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
X+Y=Y ; Cancel out (X-Y) term
2Y=Y ; Substitute X for Y, by equation 1
2=1 ; Divide both sides by Y
-- Proof that 2 equals 1
whats the problem here , it left me totally dumbfounded
X=Y is given.
Therefore X-Y=0.
You therefore are dividing by zero between steps 4 and 5.
LL
Yeah… what he said!
You can prove just about anything if you allow yourself to divide by zero.
100 = 50
Well, that’s true enough (0=0). However, it most obviously does not follow that:
10 = 5
According to the given statement, by step (3) that “proof” reduces itself to 0=0. Try performing the same manipulations with an actual number (I think it’s more clear that way) - say X=Y=5:
You can’t just “cancel” anything in reality. That’s a shorthand for multiplying both sides by an inverse of something and reducing some factors to unity. In this case, that would require multiplying both sides of the equation by 1/0 to “reduce” (5-5)/(5-5) to unity.
That’s a no-no because 1/0 is undefined. (0/0 is not equal to 1) This “proof” is a pretty good illustration of just why that’s so.
If X=Y in step (1), then why doesn’t step (4) become (x+y)(x-y)= 2Y(x-y)?
This seems blindingly obvious to be, but the reason you come up with 2=1 is because the equation was totally (and incorrectly) changed at Line 3. Line 3 doesn’t become line 4 by subtracting Y^2. If you subtract Y^2 from both sides, you get X^2=XY, which is just a restatement of the given fact that X=Y. Division by zero isn’t the reason this si screwd up; it’s just an arithmetic error in line 4. Division by zero’s still impossible, of course.
Where the heck does line 4, (X+Y)(X-Y)=Y(X-Y), come from? That’s equal to X^2+XY-Y^2=XY-Y^2, which is where the extra 1 comes from; that X^2 on the left side is what makes it 2 rather than 1 at the end of the equation. But it shouldn’t be there. This is in NO way equivalent to line 3, X^2=XY. There’s no arithmetic way to get from line 3 to line 4; the author of this joke just made up line 4 in the hopes you wouldn’t see that he was making up a completely different equation. Eohippus’s equation would solve this problem and you’d end up with 2=2.
You can prove anything if you just completely change the equation without any regard to arithmetic or common sense. 
Looking at both sides individually:
(X+Y)(X-Y) = X[sup]2[/sup]-Y[sup]2[/sup]
Y(X-Y) = XY-Y[sup]2[/sup]
That’s legal
You’re wrong here, RickJay. In your second paragraph, you say that (X+Y)(X-Y)=Y(X-Y) goes to X^2+XY-Y^2=XY-Y^2. That’s wrong. It goes to X^2+XY-XY+Y^2=XY-Y^2 The XY and -XY cancel, leaving X^2+Y^2=XY-Y^2. That’s exactly Line 3, which is where he got it from. The point is that at this point, the factoring leaves it at 0=0, which is why the proof fails. His arithmetic at the beginning is correct. He simply factored the polynomial in line 3 to get to line 4, and he did it correctly.
Jman
You’re right; I didn’t expand the expression correctly. (I assume you meant -Y^2 on the left side of the equation.) Stupid error.
Because 1 = 2
1+1=2+1
2=3
But 1=2 so
1=3
By mathematical induction we see that every natural number equals every other natural number. We can do it by subtraction, too, and come up with all the negative numbers equalling everything.
Indeed, through other manipulations we can get irrational numbers to equal 1, and so on and so on.
sqrt1=sqrt2=1=3=10…
Yes…see, we all makes stupid mistakes. 
(note to self: quadruple check typing when correcting someone else). See ya 'round, RickJay.
Jman
Just for kicks, your “proof” of 1=2 can be found here along with several other similar puzzlers.
quit it. Everybody knows 42 is the answer to life, the universe, and everything.
Alright, Alright, that proof was wrong and I’m no better at math than the one who wrote that proof, but see I got this calculus book here which seems to prove 1 = 2 to me, and while were looking if you add more dimensions you can get 1 = 3 and so on.
ok it’s all about the chain rule, I’m going to use “d” as the partial sign because I can think of no better way.
dw/dx = (dw/du)(du/dx) + (dw/dv)(dv/dx)
well if you go ahead and cancel the “du” and the “dv” on the right side you end up with
dw/dx = dw/dx + dw/dx
and now 1 = 2
there is a proof listed but it has lots of symbols that I don’t really understand, I’m pretty sure the answer lies in those. But when learning this chain rule I thought it nicely proved 1=2
Well, you need to supply more info, specifically that w is a function of u and v, and that both u and v are functions of x.
Then, you have a problem of too few symbols. dw/dx = dw/dx + dw/dx, true, but the three partials are all different! The first is partial with respect to x as seen by both u and v, the next two partials are the partials of w with respect to x as seen by u and v, separately.
I could lug out my notebook where my calc prof showed us exactly that, and work out the coding to show it to you well, but, quite frankly, it’s saturday night and I can hear my beer calling from the fridge.
Which is exactly why your calculus book said that you can’t just cancel out "du"s and "dv"s. If you do it with total derivatives (the ones written with normal ds), you’ll generally end up with something that makes sense, but if you try it with partials (the curly ds), you’ll get into trouble.
I’ve also seen a few of these “proofs” that involve leaving off a constant of integration, but those need a bit more education to be understood.