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Euler's identity

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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>

and

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

References

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



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