1 = 2?? must be wrong but how?

[Yorkie580]

I've been shown this problem and I can't work out where it goes wrong...

x = y

x^2 = xy [multiply by x]

x^2 - y^2 = xy - y^2 [subtract y^2]

(x+y)(x-y) = y(x-y) [factorise]

(x+y) = y [cancel (x-y)]

y + y = y [x = y so x + y = y + y]

2y = y

2 = 1 [divide by y]

[ugly_bob]

mathemagical

[gen2]

The error in the maths comes between steps 4 and 5.
You have the following:

(x+y)(x-y) = y(x-y) [factorise]
(x+y) = y [cancel (x-y)]

You have concealed what you are actually doing by using the non-mathematical operation you have called 'cancel'
What you have in fact done to get from step 4 to 5 is to DIVIDE by (x-y), so it should read as follows:

x+y)(x-y) = y(x-y) [factorise]
(x+y) = y [divide by (x-y)]

Now, since x = y (step 1), (x-y) = 0.

As all good mathematicians should know, you cannot divide by zero, it is undefined. Try it on a calculator and it will generate an error.

So there is the error in your"proof", you have divided by zero.

[scotstone]

Some more of these dubious proofs see here

[Yorkie580]

Thanks gen2 - I'd been trying to work it out for days.

cheers for the link scotstone

[potterfan3]

how about x doesn't equal y, There are two different values

[sddsddean]

In fact you can see the error in the 'factorise' line. As gen2 says x-y=0, you have (x+y)*0 =y*0 , which of course is 0=0.