dimensional analysis

Dimensional analysis is the habit of checking the 'kind' of each quantity (is it a length, a time, a mass?) to make sure an equation makes sense before you trust its numbers. You would never add 3 apples to 2 hours; dimensional analysis is the same common sense made into a tool. Every physical quantity has a dimension: length [L], mass [M], time [T], and combinations of them.

The core rule: both sides of any correct equation must have the same dimensions, and you can only add or subtract quantities of the same dimension. Speed has dimension length over time, [L]/[T]; multiply by a time [T] and you get a length [L], which matches distance = speed times time. You can even guess the form of a law: if a pendulum's period T depends on length L and gravity g (units m/s^2), the only combination with the dimension of time is sqrt(L/g), and indeed T = 2 pi sqrt(L/g).

Dimensional analysis catches many mistakes for free and even lets you reconstruct half-forgotten formulas, but it has limits: it cannot find pure numbers like the 2 pi or the 1/2, since those are dimensionless. It tells you the shape of the answer, not its exact numerical coefficient.