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?

  • $\begingroup$ No, you don't need a unit conversion to state that $6$ apples and $6$ bananas are the same number of fruits. There's also no choice of which unit to choose here: while length, mass etc. don't come with a unit and one needs to be defined in order to specify a length or mass, the number of coins doesn't need a unit to be specified. $\endgroup$
    – Christoph
    May 21, 2021 at 7:10

1 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 $\alpha\approx\frac1{137}$.


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