Part of a number line continues to be split into equal parts of this size how many equal parts will be between 0 and 1?

I see from the drawing that the first part is of length 1/4.

You can split the interval [0, 1] into 2 intervals of length 1/2 each;
from that point you can split each of the intervals of length 1/2 into 2 equal parts of length 1/4 each.  

Thus, the interval [0, 1] can be subdivided into four = intervals of length 1/4 each.