Physical quantities carry both a numerical value and a physical dimension. Dimensions describe the kind of quantity independently of the chosen units: length has dimension $L$, time $T$, mass $M$, electric current $I$, and derived quantities combine these powers. For example, velocity has dimension $LT^{-1}$ and force has dimension $MLT^{-2}$.

A physically meaningful equation must be dimensionally homogeneous: terms that are added or equated must have the same dimensions. This gives a powerful error check before any numerical calculation.

Suppose a proposed formula for the distance fallen from rest is $$x=\frac12 gt,$$ where $g$ is acceleration. The right-hand side has dimensions $$[g][t]=(LT^{-2})T=LT^{-1},$$ which is velocity, not length. The expression therefore cannot be correct. Replacing $t$ by $t^2$ gives $$x=\frac12 gt^2,$$ with dimensions $LT^{-2}T^2=L$.

Dimensionless quantities have no physical dimension. Ratios such as $v/c$, angles in radians, strain, and many normalized parameters are dimensionless even though their numerical values may still depend on conventions. Functions such as $\exp$, $\sin$, and $\log$ require dimensionless arguments: writing $e^{-t}$ is incomplete if $t$ has units, whereas $e^{-t/\tau}$ is valid when $\tau$ is a time.

Dimensional consistency is necessary but not sufficient for correctness: both $x=gt^2$ and $x=\tfrac12gt^2$ have the right dimensions. It can reject impossible formulas and constrain possible forms, but numerical constants and detailed functional relationships require additional physics.