International System of Quantities

The conventional symbolic representation of the dimension of a derived quantity is the product of powers of the dimensions of the base quantities according to the definition of the derived quantity. The dimension of a quantity is denoted by ${\displaystyle {\mathsf {L}}^{a}{\mathsf {M}}^{b}{\mathsf {T}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}}$, where the dimensional exponents are positive, negative, or zero. The symbol may be omitted if its exponent is zero. For example, in the ISQ, the quantity dimension of velocity is denoted ${\displaystyle {\mathsf {LT}}^{-1}}$. The following table lists some quantities defined by the ISQ.

A quantity of dimension one is historically known as a dimensionless quantity (a term that is still commonly used); all its dimensional exponents are zero and its dimension symbol is ${\displaystyle {\mathsf {1}}}$. Such a quantity can be regarded as a derived quantity in the form of the ratio of two quantities of the same dimension.

Derived quantity	Expression in SI base dimensions
plane angle	${\displaystyle {\mathsf {1}}}$
solid angle	${\displaystyle {\mathsf {1}}}$
frequency	${\displaystyle {\mathsf {T}}^{-1}}$
force	${\displaystyle {\mathsf {LMT}}^{-2}}$
pressure	${\displaystyle {\mathsf {L}}^{-1}{\mathsf {MT}}^{-2}}$
velocity	${\displaystyle {\mathsf {LT}}^{-1}}$
area	${\displaystyle {\mathsf {L}}^{2}}$
volume	${\displaystyle {\mathsf {L}}^{3}}$
acceleration	${\displaystyle {\mathsf {LT}}^{-2}}$

While not included as a SI Unit in the International System of Quantities, several ratio measures are included by the International Committee for Weights and Measures (CIPM) as acceptable in the "non-SI unit" category. The level of a quantity is a logarithm of the ratio of the quantity with a stated reference value of that quantity. It is differently defined for a root-power quantity (also known by the deprecated term field quantity) and for a power quantity. It is not defined for ratios of quantities of other kinds.

The level of a root-power quantity ${\textstyle F}$ with reference to a reference value of the quantity ${\textstyle F_{0}}$ is defined as

${\displaystyle L_{F}=\ln {\frac {F}{F_{0}}},}$

where ${\displaystyle \ln }$ is the natural logarithm. The level of a power quantity ${\textstyle P}$ with reference to a reference value of the quantity ${\textstyle P_{0}}$ is defined as

${\displaystyle L_{P}=\ln {\sqrt {\frac {P}{P_{0}}}}={\frac {1}{2}}\ln {\frac {P}{P_{0}}}.}$

When the natural logarithm is used, as it is here, use of the neper (symbol Np) is recommended, a unit of dimension 1 with Np = 1. The neper is coherent with SI. Use of the logarithm base 10 in association with a scaled unit, the bel (symbol B), where ${\textstyle {\text{B}}=\left({\frac {1}{2}}\ln 10\right){\text{Np}}\approx 1.151293~{\text{Np}}}$.

An example of level is sound pressure level. Within the ISQ, all levels are treated as derived quantities of dimension 1 and thus are not approved SI units per se, but rather are included in Table 8 of non-SI units that are approved for use outside the SI.[7]

The ISQ recognizes another logarithmic quantity: information entropy, for which the coherent unit is the natural unit of information (symbol nat).