# 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.[1] 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.[2][3] 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[4] and also includes many other quantities in modern science and technology.[5]

## 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.[6]

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[7] type.

Base quantity Symbol for quantity[8] Symbol for dimension SI base unit[8] SI unit symbol[8]
length ${\displaystyle l}$  ${\displaystyle {\mathsf {L}}}$  metre m
mass ${\displaystyle m}$  ${\displaystyle {\mathsf {M}}}$  kilogram kg
time ${\displaystyle t}$  ${\displaystyle {\mathsf {T}}}$  second s
electric current ${\displaystyle I}$  ${\displaystyle {\mathsf {I}}}$  ampere A
thermodynamic temperature ${\displaystyle T}$  ${\displaystyle {\mathsf {\Theta }}}$  kelvin K
amount of substance ${\displaystyle n}$  ${\displaystyle {\mathsf {N}}}$  mole mol
luminous intensity ${\displaystyle I_{\text{v}}}$  ${\displaystyle {\mathsf {J}}}$  candela cd

## Derived quantities

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 ${\displaystyle {\mathsf {L}}^{a}{\mathsf {M}}^{b}{\mathsf {T}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}}$ , 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 ${\displaystyle {\mathsf {LT}}^{-2}}$

### 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

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

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.[9]

#### Information entropy

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

## References

1. ^ ISO 80000-1:2009, The system of quantities, including the relations among them the quantities used as the basis of the units of the SI, is named the International System of Quantities, denoted 'ISQ', in all languages. [...] It should be realized, however, that ISQ is simply a convenient notation to assign to the essentially infinite and continually evolving and expanding system of quantities and equations on which all of modern science and technology rests. ISQ is a shorthand notation for the 'system of quantities on which the SI is based', which was the phrase used for this system in ISO 31.
2. ^ The International System of Units (SI) (PDF) (8th ed.), 2006, p. 104, retrieved 2020-08-02, The system of quantities, including the equations relating the quantities, to be used with the SI, is in fact just the quantities and equations of physics that are familiar to all scientists, technologists, and engineers. They are listed in many textbooks and in many references, but any such list can only be a selection of the possible quantities and equations, which is without limit.
3. ^ The International System of Units (PDF) (9th ed.), BIPM, 2019, p. 125, retrieved 2020-08-02, The system of quantities underlying the SI and the equations relating them are based on the present description of nature and are familiar to all scientists, technologists and engineers.
4. ^ "1.16" (PDF). International vocabulary of metrology – Basic and general concepts and associated terms (VIM) (3rd ed.). International Bureau of Weights and Measures (BIPM):Joint Committee for Guides in Metrology. 2012. Retrieved 28 March 2015.
5. ^ ISO 80000-1 Quantities and units. Part 1: General (1st ed.). Switzerland: ISO (the International Organization for Standardization). 2009-11-15. p. vi. Retrieved 23 May 2015.
6. ^ ISO 80000-1:2009
7. ^ The status of the requirement for sans-serif is not as clear, since ISO 80000-1:2009 makes no mention of it ("The conventional symbolic representation of the dimension of a base quantity is a single upper case letter in roman (upright) type.") whereas the secondary source BIPM JCGM 200:2012 does ("The conventional symbolic representation of the dimension of a base quantity is a single upper case letter in roman (upright) sans-serif type.").
8. ^ a b c The associated symbol and SI unit are given here for reference only; they do not form part of the ISQ.
9. ^ Chapter 4 – Units outside the SI