Physical Quantity
Definition1
A physical quantity refers to a property of a material or system that can be measured and quantified.
Explanation
Any physical quantity is expressed as the product of a numerical value and a unit of measurement, which can be abbreviated to value and unit, respectively. Specifically, for an arbitrary physical quantity $Z$, its value is denoted by $\left\{ Z \right\}$ enclosed in braces, and its unit by $\left[ Z \right]$ enclosed in brackets.
$$ Z = \left\{ Z \right\} \times [Z] = \left\{ Z \right\} [Z] $$
The product of two physical quantities $Z$ and $W$ is expressed as follows.
$$ ZW = (\left\{ Z \right\} \left\{ W \right\}) \times ([Z][W]) $$
Dimension
Dimension serves to distinguish different physical quantities; physical quantities that have different dimensions are of different kinds. It is a way of expressing the fundamental nature of a physical quantity, and it is also used to define the base quantities, derived quantities, and dimensionless quantities discussed below. There are seven fundamental dimensions as follows, and every physical quantity is expressed as a combination of them.
$$ \text{질량: } \mathsf{M} \quad \text{길이: } \mathsf{L} \quad \text{시간: } \mathsf{T} \quad \text{전류: } \mathsf{I} $$ $$ \text{온도: } \mathsf{\Theta} \quad \text{물질의 양: } \mathsf{N} \quad \text{광도: } \mathsf{J} $$
Unit
Unit refers to the standard used to assign a value to a physical quantity. It is indispensable for gauging the magnitude of a physical quantity.
In summary, dimension is a concept for distinguishing physical quantities, while unit is a concept for assigning and expressing the values of physical quantities.
Base Quantities
base quantities are a subset of physical quantities that cannot be expressed in terms of any other physical quantity. All other physical quantities can be derived from the base quantities. The international system of quantities (ISQ) specifies the following $7$ base quantities.
| Physical quantity | SI unit | Dimension symbol | ||
|---|---|---|---|---|
| Name | Symbol | Name | Symbol | |
| Length | $l$, $x$, $r$ | metre | $\mathrm{m}$ | $\mathsf{L}$ |
| Time | $t$ | second | $\mathrm{s}$ | $\mathsf{T}$ |
| Mass | $m$ | kilogram | $\mathrm{kg}$ | $\mathsf{M}$ |
| Temperature | $T$ | kelvin | $\mathrm{K}$ | $\mathsf{\Theta}$ |
| Amount of substance | $n$ | mole | $\mathrm{mol}$ | $\mathsf{N}$ |
| Electric current | $i$, $I$ | ampere | $\mathrm{A}$ | $\mathsf{I}$ |
| Luminous intensity | $I_{\mathrm{v}}$ | candela | $\mathrm{cd}$ | $\mathsf{J}$ |
Therefore the dimension of an arbitrary physical quantity $Z$ is expressed as a product of powers of the base quantities as follows.
$$ [Z] = \left[ \mathsf{L} \right]^{\alpha} \left[ \mathsf{T} \right]^{\beta} \left[ \mathsf{M} \right]^{\gamma} \left[ \mathsf{\Theta} \right]^{\delta} \left[ \mathsf{N} \right]^{\epsilon} \left[ \mathsf{I} \right]^{\zeta} \left[ \mathsf{J} \right]^{\eta} $$
From an algebraic viewpoint, the base quantities can be likened to a basis of a vector space. Any vector $v \in V$ of a vector space $V$ is expressed using a basis $\left\{ e_{i} \right\}$ as follows. $$ v = \sum_{i} c_{i} e_{i} $$ Just as a basis of a vector space can express every vector through combinations of its elements, the base quantities can likewise express all other physical quantities through their combinations.
Logically, they can be likened to axioms. In mathematics, an axiom is a proposition that is accepted as true without being derived or proved from any other proposition. While velocity can be explained in terms of length and time, and momentum in terms of mass and velocity, the base quantities mass and time must be understood and accepted in themselves.
Derived Quantities
derived quantities are physical quantities defined as combinations of the base quantities.
Scalar
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| Mass | $m$ | kilogram $\mathrm{kg}$ | $\mathsf{M}$ |
| Time | $t$ | second $\mathrm{s}$ | $\mathsf{T}$ |
| Volume | $V$ | $\mathrm{m}^{3}$ | $\mathsf{L}^{3}$ |
| Density | $\rho$ | $\mathrm{kg/m^{3}}$ | $\mathsf{ML}^{-3}$ |
Vector
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| Position | $x$, $\mathbf{x}$ | - | - |
| Velocity | $v$, $\mathbf{v}$ | - | - |
| Acceleration | $a$, $\mathbf{a}$ | - | - |
| Force | $F$, $\mathbf{F}$ | - | - |
| Momentum | $p$, $\mathbf{p}$ | - | - |
Tensor
| Name | Symbol | SI unit | Dimension |
|---|---|---|---|
| Cauchy stress tensor | $\sigma$ | - | - |
| Maxwell stress tensor | $\mathbf{T}$ | - | - |
Dimensionless Quantities
Dimensionless quantities are physical quantities whose dimension is $1$, that is, whose exponents for all constituent dimensions are $0$. If $A$ is a dimensionless quantity, then
$$ \left[ A \right] = \left[ \mathsf{L} \right]^{0} \left[ \mathsf{T} \right]^{0} \left[ \mathsf{M} \right]^{0} \left[ \mathsf{\Theta} \right]^{0} \left[ \mathsf{N} \right]^{0} \left[ \mathsf{I} \right]^{0} \left[ \mathsf{J} \right]^{0} = 1 $$