Encyclopedia > The most remarkable formula in the world

  Article Content

Euler's identity

Redirected from The most remarkable formula in the world

Euler's identity, a special case of Euler's formula, is the following:

<math>e^{i \pi} + 1 = 0</math>

The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, <math>i</math> is the imaginary unit (an imaginary number with the property i² = -1), and <math>\pi</math> is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

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 numbers 0 and 1 are elementary for counting and arithmetic.
  • The number <math>\pi</math> is a constant related to our world being 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 <math>e</math> is important in describing growth behaviors, as the simplest solution to the simplest growth equation <math>dy / dx = y</math> is <math>y = e^x</math>.
  • Finally, the imaginary unit <math>i</math> was introduced to ensure that all non-constant polynomial equations would have solutions (see Fundamental Theorem of Algebra).

The formula also involves the fundamental arithmetical operations of addition, multiplication and exponentiation.

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

<math>e^{ix} = \cos x + i \sin x</math>

for any real number <math>x</math>. If we set <math>x = \pi</math>, then

<math>e^{i \pi} = \cos \pi + i \sin \pi,</math>

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

<math>e^{i \pi} = -1</math>


<math>e^{i \pi} + 1 = 0.</math>


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

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
September 11, 2001 Terrorist Attack City of New York casualties

... Donald Bittner[?], 27 Albert Balewa Blackman, Jr.[?], 26 Susan Leigh Blair[?], 35 Harry Blanding, Jr.[?], 38 Janice Lee Blaney[?], 55 Craig Michael Blass[?], ...