# Two = One

### A Proof That 2 = 1

```     X = Y                  Given
X^2 = XY               Multiply both sides by X
X^2 - Y^2 = XY - Y^2   Subtract Y^2 from both sides
(X+Y)(X-Y) = Y(X-Y)    Factor both sides
(X+Y) = Y              Cancel out common factors
Y+Y = Y                Substitute in from line 1
2Y = Y                 Collect the Y's
2 = 1                  Divide both sides by Y
```
Where is the flaw?

### When you've sorted that out, how about a proof that -1 = 1

```     -1 = -1
-1 / 1 = 1 / -1
sqrt( -1 / 1 ) = sqrt( 1 / -1 )   Take square root of both sides
sqrt( -1 ) = 1 / sqrt( -1 )       Simpify
sqrt( -1 ) ^ 2 = 1                Multiply each side by sqrt(-1)
-1 = 1
```

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