International System of Quantities

The International System of Quantities (ISQ) is a set of quantities and the equations that relate them describing physics and nature, as used in modern science, officialized by the International Organization for Standardization (ISO) by year 2009. This system underlies the International System of Units (SI), being more general: it does not specify the unit of measure chosen for each quantity. The name is used by the General Conference on Weights and Measures (CGPM) and standards bodies such as the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) to refer to this system, in particular with reference to a system that is consistent with the SI. The ISO standard describing the ISQ by 2009 is ISO/IEC 80000, which replaces the preceding standards ISO 31 and ISO 1000 published in 1992.

The ISQ is an incompletely defined system. Working jointly, ISO and IEC have formalized use of parts of it by giving information and definitions concerning quantities, systems of quantities, units, quantity and unit symbols, and coherent unit systems, with particular reference to the ISQ. The ISO/IEC 80000 standard defines physical quantities that are measured with the SI units and also includes many other quantities in modern science and technology.

Base quantities

A base quantity is a physical quantity in a subset of a given system of quantities that is chosen by convention, where no quantity in the set can be expressed in terms of the others. The ISQ defines seven base quantities. The symbols for them, as for other quantities, are written in italics.

The dimension of a physical quantity does not include magnitude or units. The conventional symbolic representation of the dimension of a base quantity is a single upper-case letter in roman (upright) sans-serif type.

Base quantity	Symbol for quantity	Symbol for dimension	SI base unit	SI unit symbol
length	$l$	${\mathsf {L}}$	metre	m
mass	$m$	${\mathsf {M}}$	kilogram	kg
time	$t$	${\mathsf {T}}$	second	s
electric current	$I$	${\mathsf {I}}$	ampere	A
thermodynamic temperature	$T$	${\mathsf {\Theta }}$	kelvin	K
amount of substance	$n$	${\mathsf {N}}$	mole	mol
luminous intensity	$I_{\text{v}}$	${\mathsf {J}}$	candela	cd

Derived quantities

See also: Dimensional analysis and SI derived unit

A derived quantity is a quantity in a system of quantities that is a defined in terms of the base quantities of that system. The ISQ defines many derived quantities.

Dimensional expression of derived 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 ${\mathsf {L}}^{a}{\mathsf {M}}^{b}{\mathsf {T}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}$, 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 ${\mathsf {LT}}^{-1}$. 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 ${\mathsf {1}}$. 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	${\mathsf {1}}$
solid angle	${\mathsf {1}}$
frequency	${\mathsf {T}}^{-1}$
force	${\mathsf {LMT}}^{-2}$
pressure	${\mathsf {L}}^{-1}{\mathsf {MT}}^{-2}$
velocity	${\mathsf {LT}}^{-1}$
area	${\mathsf {L}}^{2}$
volume	${\mathsf {L}}^{3}$
acceleration	${\mathsf {LT}}^{-2}$

Logarithmic quantities

Level

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 logarithmic quantification 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

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

where $\ln$ 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

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

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.

Information entropy

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

See also

• Dimensional analysis
• ISO/IEC 80000
• List of physical quantities
• Quantity
  • Observable quantity
• International System of Units
• SI base unit

Further reading

• B. N. Taylor, Ambler Thompson, International System of Units (SI), National Institute of Standards and Technology 2008 edition, ISBN 1-4379-1558-2.