Euler's Identity

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Theorem

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


Proof

Follows directly from Euler's Formula $e^{i z} = \cos z + i \sin z$, by plugging in $z = \pi$:

$e^{i \pi} + 1 = \cos \pi + i \sin \pi + 1 = -1 + i \times 0 + 1 = 0$

$\blacksquare$


Also presented as

This result can also be presented as:

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

or:

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


Source of Name

This entry was named for Leonhard Paul Euler.


Sources

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