I've been trying to understand how a ratio of two of the same units gives a dimensionless quantity. It makes no sense to me. For example, density in terms of mass and volume:
$d = \frac{m}{V}$
That I can interpret from the definition of division that on every unit of volume comes $n$ units of mass. But here's something that bugs me. If I wanted to discertain what is the mass in $5$ units of volume (whatever they are, laws of physics should be independent from the underlying units if they are correctly defined), that gives a ratio of $5V/1V$ and gives me a dimensionless number $5$.
So, that can be stated as: On every unit of volume there comes $5$... What? $5$ what? How can it be dimensionless? I understand that units have to be defined in terms of pointing a finger to something (such as a meter, they cannot be described as a singular number, that would make no sense). So, by assuming they are the same, we can state that is equal to 1, or that the value is dimensionless (unit of 1). But how to interpret that intuitively?
My primary question is: "So, that can be stated as: On every unit of volume there comes $5$... What? $5$ what?" If we drop the units, it seems to me that this can be replaced by anything that is dimensionless ($Pa/Pa$, $kg/kg$, $m^2/m^2$).
This seems to fall into dimensional analysis, but I can't create the tag.