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

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