To cut that line into thirds, notice that we cut the line only twice. One third, two thirds. To divide 1 into fourths , we cut the line three times. We are speaking of something we imagine to have any part at all. The model for such a thing is a length or a distance or a time.
Fractions have nothing to do with counting.
They belong only to measuring. Moreover, our concept of number has now evolved. That is called measuring the line. The problem of an irrational number is that its relationship to 1 is inexpressible, and ultimately unknowable. We cannot say .