Quote Originally Posted by Mario F. View Post
They don't really. Not for the "last" decimal place, represented by an infinitesimal. If we replace the above for a finite quantity it's easy to see why:

Let X = 0.999
Thus 10 * x = 9.99.
Thus 10 * x - x = 9.99 - 0.999
Oops!
I'm not sure I understand how there can be a last digit in an infinite string of digits. I also have heard that this is a common sort of misunderstanding when thinking about this problem (imagining that there can somehow be a "last" digit even if arbitrarily chosen).

How about a proof that 1 + 2 + 4 + 8 + 16 + ... is equal to -1?

X = 1 + 2 + 4 + 8 + 16 + ...
X = 1 + 2 * (1 + 2 + 4 + 8 + 16 + ...)
hey, the thing in parentheses is X again
X = 1 + 2 * X
X - 2 * X = 1
-X = 1
X = -1

There are such proofs for absurdities like "0 = 1" but they mostly involve on steps which are obviously problematic such as dividing by zero. There's nothing obviously wrong with the above... Or is there?