TITLE: Squarring the Euler's formula
QUESTION [0 upvotes]: $$\begin{align}&e^{\pi i} + 1 = 0 &\text{ (Euler's Formula)}\\
\implies &e^{\pi i} = -1&\\
\implies &e^{2\pi i} = 1& \text{ (Squaring both sides)}\\
\implies &e^{2\pi i} = e^0 (e^0 = 1)&\\
\implies &2\pi i = 0&\end{align}$$
how is this possible?

REPLY [3 votes]: In the complex setting, 
$$e^{z_1}=e^{z_2}\implies z_1=z_2$$ is a false conclusion because the function $$z\mapsto e^z$$ is not injective. In fact, for every $z\in \mathbb C$ and every $k\in\mathbb Z$, you have
$$e^{z+2k\pi\cdot  i} = e^z$$
and,  if $k\neq 0$, then $z\neq z+2k\pi\cdot i$.
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