International System of Quantities

Metrological dependencies be­tween the base units of the SI

The International System of Quantities (ISQ) is a standard system of quantities used in physics and in modern science in general. It includes seven ISQ base quantities – length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity – and the relationships between those quantities in derived quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities. The system is formally described in a multi-part standard ISO/IEC 80000, which also defines many other derived quantities used in science and technology, first completed in 2009 and subsequently revised and expanded.

Base quantities

The base quantities of a given system of physical quantities form a subset in which no base quantity can be expressed in terms of the others, while every other quantity in the system can be expressed in terms of the base quantities. Within this constraint, the set of base quantities is chosen by convention. There are seven ISQ 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 dimension	Symbol for quantity	SI base unit	SI unit symbol
length	${\mathsf {L}}$	$l$	metre	m
mass	${\mathsf {M}}$	$m$	kilogram	kg
time	${\mathsf {T}}$	$t$	second	s
electric current	${\mathsf {I}}$	$I$	ampere	A
thermodynamic temperature	${\mathsf {\Theta }}$	$T$	kelvin	K
amount of substance	${\mathsf {N}}$	$n$	mole	mol
luminous intensity	${\mathsf {J}}$	$I_{\text{v}}$	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 defined in terms of only the base quantities of that system. The ISQ defines many derived quantities and corresponding derived units.

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 dimension 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. For more, see List of physical quantities.

Derived quantity	Expression in SI base dimensions
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	${\mathsf {LT}}^{-2}$

Dimensionless quantities

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 $1$. Such a quantity can be regarded as a derived quantity in the form of the ratio of two quantities of the same dimension. The named dimensionless units "radian" (rad) and "steradian" (sr) are acceptable for distinguishing dimensionless quantities of different kind, respectively plane angle and solid angle.

Logarithmic quantities

Level

Main article: Level (logarithmic quantity)

Further information: Power, root-power, and field quantities

The level of a quantity is defined as the logarithm of the ratio of the quantity with a stated reference value of that quantity. Within the ISQ 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. Several units for levels are defined by the SI and classified as "non-SI units accepted for use with the SI units". An example of level is sound pressure level, with the unit of decibel.

Other logarithmic quantities

Units of logarithmic frequency ratio include the octave, corresponding to a factor of 2 in frequency (precisely) and the decade, corresponding to a factor 10.

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

Documentation

The system is formally described in the multi-part standard ISO/IEC 80000, first completed in 2009 but subsequently revised and expanded, which replaced standards published in 1992, ISO 31 and ISO 1000. Working jointly, International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC) have formalized parts of the ISQ 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. ISO/IEC 80000 defines physical quantities that are expressed in terms SI units and also includes many other quantities in modern science and technology. The name "International System of Quantities" is used by the General Conference on Weights and Measures (CGPM) to describe the system of quantities that underlie the International System of Units.

See also

• Dimensional analysis
• List of physical quantities
• International System of Units
• SI base unit