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# Euler's identity

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

$e^{i \pi} + 1 = 0$

The equation appears in Leonhard Euler's Introductio, 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 i² = -1), and $\pi$ 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 $\pi$ 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 $e$ is important in describing growth behaviors, as the simplest solution to the simplest growth equation $dy / dx = y$ is $y = e^x$.
• Finally, the imaginary unit $i$ 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

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

for any real number $x$. If we set $x = \pi$, then

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

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

$e^{i \pi} = -1$

and

$e^{i \pi} + 1 = 0.$
• Feynman RP - The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)

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