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{{short description|System of quantities used in science and their interrelationships}}

The '''International System of Quantities''' ('''ISQ''') is a standard system of [[Quantity|quantities]] used in [[physics]] and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities.{{efn|"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."<ref name="ISO 80000-1:2009">{{citation|title=ISO 80000-1:2009 Quantities and units. Part 1: General |date=2009-11-15 |publisher=ISO (the International Organization for Standardization) |location=Switzerland |page=vi |edition=1st |url=https://www.iso.org/obp/ui/#iso:std:iso:80000:-1:ed-1:v1:en |access-date=23 May 2015}}</ref>}}  This system underlies the [[International System of Units]] (SI){{efn|"The revised harmonized standard will be known as ISO/IEC 80000, ''Quantities and Units'', in which it is proposed that the quantities and equations used with the SI will be known as the International System of Quantities."<ref>{{citation |title=NIST Special Publication 330 – 2008 edition |url=https://www.nist.gov/publications/international-system-units-si-2008-edition |page=10 }}</ref>}} but does not itself determine the units of measurement used for the quantities.

The system is formally described in a multi-part [[ISO standard]] [[ISO/IEC&nbsp;80000]] (which also defines many other quantities used in science and technology), first completed in 2009 and subsequently revised and expanded.
[[File:Unit_relations_in_the_new_SI_planar.svg|thumb|Metrological dependencies be&shy;tween the base units of the SI system]]

==Base quantities==
The '''base quantities''' of a given system of [[physical quantity|physical quantities]] is a subset of those quantities, where no base quantity can be expressed in terms of the others, but where every 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.<ref name="ISO 80000-1:2009"/>

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 (typeface)|roman]] (upright) [[sans-serif]]{{efn|The status of the requirement for sans-serif is not as clear, since [[ISO 80000-1|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.").}} type.

{| class="wikitable"
|-
! Base quantity
! Symbol for quantity{{efn|name=ref_only|The associated quantity symbol, the SI unit name and SI unit symbol are given here for reference only; they do not form part of the ISQ.}}
! Symbol for dimension
! [[SI base unit]]{{efn|name=ref_only}}
! SI unit symbol{{efn|name=ref_only}}
|-
| [[length]]
| align="center" | <math>l</math>
| align="center" | <math>\mathsf L</math>
| [[metre]]
| m
|-
| [[mass]]
| align="center" | <math>m</math>
| align="center" | <math>\mathsf M</math>
| [[kilogram]]
| kg
|-
| [[time]]
| align="center" | <math>t</math>
| align="center" | <math>\mathsf T</math>
| [[second]]
| s
|-
| [[electric current]]
| align="center" | <math>I</math>
| align="center" | <math>\mathsf I</math>
| [[ampere]]
| A
|-
| [[thermodynamic temperature]]
| align="center" | <math>T</math>
| align="center" | <math>\mathsf \Theta</math>
| [[kelvin]]
| K
|-
| [[amount of substance]]
| align="center" | <math>n</math>
| align="center" | <math>\mathsf N</math>
| [[mole (unit)|mole]]
| mol
|-
| [[luminous intensity]]
| align="center" | <math>I_\text{v}</math>
| align="center" | <math>\mathsf J</math>
| [[candela]]
| cd
|}

==Derived quantities==
{{see also|Dimensional analysis|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 unit]]s.

===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 <math>\mathsf L^a \mathsf M^b \mathsf T^c \mathsf I^d \mathsf \Theta^e \mathsf N^f \mathsf J^g</math>, 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 <math>\mathsf{LT}^{-1}</math>. The following table lists some quantities defined by the ISQ.

{| class="wikitable"
|-
! Derived quantity
! Expression in SI base dimensions
|-
| [[frequency]]
| <math>\mathsf T^{-1}</math>
|-
| [[force]]
| <math>\mathsf{LMT}^{-2}</math>
|-
| [[pressure]]
| <math>\mathsf L^{-1} \mathsf {MT}^{-2}</math>
|-
| [[velocity]]
| <math>\mathsf{LT}^{-1}</math>
|-
| [[area]]
| <math>\mathsf L^2</math> 
|-
| [[volume]]
| <math>\mathsf L^3</math> 
|-
| [[acceleration]]
| <math>\mathsf{LT}^{-2}</math> 
|}

====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 <math>1</math>. 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]].<ref name=80000-3:2019>{{cite web|title=ISO 80000-3:2019|url=https://www.iso.org/standard/64974.html|publisher=[[International Organization for Standardization]]|access-date=2019-10-23}}</ref>

===Logarithmic quantities===
====Level====
{{main|Level (logarithmic quantity)}}
{{further|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.  Within the ISQ, all levels are treated as derived quantities of dimension 1.{{citation needed|reason=is this really true?|date=February 2022}}  Several units for levels are defined by the SI and classified as "non-SI units accepted for use with the SI units".<ref>{{SIbrochure9th|page=145}}</ref>
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 (log scale)|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).{{cn|date=December 2017}}

==Documentation==
The system is formally described in a multi-part [[ISO standard]] [[ISO/IEC&nbsp;80000]], first completed in 2009 but subsequently revised and expanded, which replaced standards published in 1992, [[ISO&nbsp;31]] and [[ISO&nbsp;1000]].  Working jointly, ISO and 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&nbsp;80000 defines physical [[quantity|quantities]] that are measured with the [[International System of Units|SI units]]<ref>{{cite book|title=International vocabulary of metrology – Basic and general concepts and associated terms (VIM)|year=2012|publisher=International Bureau of Weights and Measures (BIPM):Joint Committee for Guides in Metrology|edition=3rd|chapter-url=http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf|access-date=28 March 2015|chapter=1.16}}</ref> and also includes many other quantities in modern science and technology.<ref name="ISO 80000-1:2009"/> 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]]

==Notes==
{{notelist}}

==References==
{{reflist|30em}}

==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}}.

{{SI base quantities}}

[[Category:Physical quantities|+]]
[[Category:Measurement]]
[[Category:International standards]]