How many partitions of a number line are possible?
To use a rational number to describe how far a point on the number line is from 0, you can begin by partitioning the unit interval into an arbitrary number of equal parts. Each of those parts can then be partitioned into an arbitrary number of equal parts, and those, in turn, can be partitioned again.
This process is actually a composition of operations. You can use arrow notation to keep a record of your partitioning actions, as well as the size of the subintervals being produced.
For example, what if you wanted to locate 17/48 on a number line from 0 to 1? You would start by drawing the number line on a piece of paper and repeatedly folding it, making sure to mark the locations of 0 and 1 before you start folding:
Here's one set of partitioning actions to find 17/48: