"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
|
Here is a comment, exposing the error: All of that's fine, but 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 3^2 = (-3)^2.