Squarring the Euler's formula

Question

$$\begin{align}&e^{\pi i} + 1 = 0 &\text{ (Euler's Formula)}\\ \implies &e^{\pi i} = -1&\\ \implies &e^{2\pi i} = 1& \text{ (Squaring both sides)}\\ \implies &e^{2\pi i} = e^0 (e^0 = 1)&\\ \implies &2\pi i = 0&\end{align}$$

how is this possible?

Answer 1

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 $z\neq z+2k\pi\cdot i$.