Proof that 2=1 (<- and proof that this is false)

[notLuke]

2 = 1 Proof
Let us use basic algebraic proofs. We begin by using X and Y as any rational value. Let X and Y be equal values: X = Y
Multiply boths sides by X: X2 = XY
Subtract boths sides with Y2: X2 - Y2 = XY - Y2
Factor each side: (X - Y)(X + Y) = Y(X - Y)
Divide by (X - Y) X + Y = Y
Since X = Y, substitute Y with X: X + X = X
Simplify 2X = X
Divide by X 2 = 1

I got sent this by one of my A-level students. My reply was as follows;

Unfortunately this is not true.
The error arises when you have to divide both sides by (X – Y).
If you go back to your original assumption. That X and Y are equal, the X – Y must equal zero.
You are therefore dividing by zero, which is something you cannot do.
Try doing something divided by zero on your calculator and it will come up with ‘syntax error’.


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