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Euler's identity

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In mathematics, Euler's identity is the following equation, which is true by definition:

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

The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i ² = -1), and π is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

The identity is a special case of Euler's formula from complex analysis, which states that

e^{ix} = \cos x + i \sin x \,\!

for any real number x. If we set x = π, then

e^{i \pi} = \cos \pi + i \sin \pi \,\!

and since cos(π) = −1 and sin(π) = 0, we get

e^{i \pi} = -1 \,\!

Perceptions of the identity

Benjamin Pierce, after proving the formula in a lecture, said, "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."

It was called "the most remarkable formula in mathematics" by Richard Feynman. Feynman found this formula remarkable because it links some very fundamental mathematical constants:

  • The number 0, the identity element for addition (for all a, a+0=0+a=a). See Group (mathematics).
  • The number 1, the identity element for multiplication (for all a, a×1=1×a=a).
  • The number π is fundamental in trigonometry, π is a constant in a world which is Euclidean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).
  • The number e is a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation dy / dx = y with initial condition y(0) = 1 is y = ex).
  • The imaginary unit i (where i 2 = −1) is a unit in the complex numbers. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers (see fundamental theorem of algebra).

Furthermore, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentiation. All the fundamental assumptions of complex analysis are present, and the integers 0 and 1 are related to the field of complex numbers.

In addition, the result is remarkable to most students learning it for the first time because it is so highly counter-intuitive. Consider that

e^{\pi} \approx 23.14069 \,\! while e^{i \pi} = -1 \,\!

The simple insertion of i changes the result dramatically.


  • Feynman, Richard P. The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)

External link

  • Proof of Euler's Identity by Julius O. Smith III
  • Proof of Euler's Identity for a Layman by Ian Henderson

Last updated: 03-18-2005 11:16:12