Impossible Numbers

A doodle from my whiteboard.  I'm not really sure what to do with this one.

The work in blue has always bugged me - it appears to show that 0.999999999... is actually equal to 1 even though common sense says it isn't.  In these cases, it's usually common sense that's wrong so I've always just assumed it was one of those fun little oddities of math and left it at that.

Alas, anyone who plays with math long enough will have their concept of infinity screwed with.  So one day I started playing around with these infinitely-repeating decimals to see if I could get 0.9999... to come full circle back to its original value.  I don't know if this sort of thing already has a name so I'm calling them "impossible" numbers until I hear differently.