$1+1=0$ What am I doing wrong???!

Question

Does someone know what I'm doing wrong? I'm struggling with this for a while now and I don't see what I do wrong!

$$1+1=$$ $$1+\sqrt{1}=$$ $$1+\sqrt{-1*-1}=$$ $$1+\sqrt{-1}*\sqrt{-1}=$$ $$1+i*i=$$ $$1-1=0$$

Answer 1

$$\sqrt {ab}=\sqrt {a} .\sqrt b$$ For $a,b\ge 0$

Answer 2

$1+\sqrt{-1\cdot-1}\neq1+\sqrt{-1}\cdot\sqrt{-1}$. That is the issue with your "proof" that $1+1=0$.

Answer 3

You've run into a common problem of the square root. It can take on two value. usually we use the principal root, that is to say the positive square root of a number. In this case, you've split the root into two part, however, you can't do that because it could then have 4 combinations of signs, 2 visible combinations. Most of the time this isn't a problem, because splitting two positive roots results in 2 combinations of signs, but only one visible sign.

I should also mention that when you say $1=\sqrt n$, this equality has to hold true even after less than truthful manipulations. If you end up with $-1=1$ its your own fault, because you neglected the original equality.