I see what you mean, nawi. The same thing could be done with any equasion:
e = mc^2
0 = mc^2-e {subtract e from each side}
3*0 = 3mc^2-3e {multiply 3x each side}
0 = 3mc^2-3e {3*0 is 0}
mc^2-e = 3mc^2-3e {substituting 0 for mc^2-e, which is 0}
1(mc^2-e) = 3(mc^2-e) {factor}
1 = 3 {dividing by mc^2-e}
...
What??? :)
As Ryan Moody said, the problem is when you divide both sides by (X-Y), or 0.
In your example, this:
Is true provided that X=Y, as stated above.
Now, when you derive the 1=2 by removing the (X-Y), you are forgetting that you may divide both sides of an equasion by any number other than zero. Since (X-Y) = 0, you can't do this.
The reason things like this don't occur very often (for me, at least) is that when I see something like "X = Y", I immediately substitute. For example, this:
Immediately becomes this:
.. when you know that Y=X.
e = mc^2
0 = mc^2-e {subtract e from each side}
3*0 = 3mc^2-3e {multiply 3x each side}
0 = 3mc^2-3e {3*0 is 0}
mc^2-e = 3mc^2-3e {substituting 0 for mc^2-e, which is 0}
1(mc^2-e) = 3(mc^2-e) {factor}
1 = 3 {dividing by mc^2-e}
...
What??? :)
As Ryan Moody said, the problem is when you divide both sides by (X-Y), or 0.
In your example, this:
1(X-Y) = 2(X-Y)
Is true provided that X=Y, as stated above.
Now, when you derive the 1=2 by removing the (X-Y), you are forgetting that you may divide both sides of an equasion by any number other than zero. Since (X-Y) = 0, you can't do this.
The reason things like this don't occur very often (for me, at least) is that when I see something like "X = Y", I immediately substitute. For example, this:
X-Y = 2X-2Y
Immediately becomes this:
X-X = 2X-2X
or
0 = 0
or
0 = 0
.. when you know that Y=X.