Novel number representations are cool. I came across Paul Tarau's "hereditarily binary natural numbers" a few years ago, and via them, Knuth's earlier TCALC representation (one of the many small side projects Knuth has done over the years, which for some reason hasn't gotten much attention).
They go pretty much the other way than this representation: they allow huge numbers, much larger than can be represented in regular notation (incl. floating point) to be represented and calculated with efficiently and exactly. Tarau's system improves slightly on Knuth's in that representations are unique. He also proves that they at worst require twice as many bits as regular representation.
Knuth's and Tarau's system are variable-length and limited to naturals, but it seems like it would be easy enough to extend them to rationals and fix the representation length.
They go pretty much the other way than this representation: they allow huge numbers, much larger than can be represented in regular notation (incl. floating point) to be represented and calculated with efficiently and exactly. Tarau's system improves slightly on Knuth's in that representations are unique. He also proves that they at worst require twice as many bits as regular representation.
Knuth's and Tarau's system are variable-length and limited to naturals, but it seems like it would be easy enough to extend them to rationals and fix the representation length.