# 歐拉恆等式

e ^ (πi) + 1 = 0

${\displaystyle {{e}^{0}}=1}$開始，以相對速度i，走π長時間，加1，則到達原點
${\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}$

## 證明

${\displaystyle e^{ix}=\cos x+i\sin x\,\!}$ 歐拉公式
${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi \,}$ （代入${\displaystyle x=\pi \,}$
${\displaystyle {{e}^{{i}\,{\pi }}}=-1}$ （因${\displaystyle \cos \pi =-1}$ ${\displaystyle \sin \pi =0}$
${\displaystyle {{{e}^{{i}\,{\pi }}}+{1}}=0}$

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