I thought my earlier essay on .999… did a pretty good of explaining why I (along with 99.999…% of mathematicians) say that it equals 1, until I asked some of my students what they got out of it; then I got a humbling jolt of pedagogical reality. The students agreed that .999… is the limit of the sequence .9, .99, .999, etc., and they also agreed that the limit of that sequence is 1. So you might think that they would have agreed that .999… equals 1, but no: they couldn’t swallow that conclusion.

I’ve decided that part of what’s going on is that my students arrive at college with a number-sense that’s so deeply grounded in their experience with terminating and non-terminating decimals, in worksheet after worksheet, that these concrete representations have taken on an independent reality for them. At that point, it does little good to tell them “.999… doesn’t mean anything till we assign it a meaning” or “We’re going to define .999… as a limit”, because they already “know” what .999… is: a dot followed by infinitely many 9s! No attempt to redefine .999… can shake loose their sense of what it already means to them.

So today I’m going to come at the problem of .999… from a totally different direction. Continue reading