Q:

what do you call a circle with every number around it?

What do you call this Zeno-ish circle? It has every single number around it.
The numbers are placed at $1/4, 3/8, 7/16, 15/36$ and so on. 
Basically $2^{n-1}/2^{n+1}$ for the positives, and $-2^{-n-1}/2^{-n+1}$ for the negatives.

A:

I think you're talking about the projectively extended real line.

