For example: "one out of every 10 apples I gather is rotten." The rotten-to-gathered ratio is [1 apple] / [10 apples] = 0.1, which is a dimensionless quantity.
Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Buckingham π-theorem of dimensional analysis, the functional dependence between a certain number (e.g.: n) of variables can be reduced by the number (e.g. k) of independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. For the purposes of the experimenter, different systems which share the same description by dimensionless numbers are equivalent.
The power-consumption of a stirrer with a particular geometry is a function of the density and the viscosity of the fluid to be stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n=5 variables representing our example.
Those n=5 variables are built up from k=3 dimensions which are:
According to the π-theorem, the n=5 variables can be reduced by the k=3 dimensions to form p=n-k=5-3=2 independent dimensionless numbers which are in case of the stirrer
There are literally thousands (to be precise: infinite) dimensionless numbers including those being used most often: (in alphabetical order, indicating their field of use)