$$1=1$$ $$\Rightarrow\frac{-1}{1}=\frac{1}{-1}$$ $$\Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}$$ $$\Rightarrow\frac{i}{1}=\frac{1}{i}$$ $$\Rightarrow\frac{i}{2}=\frac{1}{2i}$$ $$\Rightarrow\frac{i}{2}+\frac{3}{2i} = \frac{1}{2i} +\frac{3}{2i}$$ $$\Rightarrow i(\frac{i}{2}+\frac{3}{2i} ) =  i(\frac{1}{2i} +\frac{3}{2i})$$ $$\Rightarrow\frac{-1}{2}=\frac{1}{2}$$ $$\Rightarrow1=2$$  What is wrong in this?
I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &amp;= i \\\\\ \frac1{\sqrt{-1}} &amp;= \frac1i \\\\  \frac{\sqrt1}{\sqrt{-1}} &amp;= \frac1i \\\\  \sqrt{\frac1{-1}} &amp;= \frac1i \\\\  \sqrt{\frac{-1}1} &amp;= \frac1i \\\\  \sqrt{-1} &amp;= \frac1i \\\\  i &amp;= \frac1i \\\\  i^2 &amp;= 1 \\\\  -1 &amp;= 1 \quad !!? \end{align}
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