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Euler's identity

From Wikipedia, the free encyclopedia
Mathematical equation linking e, i and π
For other uses, seeList of topics named after Leonhard Euler § Identities.
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Inmathematics,Euler's identity[note 1] (also known asEuler's equation) is theequalityeiπ+1=0{\displaystyle e^{i\pi }+1=0}where

e{\displaystyle e} isEuler's number, the base ofnatural logarithms,
i{\displaystyle i} is theimaginary unit, which by definition satisfiesi2=−1{\displaystyle i^{2}=-1}, and
π{\displaystyle \pi } ispi, the ratio of thecircumference of a circle to itsdiameter.

Euler's identity is named after the Swiss mathematicianLeonhard Euler. It is a special case ofEuler's formulaeix=cos⁡x+isin⁡x{\displaystyle e^{ix}=\cos x+i\sin x} when evaluated forx=π{\displaystyle x=\pi }. Euler's identity is considered an exemplar ofmathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used ina proof[3][4] thatπ istranscendental, which implies the impossibility ofsquaring the circle.

Mathematical beauty

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Euler's identity is often cited as an example of deepmathematical beauty.[5] Three of the basicarithmetic operations occur exactly once each:addition,multiplication, andexponentiation. The identity also links five fundamentalmathematical constants:[6]

The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professorKeith Devlin has said, "like a Shakespeareansonnet 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]Paul Nahin, a professor emeritus at theUniversity of New Hampshire who wrote a book dedicated toEuler's formula and its applications inFourier analysis, said Euler's identity is "of exquisite beauty".[8]

Mathematics writerConstance Reid has said that Euler's identity is "the most famous formula in all mathematics".[9]Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor atHarvard University, after proving Euler's identity during a lecture, said that 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".[10]

A 1990 poll of readers byThe Mathematical Intelligencer named Euler's identity the "most beautiful theorem in mathematics".[11] In a 2004 poll of readers byPhysics World, Euler's identity tied withMaxwell's equations (ofelectromagnetism) as the "greatest equation ever".[12]

At least three books inpopular mathematics have been published about Euler's identity:

  • Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, byPaul Nahin (2011)[13]
  • A Most Elegant Equation: Euler's formula and the beauty of mathematics, by David Stipp (2017)[14]
  • Euler's Pioneering Equation: The most beautiful theorem in mathematics, byRobin Wilson (2018).[15]

Explanations

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Imaginary exponents

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Main article:Euler's formula
See also:Complex exponents with a positive real base
In this animationN takes various increasing values from 1 to 100. The computation of(1 +⁠iπ/N⁠)N is displayed as the combined effect ofN repeated multiplications in thecomplex plane, with the final point being the actual value of(1 +⁠iπ/N⁠)N. It can be seen that asN gets larger(1 +⁠iπ/N⁠)N approaches a limit of −1.

Euler's identity asserts thateiπ{\displaystyle e^{i\pi }} is equal to −1. The expressioneiπ{\displaystyle e^{i\pi }} is a special case of the expressionez{\displaystyle e^{z}}, wherez is anycomplex number. In general,ez{\displaystyle e^{z}} is defined for complexz by extending one of thedefinitions of the exponential function from real exponents to complex exponents. For example, one common definition is:

ez=limn→∞(1+zn)n.{\displaystyle e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.}

Euler's identity therefore states that the limit, asn approaches infinity, of(1+iπn)n{\displaystyle (1+{\tfrac {i\pi }{n}})^{n}} is equal to −1. This limit is illustrated in the animation to the right.

Euler's formula for a general angle

Euler's identity is aspecial case ofEuler's formula, which states that for anyreal numberx,

eix=cos⁡x+isin⁡x{\displaystyle e^{ix}=\cos x+i\sin x}

where the inputs of thetrigonometric functions sine and cosine are given inradians.

In particular, whenx =π,

eiπ=cos⁡π+isin⁡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

eiπ=−1+0i,{\displaystyle e^{i\pi }=-1+0i,}

which yields Euler's identity:

eiπ+1=0.{\displaystyle e^{i\pi }+1=0.}

Geometric interpretation

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Any complex numberz=x+iy{\displaystyle z=x+iy} can be represented by the point(x,y){\displaystyle (x,y)} on thecomplex plane. This point can also be represented inpolar coordinates as(r,θ){\displaystyle (r,\theta )}, wherer is the absolute value ofz (distance from the origin), andθ{\displaystyle \theta } is the argument ofz (angle counterclockwise from the positivex-axis). By the definitions of sine and cosine, this point has cartesian coordinates of(rcos⁡θ,rsin⁡θ){\displaystyle (r\cos \theta ,r\sin \theta )}, implying thatz=r(cos⁡θ+isin⁡θ){\displaystyle z=r(\cos \theta +i\sin \theta )}. According to Euler's formula, this is equivalent to sayingz=reiθ{\displaystyle z=re^{i\theta }}.

Euler's identity says that−1=eiπ{\displaystyle -1=e^{i\pi }}. Sinceeiπ{\displaystyle e^{i\pi }} isreiθ{\displaystyle re^{i\theta }} forr = 1 andθ=π{\displaystyle \theta =\pi }, this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positivex-axis isπ{\displaystyle \pi } radians.

Additionally, when any complex numberz ismultiplied byeiθ{\displaystyle e^{i\theta }}, it has the effect of rotatingz{\displaystyle z} counterclockwise by an angle ofθ{\displaystyle \theta } on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any pointπ{\displaystyle \pi } radians around the origin has the same effect as reflecting the point across the origin. Similarly, settingθ{\displaystyle \theta } equal to2π{\displaystyle 2\pi } yields the related equatione2πi=1,{\displaystyle e^{2\pi i}=1,} which can be interpreted as saying that rotating any point by oneturn around the origin returns it to its original position.

Generalizations

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Euler's identity is also a special case of the more general identity that thenthroots of unity, forn > 1, add up to 0:

∑k=0n−1e2πikn=0.{\displaystyle \sum _{k=0}^{n-1}e^{2\pi i{\frac {k}{n}}}=0.}

Euler's identity is the case wheren = 2.

A similar identity also applies toquaternion exponential: let{i,j,k} be the basisquaternions; then,

e13(i±j±k)π+1=0.{\displaystyle e^{{\frac {1}{\sqrt {3}}}(i\pm j\pm k)\pi }+1=0.}

More generally, letq be a quaternion with a zero real part and a norm equal to 1; that is,q=ai+bj+ck,{\displaystyle q=ai+bj+ck,} witha2+b2+c2=1.{\displaystyle a^{2}+b^{2}+c^{2}=1.} Then one has

eqπ+1=0.{\displaystyle e^{q\pi }+1=0.}

The same formula applies tooctonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, sincei{\displaystyle i} and−i{\displaystyle -i} are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.

History

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Euler's identity is a direct result ofEuler's formula, published in his monumental 1748 work of mathematical analysis,Introductio in analysin infinitorum,[16] but it is questionable whether the particular concept of linking five fundamental constants in a compact form can be attributed to Euler himself, as he may never have expressed it.[17]

Robin Wilson writes:[18]

We've seen how [Euler's identity] can easily be deduced from results ofJohann Bernoulli andRoger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly—and certainly it doesn't appear in any of his publications—though he must surely have realized that it follows immediately from his identity [i.e.Euler's formula],eix = cosx +i sinx. Moreover, it seems to be unknown who first stated the result explicitly

See also

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Notes

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  1. ^The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formulaeix = cosx +i sinx,[1] and theEuler product formula.[2] See alsoList of topics named after Leonhard Euler.

References

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  1. ^Dunham, 1999,p. xxiv.
  2. ^Stepanov, S.A. (2001) [1994],"Euler identity",Encyclopedia of Mathematics,EMS Press
  3. ^Milla, Lorenz (2020),The Transcendence of π and the Squaring of the Circle,arXiv:2003.14035
  4. ^Hines, Robert."e is transcendental"(PDF).University of Colorado.Archived(PDF) from the original on 2021-06-23.
  5. ^Gallagher, James (13 February 2014)."Mathematics: Why the brain sees maths as beauty".BBC News Online. Retrieved26 December 2017.
  6. ^Paulos, 1992, p. 117.
  7. ^Nahin, 2006,p. 1.
  8. ^Nahin, 2006, p. xxxii.
  9. ^Reid, chaptere.
  10. ^Maor,p. 160, and Kasner & Newman,p. 103–104.
  11. ^Wells, 1990.
  12. ^Crease, 2004.
  13. ^Nahin, Paul (2011).Dr. Euler's fabulous formula : cures many mathematical ills. Princeton University Press.ISBN 978-0-691-11822-2.
  14. ^Stipp, David (2017).A Most Elegant Equation : Euler's Formula and the Beauty of Mathematics (First ed.). Basic Books.ISBN 978-0-465-09377-9.
  15. ^Wilson, Robin (2018).Euler's pioneering equation : the most beautiful theorem in mathematics. Oxford: Oxford University Press.ISBN 978-0-19-879493-6.
  16. ^Conway & Guy, p.254–255.
  17. ^Sandifer, p. 4.
  18. ^Wilson, p. 151-152.

Sources

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External links

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