How Can 0.999... = 1?

There are many different proofs of the fact that "0.9999..." does indeed equal 1. So why does this question keep coming up?
Students don't generally argue with "0.3333..." being equal to 1/3, but then, one-third is a fraction. Maybe it's just that it "feels" "wrong" that something as nice and neat and well-behaved as the number "1" could also be written in such a messy form as "0.9999..." Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
One of the major sticking points seems to be notational, so let me get that out of the way first: When I say "0.9999...", I don't mean "0.9" or "0.99" or "0.9999" or "0.999" followed by some large but finite (limited) number of 9's". The ellipsis (the "dot, dot, dot" after the last 9) means "goes on forever in like manner". In other words, "0.9999..." never ends. There will always be another "9" to tack onto the end of 0.9999.... So don't object to 0.9999... = 1 on the basis of "however far you go out, you still won't be equal to 1", because there is no "however far" to "go out" to; you can always go further.
"But", some say, "there will always be a difference between 0.9999... and 1." Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999...9 and 1. That is, if you do the subtraction, 1 – 0.999...9 will not equal zero. But the point of the "dot, dot, dot" is that there is no end; 0.9999... is inifinte. There is no "last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite. (That's not a "criticism", per se; infinity is a messy topic.)
Category:Notation
Category:Math Tips