In physics and all applied sciences, quantities are measured all the time. Most of the time, quantities are not represented by pure numbers, rather they are expressed with a physical dimension relevant to the context from which they arose. As a simple example, if you were to ask someone for his height, you would expect to get a number with some unit that measures the physical dimension of length, as in 1.82 meters or 6 foot or 182 cm or 0.00182 km. The commonality between all of these answers is that they are measuring length, and this physical dimension can be measured in many different units. One will be very surprised and confused if the response to that question were “I am 23 kilograms long”, since a kilogram is another unit that measures a different physical dimension, we call that other dimension mass.
It is logical to add or subtract equal dimensions, say 4 kg + 10 kg is 14 kg, but adding different dimensions provides no meaning. What could be the interpretation of 10 meters + 20 seconds? However, multiplying and dividing dimensions does have meaning, when we say that the speed of a car is 100 km/h we are taking the dimension of length and dividing it by the dimension of time.
In the study of motion and forces (mechanics), there are three basic dimensions and from them, all other dimensions are derived. These are Length [ L ], Time [ T ] and Mass [ M ]. Let us consider the dimensions of Speed. To measure that quantity, we need to measure a distance and a time; thus, we can say that the dimensions of speed are
Likewise, all the other physical quantities in mechanics (momentum, energy, torque, acceleration, gravitational constant, density and so on) can be constructed by multiplying powers of the three basic dimensions, that is if X is a quantity, then
By now, you can probably think of other quantities that cannot be expressed in terms of these three basic dimensions. Some of these include electric current and magnetic fields. In order to express these, we need another fundamental dimension called Charge [ Q ].
Knowing the dimensions of quantities can often help us discard wrong expressions. After all, if you are solving a problem that asks for a quantity that has dimensions of length squared per mass and your expression yields dimensions of Time per Length, then you know for certain that the expression is wrong as the dimensions are in disagreement. The process of checking dimensional consistency in expressions is called dimensional analysis.