why it does: first,
(.999~)-((1/10)*(.999~))=
(.999~)-(.0999~)=
.9
but this can be summarized as:
x - (1/10)x = (9/10)x
so that means that what we just did was:
(9/10)*(.999~)=.9
multiply both sides of this equation by (10/9):
(10/9)(9/10)(.999~)= (10/9)*(.9)
and simplify:
.999~ = 1
QED
A couple of things. First, pnorris, welcome to the Straight Dope Message Boards, glad to have you with us!
When you start a new thread, it’s helpful to provide a link to the column you are discussing. Saves duplication of effort and helps keep everyone on the same page. No big deal, you’ll know for next time. In this case: An infinite question: why doesn’t .999^ = 1?
You’ve offered the standard mathematical proof. Frankly, anyone who would be convinced by a mathematical proof wouldn’t need it, they’d already “know” that .999… = 1. Cecil clearly says he’s trying to come up with a “common sense” argument, rather than an algebraic “proof.” What you’ve stated is not WHY .999… = 1, you’ve provided an algebraic manipulation that “proves” it true. That’s not different from the .333… = 1/3 and 3 x .333 = .999… = 3 x 1/3 = 1 argument that Cecil offers. It’s algebra. Such proofs are not the “reason”, if you follow my distinction. Cecil was trying to provide the “reason”, not the “proof.”
By the way, there a few other threads on this subject, several of which offer the same proof that you did, and some other comments. F’rinstance: An infinite question… and “.999”