Euler's identity

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E-to-the-i-pi.svg

Euler's identity, sometimes called Euler's equation, is this equation:

e^{i\pi} + 1 = 0

Euler's identity is named after the Swiss mathematician Leonard Euler. It is not clear that he did invent it.[1]

Respondents to a Physics World poll called the identity "the most profound mathematical statement ever written", "uncanny and sublime", "filled with cosmic beauty" and "mind-blowing".[2]

Mathematical proof using Euler's formula[change | change source]

There are many equations named after Euler. One of them is Euler's formula:

e^{ix} = \cos(x) + i \sin(x)

Write Euler's formula replacing x with the value

  • x = \pi
  • e^{i\pi} = \cos(\pi) + i \sin(\pi)

From trigonometry

  • \cos(\pi) = -1

and

  • \sin(\pi) = 0
  • e^{i\pi} = 0-1
  • e^{i\pi} + 1 = 0

References[change | change source]

  1. Sandifer, C. Edward 2007. Euler's greatest hits. Mathematical Association of America, p. 4. ISBN 978-0-88385-563-8
  2. Crease, Robert P. (2004-10-06). "The greatest equations ever". IOP. http://physicsworld.com/cws/article/print/2004/oct/06/the-greatest-equations-ever. Retrieved 2016-02-20.