As said in the title.

Of cause something goes wrong here. Hint: It is a common mistake that we always make!

$-1=-1$

$\frac{-1}{1} = \frac{1}{-1}$

$\sqrt{\frac{-1}{1}}=\sqrt{ \frac{1}{-1}}$

$\frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}$

$\frac{i}{1}=\frac{1}{i}$

$\frac{i}{2}=\frac{1}{2i}$

$\frac{i}{2}+\frac{3}{2i}=\frac{1}{2i}+\frac{3}{2i}$

$i(\frac{i}{2}+\frac{3}{2i})=i(\frac{1}{2i}+\frac{3}{2i})$

$\frac{i^2}{2}+\frac{3i}{2i}=\frac{i}{2i}+\frac{3i}{2i}$

$\frac{-1}{2}+\frac{3i}{2i}=\frac{i}{2i}+\frac{3i}{2i}$

$1=2$

This fallacious proof is taken from http://www.math.toronto.edu/mathnet/falseProofs/second1eq2.html.