What is a dimension in physics? 📂Physics

What is a dimension in physics?

Definition¹

In physics, a dimension denotes the kind (or fundamental nature) of a physical quantity, such as mass, length, or time.

Explanation

The most representative and basic physical quantities, as stated in the definition, are mass, length, and time. In particular, most physical quantities encountered in classical mechanics are combinations of these three dimensions.

If two values have different dimensions, it means they are different kinds of physical quantities. Different kinds of physical quantities cannot be added or subtracted. The dimensions of mass, length, and time are denoted by their English initial letters $\mathsf{M}$, $\mathsf{L}$, $\mathsf{T}$, respectively. It is common to use uppercase sans-serif type for these. They are also sometimes written enclosed in square brackets. For example, the dimension of acceleration $a$ is shown below.

$$ \dim a = \dfrac{\mathsf{L}/\mathsf{T}}{\mathsf{T}} = \mathsf{L}\mathsf{T}^{-2} $$

A related concept is 🔒(26/04/19)unit. The main difference in notation is that units are written in lowercase roman type, while dimensions are written in uppercase sans-serif. (Of course, units derived from personal names are sometimes capitalized.) Be careful not to confuse physical quantities, dimensions, and units.

Physical quantity	Dimension	SI unit
Mass	$\mathsf{M}$	kilogram $\mathrm{kg}$
Length	$\mathsf{L}$	meter $\mathrm{m}$
Time	$\mathsf{T}$	second $\mathrm{s}$
Electric current	$\mathsf{I}$	ampere $\mathrm{A}$
Temperature	$\mathsf{\Theta}$	kelvin $\mathrm{K}$
Amount of substance	$\mathsf{N}$	mole $\mathrm{mol}$
Luminous intensity	$\mathsf{J}$	candela $\mathrm{cd}$

Units vs. dimensions

If dimensions indicate the kind of physical quantity, units indicate the scale of the quantity. Dimensions do not change when the measuring instrument changes, but units (and the numerical values expressed) change depending on the measuring tool. For example, if a rod of length $1\mathrm{m}$ is measured with a $1\mathrm{m}$-long ruler, the measured value is $1\mathrm{m}$; if measured with a $1\mathrm{cm}$-long ruler, it is $100\mathrm{cm}$. However, the fact that its dimension is $\mathsf{L}$ does not change.

In summary, dimensions are a concept for distinguishing kinds of physical quantities, while units are a concept for assigning and expressing numerical values of those quantities.