Euler's identity

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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 "e^{i \pi} + 1 = 0. \,\!"

Quotes[edit]

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, e^{i \pi} + 1 = 0. \,\! 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.
  • Addition and multiplication In our study of oscillating systems we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicist's point of view we could bring forth this formula in two minutes or so, and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics called elementary algebra.
  • There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: e^{i \pi} + 1 = 0. \,\! 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
e^{ix} = \cos x +  i\sin x \,\!
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  i =  \sqrt{-1} 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

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