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.