Do there exist dimensionless quantities in dimensional analysis?

Question

In physics and science in general researchers use dimensional analysis and they often state that some quantities are dimensionless like for example the number of some objects but I don't think this is true here's why.

Consider a typical problem, a coin has diameter $x\,\textrm m$ and we place a number of coins next to each other in a straight line which has length $y\,\textrm m$, find the number of coins.

The answer is fairly simple we just do $$\frac{y\,\textrm m}{x\,\textrm m}=\frac{y}{x}$$ which looks like it's a dimensionless quantity, but now let's consider the unit $[c]$ that measures the number of coins, so $1 [c]$ means one coin and so on, then $x$ is not measured in meters but in meters per coin, so $$\frac{y\,\textrm m}{x\,\textrm m/[c]}=\frac{y}{x} [c]$$ which is not dimensionless.

The same thing can be done to any problem, we can define a unit of measurement to measure anything.

Is my reasoning correct?

Answer 1

Dimensionality is always with respect to some unit system – and the typical one, SI, does not include "coin" as a unit. Thus if you add "coin" to SI, the number of coins is no longer a dimensionless quantity. However, the SI does include a unit which is very similar to your "coin" unit, in that it merely measures a count of objects, and which some scientists have argued should not be in SI – the mole.

Another example of the relativity of dimension lies in electromagnetism: the constant of proportionality in Coulomb's law has a dimension in SI but is simply $1$ in electrostatic cgs (centimetre/gram/second) and $\frac1{4\pi}$ in Gaussian cgs.

Regardless, there do exist physical quantities that are inherently dimensionless – it is simply not sensible to define a unit from them. The prime example is the fine-structure constant $\alpha\approx\frac1{137}$.