# Dimensions

In real-world problems, variables will often represent quantities such as length, area, time, speed, and so on. It is meaningless to add or subtract different types of quantities, though we can multiply or divide them. If we replace every variable in a purported formula by the units that would be used for the quantity it represents, we can check that every term we are adding or subtracting in the formula has the same units. We can also use this technique to work out some physical formulae, at least up to a constant factor.

Quantities which represent pure numbers, such as $\pi$ or angles, are called “dimensionless”. (Angles measured in radians are calculated as one length divided by another, so have no units.)