### "Proof" that One Equals Two

 Statement Comment: -20 = -20 Hard to argue with! 16 - 36 = 25 - 45 Rewriting 16 - 36 + 81/4 = 25 - 45 + 81/4 Add 81/4 to each side (4 - 9/2)2 = (5 - 9/2)2 Rewriting 4 - 9/2 = 5 - 9/2 Sqrt of each side 4 = 5 Add 9/2 to each side 1 = 2 Subtract 3 from each side

 Statement Comment: 16 = 16 Reflexive property of "=" 4 + 12 = 16 From definition of "+" 4 - 12 = 16 - 24 Subtract 24 from both sides 4 - 12 + 9 = 16 - 24 + 9 Add 9 to both sides, completing squares (2 - 3)2 = (4 - 3)2 Factor, rewriting as squares 2 - 3 = 4 - 3 Sqrt of each side 2 = 4 Add 3 to each side 1 = 2 Divide by 2

Exposing the error: one side squares a negative number and the other squares its additive inverse (I think that's correct term, it's been 30 years). That's equivalent to saying that 3 = -3 because 32 = (-3)2.

Here's a different one, with a different flaw:

x = y
x2 = xy
x2 - y2 = xy - y2
(x - y)(x + y) = y(x - y)
(x - y)(x + y)/(x - y) = y(x - y)/(x - y)
x + y = y
y + y = y
2y = y
2 = 1

The flaw is this:

x = y
x - y = 0
(x - y)(x + y)/(x - y) is division by zero.