Is this really true?

The idea is that 0.9 recurring
(0.999..., with the digits going on forever)
is actually equal to 1

(For this exercise I will use the notation 0.999... as notation for 0.9 recurring,
the correct way would be to put a little dot above the 9, or a line on top like this: 0.9)

Does 0.999... = 1 ?

   
 
Let X = 0.999...
 
Then 10X = 9.999...
 
Subtract X from each side to give us:
   
 
9X = 9.999... - X
 
but we know that X is 0.999..., so:
   
 
9X = 9.999... - 0.999...
or:
9X = 9
 
Divide both sides by 9:
   
 
X = 1
 
But hang on a moment I thought we said X was equal to 0.999... ?

Yes, it does, but from our calculations X is also equal to 1. So:
   
 
X = 0.999... = 1
 
 
 
Therefore 0.999... = 1
 
 
Does anyone disagree with this? Let me know on the Math is Fun Forum.