In the figure to the left, the bar above the number 9 indicates that it is to be repeated forever. For the remainder of this post, we will represent that concept by several nines with an ellipsis (.999...).
Now, here is the conundrum. .9 repeating is EQUAL TO ONE. Not CLOSE to one, mind you, but EQUAL to one.
Nonsense, you reply. It is obviously less than one. Not by much - by an infinitely small amount, in fact. But the simple fact (?) that it is not one is enough to demonstrate that it can't be equal to one. It's as close as you can get to one without being one.
Wrong. It is in fact equal to one, and that fact can be demonstrated mathematically in several ways.
The most easily understood is to revert to other familiar repeating digits. Everyone knows that 1/3 is 0.333... and that 2/3 is 0.666... If you add them together, you get 3/3, which is one.
But now note that the sum of the decimals on the right side of the equation is 0.999...
Therefore, one is equal to (not close to) .999...
You don't agree? Then try this. Subtract .999... from one. What you have is 0.000... An infinitely long string of zeroes, which can only be equal to zero. And if the subtraction of .999... from one leaves zero, then the .999... must be one. But, you say, there's a one at the end that string of zeroes. No, there isn't, because the string of 9s doesn't end.
There are other proofs at the Polymathematics blog, along with a long series of comments, an update refuting the counterarguments, and a final refutation of the most stubborn skeptics.