There is an awesome power here by Inigo Quilez which develops his idea of a "floating bar" number. The core idea is to create a representation of rational numbers that mimicks the "floating" property of floating points numbers.
Its an amazing article, and while he develops the concept throughly, more work on this seems like it could be fruitful. The investigation for this article is related to graphics rendering, but I can't help work happens to other applications when floating point numbers are replaced by floating bar numbers. I imagine there are certain applicatiosn out there where a floating bar could fit better then a floating point.
Anyway, I just wanted to put a link to this post on my blog because I love these alternative view of common things, and I love thinking about the alternate world they imply.