Euler's identity

thumb|right|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 Euler's identity, or Euler's equation, named after Leonhard Euler, is the equation of mathematical analysis Quotes thumb|right|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. Robert P. Crease, in "The greatest equations ever" at PhysicsWeb (October 2004) 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, “The Most Beautiful Equation,” Wabash Magazine, Winter/Spring 2002. As quoted in Dr. Euler's Fabulous Formula : Cures Many Mathematical Ills (2006) 978-0691118222 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. Edward Kasner and James R. Newman, in Mathematics and the Imagination (1940) thumb|right|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. David Darling, in The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes (2004), John Wiley & Sons, p. 111 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. Paul J. Nahin, in Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (2011), Princeton University Press, p. xiii -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 e^(pi(i))+1=0 I have no idea what it means Yet it is the most profound thing that I have ever seen External links Euler's formula Category:Mathematics