Euler's identity

e ^ (πi) + 1 = 0

Euler's identity, or Euler's equation, named after Leonhard Euler, is the equation of mathematical analysis

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. ~ Keith Devlin

QuotesEdit

 
Gentlemen, that is surely true, 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. ~ Benjamin Peirce
  • One of the most frequently mentioned equations was Euler's equation,   Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.
  • 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.
  • There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre:   It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
 
Euler's formula illustrated in the complex plane.
  • Gentlemen, that is surely true, 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.
    • Benjamin Peirce, as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880 : Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman
  • For any real number x, Euler’s formula is
 
Where e is fundamental constant (the base of natural logarithms) and i = √-1. If we now put x =π, we get e^(iπ) = cos π+i sin π, and since cos(π) =−1 and sin(π) = 0, this reduces to e^(iπ) =−1 so that e^(iπ) +1=0.
  • This most extraordinary equation first emerged in Leonard Euler’s Introductio in 1748.
    • David Darling, in The Universal Book of Mathematics : From Abracadabra to Zeno's Paradoxes (2004), p. 111
  • Since high school I have been utterly fascinated by the “mystery” of   and by the beautiful calculations that flow, seemingly without end, from complex numbers and functions of complex variables.
  • -1 = e^iπ,
    proves that Euler was a sly guy.
But ζ(2)
was totally new
And raised respect for him sky-high.
  • William C. Waterhouse, as quoted in Dr. Euler's Fabulous Formula : Cures Many Ills (2011), by Paul J. Nahin, p. xiii

External linksEdit

Wikipedia has an article about:
Wikimedia Commons has media related to:

Mathematics
Mathematicians
(by country)

AbelAnaxagorasArchimedesAristarchus of SamosAverroesArnoldBanachCantorCartanCohenDescartesDiophantusErdősEuclidEulerFourierGaussGödelGrassmannGrothendieckHamiltonHilbertHypatiaLagrangeLaplaceLeibnizMilnorNewtonvon NeumannNoetherPenrosePerelmanPoincaréPólyaPythagorasRiemannRussellSchwartzSerreTaoTarskiThalesTuringWilesWitten

Numbers

123360eπFibonacci numbersIrrational numberNegative numberNumberPrime numberQuaternion

Concepts

AbstractionAlgorithmsAxiomatic systemCompletenessDeductive reasoningDifferential equationDimensionEllipseElliptic curveExponential growthInfinityIntegrationGeodesicInductionProofPartial differential equationPrinciple of least actionPrisoner's dilemmaProbabilityRandomnessTheoremTopological spaceWave equation

Results

Euler's identityFermat's Last Theorem

Pure math

Abstract algebraAlgebraAnalysisAlgebraic geometry (Sheaf theory) • Algebraic topologyArithmeticCalculusCategory theoryCombinatoricsCommutative algebraComplex analysisDifferential calculusDifferential geometryDifferential topologyErgodic theoryFoundations of mathematicsFunctional analysisGame theoryGeometryGlobal analysisGraph theoryGroup theoryHarmonic analysisHomological algebraInvariant theoryLogicNon-Euclidean geometryNonstandard analysisNumber theoryNumerical analysisOperations researchRepresentation theoryRing theorySet theorySheaf theoryStatisticsSymplectic geometryTopology

Applied math

Computational fluid dynamicsEconometricsFluid mechanicsMathematical physics Science

History of math

Ancient Greek mathematicsEuclid's ElementsHistory of algebraHistory of calculusHistory of logarithmsIndian mathematicsPrincipia Mathematica

Other

Mathematics and mysticismMathematics educationMathematics, from the points of view of the Mathematician and of the PhysicistPhilosophy of mathematicsUnification in science and mathematics