Math Time: Why is 0.999 = 1?

Math Time: Why is 0.999 = 1?

Today I had a rather strange challenge by another person (who will remain anonymous for their sake). This person was attempting to one-up me on an argument, and threw this question at me as a final resort: "I bet you can't tell us why 1 = 0.999."

At first I thought they were surely joking, until it became apparent that they surely were not. So I proceeded to write out the following, for all to see in the holy writings known as "Trade Chat", that this was (1) a very stupid question, and (2) to finally shut the idiot up.

(...) mean repeating to infinity.

=> X = 0.999...

=> 10 * X = 9.999...

=> 10 * X = 9 + 0.999...

=> 10 * X = 9 + X

=> (-X) + 10 * X = 9 + X + (-X)

=> 9 * X = 9

=> X = 9/9

=> X = 1 = 0.999....

Boom. I don't know if he ever saw the hilarity that ensued, as this person had apparently gone offline. 

For good measure, here's another proof I tossed into Trade:

=> 1/9 = 0.111...

=> 9 * 1/9 = (0.111...) * 9

=> 9/9 = 0.999...

=> 1 = 0.999...

There are also a number of different ways to express this using more intricate methods, but the above two are probably the easiest. So whenever you have someone challenge you with this question, you know the answer. =)