I've seen all of these "proofs" being added recently for 2 = 0, 1 < 0, etc, etc...so I figured I'd add my two proofs for 2 = 1, and proof that magic exists...
I apologize if these have already been posted...I haven't seen them yet though, so I thought I would put them up. These aren't difficult, and I fully expect them to be solved very quickly, but they're fun.
This first one is the golden oldie (I learned this one quite a few years ago and still like it):
let a = b
a2 = ab (multiply both sides by a)
a2 - b2 = ab - b2(subtract b2 from both sides)
(a + b)(a - b) = b(a - b) (factor)
(a + b) = b (divide both sides by (a - b) )
b + b = b (substitution)
2b = b
2 = 1
Then this following one uses calculus (fun one at first when learning calculus):
x = 1 + 1 + ... + 1 (x times)
x2 = x + x + ... + x (multiply through by x)
2x dx = (1 + 1 + ... + 1) dx (take deriviative of both sides)
2x = (1 + 1 + ... + 1)
2x = x
2 = 1
and my last one is to show that something TRULY can come from nothing:
0 = 0 + 0 + 0 + ...
0 = (1 - 1) + (1 - 1) + (1 - 1) + ... (since 0 = 1 - 1)
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... (associative law of addition)
0 = 1 + 0 + 0 + 0 + ... (since -1 + 1 = 0)
0 = 1
So, as if by magic something appeared out of nothing! YAY!