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’.


Please post your comments

[Peter L]

My question is, how is that solving anything? You can't just multiply both sides, then factor them, then divide them, etc. For the first step, you'd divide because it's a multiplication problem.

[notLuke]

Well, exactly. That's what I told him. It's rushed.

[υℓqυισяяα]

Haha... must have cracked up aristortle. And no, when you divide by 0 the calculator gives math error. Because anything divided by 0 is infinity, which is too large for a puny calculator.

[notLuke]

Haha... must have cracked up aristortle. And no, when you divide by 0 the calculator gives math error. Because anything divided by 0 is infinity, which is too large for a puny calculator.

Exactly

[lunagron]

Try doing something divided by zero on your calculator and it will come up with ‘syntax error’.

My calculator actually says "divide by zero error" and then it shuts off. (Not a joke)

[alfonzo1955]

Haha... must have cracked up aristortle. And no, when you divide by 0 the calculator gives math error. Because anything divided by 0 is infinity, which is too large for a puny calculator.

No, anything divided by zero is undefined, it could be called infinity but in a mathematical sense it is undefined.

[lunagron]

No, anything divided by zero is undefined, it could be called infinity but in a mathematical sense it is undefined.

Technically it is a complex infinity.

[robbiecee2]

My question is, how is that solving anything? You can't just multiply both sides, then factor them, then divide them, etc. For the first step, you'd divide because it's a multiplication problem.

agreed

[notLuke]

Just thought I'd bump this thread and get it a little more attention. Any thoughts on this?