displaystyle -i\,\!} ), and
π {\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 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

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

for any real number x. If x = π {\displaystyle x=\pi \,\!} , then

e i π = cos ⁡ π + i sin ⁡ π {\displaystyle e^{i\pi }=\cos \pi +i\sin \pi \,\!}

and since cos ⁡ π = − 1 {\displaystyle \cos \pi =-1\,\!} and sin ⁡ π = 0 {\displaystyle \sin \pi =0\,\!} , it follows that

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

which gives the identity.

Perceptions of the identity

Many people find this identity remarkable for its mathematical beauty. The identity links what are arguably the most fundamental mathematical constants:

Furthermore, the most fundamental functions of arithmetic are also present exactly once: addition, multiplication, and exponentiation. As well, an equation with zero on one side is the most fundamental relation in mathematics.

Benjamin Peirce, the noted nineteenth century mathematician and Harvard professor, after proving the identity in a lecture, 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)

References

  • Maor, Eli, e: The Story of a number (Princeton University Press, 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.

See also

External links

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