Dimensional Quantities, Dimensionless Quantities, Principle of Homogeneity

Classifying by dimension. Every physical quantity falls into exactly one of four categories:

1. Dimensional variables -- have dimensions AND take variable numeric values (length, velocity, acceleration).
2. Dimensionless variables -- have no dimensions, but still take variable numeric values (specific gravity, strain, refractive index).
3. Dimensional constants -- have dimensions but a fixed numeric value (the gravitational constant $G$, Planck's constant $h$).
4. Dimensionless constants -- neither dimensioned nor variable ($\pi$, $e$, pure counting numbers). …