1=1^{2}=(1)^{2}=√1, yes? Are we all familiar with that? (What? You're not? Oh, go take a math class.)
Therefore:
√1^{2}=√(1)^{2}
=1 =1 (since square roots always cancel out square powers)
Therefore 1=1 and 2=0 (adding 1 to both sides).
So what's the problem? Well, the problem is: square roots do not cancel out square powers, although they certainly appear to. Ever heard of BODMAS or PEMDAS? Same thing, different acronym. Evaluate 1^{2} or (1)^{2} first, then find the root of the answer. It ends up coming out to 1 both times.
Also: ever heard of the modulus or piecewise function? It's defined as √x^{2} or x, and its graph looks like
y
\  /
\  /
\  /
\  /
\  /
\  /
\  /
\  /
\/
x


Note that y=x looks like:
y
 /
 /
 /
 /
 /
 /
 /
 /
/
x
/
/ 
/ 
/ 
/ 
and nothing like our friend x.
God, I love proving stuff.