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==Perceptions of the identity== ==Perceptions of the identity==
Euler's identity is remarkable for its ]. Three basic ] functions are present exactly once: ], ], and ]. As well, the identity links five fundamental ]s:
cut this shit yo
* The number 0.
* The number 1.
* The number π is ubiquitous in ], ], and ].
* The number ''e'' occurs widely in ].
* The number ''i'' generates the ]s, which contain the roots of all nonconstant polynomials and lead to deeper insights into many operators, such as ].


Equations with zero on one side are generally useful in mathematical analysis.
== '''''cut the crap''''' ==

After proving the identity in a lecture, ], a noted ] ] and ] 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."{{rf|1|Maor}}


==Notes== ==Notes==

Revision as of 15:39, 24 February 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

Euler's formula for a general angle.

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. In particular, if x = π {\displaystyle x=\pi \,\!} , then

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

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

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:

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

External links

Categories: