View source for International System of Units

{{short description|Modern form of the metric system}}
{{Redirect|SI|the chemical element with symbol Si|Silicon|other uses|SI (disambiguation)}}
{{Use British English|date=September 2024}}
{{Use dmy dates|date=September 2024}}
The '''International System of Units''', internationally known by the abbreviation '''SI''' (from French {{lang|fr|Système international d'unités}}), is the modern form of the [[metric system]] and the world's most widely used [[system of measurement]]. It is the only system of measurement with official status in nearly every country in the world, employed in science, technology, industry, and everyday commerce. The SI system is coordinated by the [[International Bureau of Weights and Measures]] which is abbreviated BIPM from {{langx|fr|Bureau international des poids et mesures}}.
[[File:SI Illustration Base Units and Constants Colour Full.svg|thumb|SI [[#SI base units|base units]] (outer ring) and [[#SI defining constants|constants]] (inner ring) ]]
{| class="wikitable floatright" style="width: 400px; text-align:center;"
|+ The seven SI base units
! scope="col" | Symbol
! scope="col" | Name
! scope="col" | Quantity
|-
! scope="row" | s
| [[second]] || [[time]] <!-- one of seven -->
|-
! scope="row" | m
| [[metre]] || [[length]] <!-- two of seven -->
|-
! scope="row" | kg
| [[kilogram]] || [[mass]] <!-- three of seven -->
|-
! scope="row" | A
| [[ampere]]  || [[electric current]] <!-- four of seven -->
|-
! scope="row" | K
| [[kelvin]] || [[thermodynamic temperature]] <!-- five of seven -->
|-
! scope="row" | mol
| [[mole (unit)|mole]] || [[amount of substance]] <!-- six of seven -->
|-
! scope="row" | cd
| [[candela]]|| [[luminous intensity]] <!-- seven of seven. That is all.  -->
|}
The SI comprises a [[Coherence (units of measurement)|coherent]] system of [[units of measurement]] starting with seven [[SI base unit|base unit]]s, which are the [[second]] (symbol s, the unit of [[time]]), [[metre]] (m, [[length]]), [[kilogram]] (kg, [[mass]]), [[ampere]] (A, [[electric current]]), [[kelvin]] (K, [[thermodynamic temperature]]), [[mole (unit)|mole]] (mol, [[amount of substance]]), and [[candela]] (cd, [[luminous intensity]]). The system can accommodate coherent units for an unlimited number of additional quantities. These are called coherent [[SI derived unit|derived units]], which can always be represented as products of powers of the base units. Twenty-two coherent derived units have been provided with special names and symbols.

The seven base units and the 22 coherent derived units with special names and symbols may be used in combination to express other coherent derived units. Since the sizes of coherent units will be convenient for only some applications and not for others, the SI provides twenty-four [[metric prefix|prefix]]es which, when added to the name and symbol of a coherent unit produce twenty-four additional (non-coherent) SI units for the same quantity; these non-coherent units are always decimal (i.e. power-of-ten) multiples and sub-multiples of the coherent unit.

The current way of defining the SI is a result of a decades-long move towards increasingly abstract and idealised formulation in which the [[realisation (metrology)|realisations]] of the units are separated conceptually from the definitions. A consequence is that as science and technologies develop, new and superior realisations may be introduced without the need to redefine the unit. One problem with artefacts is that they can be lost, damaged, or changed; another is that they introduce uncertainties that cannot be reduced by advancements in science and technology.

The original motivation for the development of the SI was the diversity of units that had sprung up within the [[centimetre–gram–second system of units|centimetre–gram–second]] (CGS) systems (specifically the inconsistency between the systems of [[electrostatic units]] and [[electromagnetic units]]) and the lack of coordination between the various [[discipline (academia)|discipline]]s that used them. The General Conference on Weights and Measures (French: ''{{lang|fr|Conférence générale des poids et mesures}}'' – CGPM), which was established by the [[Metre Convention]] of 1875, brought together many international organisations to establish the definitions and standards of a new system and to standardise the rules for writing and presenting measurements. The system was published in 1960 as a result of an initiative that began in 1948, and is based on the [[MKS system of units|metre–kilogram–second system of units]] (MKS) combined with ideas from the development of the CGS system.

== Definition ==

<!-- no wikilinks in this paragraph please -->
The International System of Units consists of a set of defining constants with corresponding base units, derived units, and a set of decimal-based multipliers that are used as prefixes.<ref name="SIBrochure9thEd"/>{{rp|page=125}}

=== SI defining constants ===

{| class="wikitable floatright" style="width: 400px; text-align:center;"
|+ SI defining constants
! scope="col" | Symbol
! scope="col" | Defining constant
! scope="col" | Exact value
|-
! scope="row" | {{math|Δ''ν''<sub>Cs</sub>}}
| [[Caesium standard|hyperfine transition frequency of <sup>133</sup>Cs]] || {{val|9192631770|u=Hz}}
|-
! scope="row" | {{mvar|c}}
| [[speed of light]] || {{val|299792458|u=m/s}}
|-
! scope="row" | {{mvar|h}}
| [[Planck constant]] || {{val|6.62607015|e=-34|u=J.s}}
|-
! scope="row" | {{mvar|e}}
| [[elementary charge]] || {{val|1.602176634|e=-19|u=C}}
|-
! scope="row" | {{mvar|k}}
| [[Boltzmann constant]] || {{val|1.380649|e=-23|u=J/K}}
|-
! scope="row" | {{Math|''N''<sub>A</sub>}}
| [[Avogadro constant]] || {{val|6.02214076|e=23|u=mol-1}}
|-
! scope="row" | {{Math|''K''<sub>cd</sub>}}
| [[luminous efficacy]] of {{val|540|u=THz}} radiation || {{val|683|u=lm/W}}
|}
The seven defining constants are the most fundamental feature of the definition of the system of units.<ref name="SIBrochure9thEd"/>{{rp|page=125}}
The magnitudes of all SI units are defined by declaring that seven constants have certain exact numerical values when expressed in terms of their SI units. These defining constants are the [[speed of light]] in vacuum {{Math|''c''}}, the [[caesium standard|hyperfine transition frequency of caesium]] {{math|Δ''ν''{{sub|Cs}}}}, the [[Planck constant]] {{Math|''h''}}, the [[elementary charge]] {{Math|''e''}}, the [[Boltzmann constant]] {{Math|''k''}}, the [[Avogadro constant]] {{Math|''N''<sub>A</sub>}}, and the [[luminous efficacy]] {{Math|''K''<sub>cd</sub>}}. The nature of the defining constants ranges from fundamental constants of nature such as {{Math|''c''}} to the purely technical constant {{Math|''K''<sub>cd</sub>}}. The values assigned to these constants were fixed to ensure continuity with previous definitions of the base units.<ref name="SIBrochure9thEd"/>{{rp|page=128}}

=== SI base units ===

{{Main|SI base unit}}

The SI selects seven units to serve as [[SI base unit|base unit]]s, corresponding to seven base physical quantities. They are the [[second]], with the symbol {{val|u=s}}, which is the SI unit of the physical quantity of [[time]]; the [[metre]], symbol {{val|u=m}}, the SI unit of [[length]]; [[kilogram]] ({{val|u=kg}}, the unit of [[mass]]); [[ampere]] ({{val|u=A}}, [[electric current]]); [[kelvin]] ({{val|u=K}}, [[thermodynamic temperature]]); [[Mole (unit)|mole]] ({{val|u=mol}}, [[amount of substance]]); and [[candela]] ({{val|u=cd}}, [[luminous intensity]]).<ref name="SIBrochure9thEd" />
The base units are defined in terms of the defining constants.  For example, the kilogram is defined by taking the Planck constant {{math|''h''}} to be {{val|6.62607015|e=-34|u=J.s}}, giving the expression in terms of the defining constants<ref name="SIBrochure9thEd" />{{rp|page=131}}
: {{nowrap|1={{val|1|u=kg}} = {{sfrac|({{val|299792458}}){{sup|2}}|({{val|6.62607015|e=-34}})({{val|9192631770}})}}{{sfrac|{{math|''h''}}{{thin space}}{{math|Δ''ν''{{sub|Cs}}}}|{{math|''c''}}{{sup|2}}}}.}}
All units in the SI can be expressed in terms of the base units, and the base units serve as a preferred set for expressing or analysing the relationships between units. The choice of which and even how many quantities to use as base quantities is not fundamental or even unique – it is a matter of convention.<ref name="SIBrochure9thEd" />{{rp|page=126|quote=[...] the choice of the base units was never unique, but grew historically and became familiar to users of the SI}}

{| class="wikitable" style="margin:1em auto 1em auto"
|+ <big>SI base units</big><ref name="SIBrochure9thEd"/>{{rp|page=136}}
|-
 ! scope="col" | Unit name
 ! scope="col" | Unit symbol
 ! scope="col" | [[Dimensional analysis#Definition|Dimension symbol]]
 ! scope="col" | [[Physical quantity|Quantity name]]
 ! scope="col" width=80 |Typical symbols
 ! scope="col" | Definition
|-
 ! scope="row" | [[second]]
 |style="text-align:center" |s
 |style="text-align:center" |<math>\mathsf{T}</math>
 | [[time]]
 |<math>t</math>
 |The duration of {{val|9192631770}} periods of the radiation corresponding to the transition between the two [[hyperfine structure|hyperfine]] levels of the [[ground state]] of the [[caesium-133]] atom.
|-
 ! scope="row" | [[metre]]
 |style="text-align:center" |m
 |style="text-align:center" |<math>\mathsf{L}</math>
 |[[length]]
 |<math>l</math>, <math>x</math>, <math>r</math>, etc.
 |The distance travelled by light in vacuum in {{sfrac|{{val|299792458}}}} second.
|-
 ! scope="row" | [[kilogram]]{{br}}<ref group="n">Despite the prefix "kilo-", the kilogram is the coherent base unit of mass, and is used in the definitions of derived units. Nonetheless, prefixes for the unit of mass are determined as if the gram were the base unit.</ref>
 |style="text-align:center" |kg
 |style="text-align:center" |<math>\mathsf{M}</math>
 |[[mass]]
 |<math>m</math>
 |The kilogram is defined by setting the [[Planck constant]] {{math|''h''}} to {{val|6.62607015|e=-34|u=J.s}} ({{nowrap|1=J = kg⋅m{{sup|2}}⋅s{{sup|−2}}}}), given the definitions of the metre and the second.<ref name="NIST 2018-11">{{cite news|url=https://www.nist.gov/news-events/news/2018/11/historic-vote-ties-kilogram-and-other-units-natural-constants|title=Historic Vote Ties Kilogram and Other Units to Natural Constants|last=Materese|first=Robin|date=16 November 2018|work=NIST|access-date=16 November 2018}}</ref>
|-
 ! scope="row" | [[ampere]]
 |style="text-align:center" |A
 |style="text-align:center" |<math>\mathsf{I}</math>
 |[[electric current]]
 |<math>I,\; i</math>
 |The flow of {{sfrac|1|{{val|1.602176634|e=-19}}}} times the [[elementary charge]] {{math|''e''}} per second, which is approximately {{val|6.2415090744|e=18}} elementary charges per second.
|-
 ! scope="row" | [[kelvin]]
 |style="text-align:center" |K
 |style="text-align:center" |<math>\mathsf{\Theta}</math>
 |[[thermodynamic temperature|thermodynamic{{br}}temperature]]
 |<math>T</math>
 |The kelvin is defined by setting the fixed numerical value of the [[Boltzmann constant]] {{math|''k''}} to {{val|1.380649|e=-23|u=J.K-1}}, ({{nowrap|1=J = kg⋅m<sup>2</sup>⋅s<sup>−2</sup>}}), given the definition of the kilogram, the metre, and the second.
|-
 ! scope="row" | [[mole (unit)|mole]]
 |style="text-align:center" |mol
 |style="text-align:center" |<math>\mathsf{N}</math>
 |[[amount of substance]]
 |<math>n</math>
 |The amount of substance of {{val|6.02214076|e=23}} elementary entities.<ref group="n">When the mole is used, the elementary entities must be specified and may be [[atom]]s, [[molecule]]s, [[ion]]s, [[electron]]s, other particles, or specified groups of such particles.</ref> This number is the fixed numerical value of the [[Avogadro constant]], {{math|''N''<sub>A</sub>}}, when expressed in the unit mol<sup>−1</sup>.
|-
 ! scope="row" | [[candela]]
 |style="text-align:center" |cd
 |style="text-align:center" |<math>\mathsf{J}</math>
 |[[luminous intensity]]
 |<math>I_{\rm v}</math>
 |The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency {{val|5.4|e=14|u=hertz}} and that has a radiant intensity in that direction of {{sfrac|1|683}} watt per [[steradian]].
|-
 | colspan="6" |
; Notes
{{reflist|group="n"}}
|}

=== Derived units ===

{{Main|SI derived unit}}
The system allows for an unlimited number of additional units, called ''[[SI derived unit|derived units]]'', which can always be represented as products of powers of the base units, possibly with a nontrivial numeric multiplier. When that multiplier is one, the unit is called a ''[[coherence (units of measurement)|coherent]]'' derived unit. For example, the coherent derived SI unit of [[velocity]] is the [[metre per second]], with the symbol {{val|u=m/s}}.<ref name=SIBrochure9thEd/>{{rp|page=139}} The base and coherent derived units of the SI together form a coherent system of units (''the set of coherent SI units''). A useful property of a coherent system is that when the numerical values of physical quantities are expressed in terms of the units of the system, then the equations between the numerical values have exactly the same form, including numerical factors, as the corresponding equations between the physical quantities.<ref name="ISO80000-1">{{cite ISO standard|csnumber=30669|title=ISO 80000-1:2009 Quantities and units – Part 1: General}}</ref>{{rp|page=6}}

Twenty-two coherent derived units have been provided with special names and symbols as shown in the table below. The radian and steradian have no base units but are treated as derived units for historical reasons.<ref name=SIBrochure9thEd/>{{rp|page=137}}

{| class="wikitable floatleft" style="margin:1em auto 1em auto;line-height:1.4"
|+ <big>The 22 SI derived units with special names and symbols</big><ref name="NIST330" />{{rp|page=15}}
|-
! scope="col" | Name
! scope="col" | Symbol
! scope="col" | Quantity
! scope="col" | In SI base units
! scope="col" | In other SI units
|-
! scope="row" | [[radian]]<ref name=":0" group="N">The radian and steradian are defined as dimensionless derived units.</ref>
| style="text-align:center;" | rad
| [[angle|plane angle]]
| style="text-align:center;" | m/m
| style="text-align:center;" | 1
|-
! scope="row" | [[steradian]]<ref name=":0" group="N" />
| style="text-align:center;" | sr
| [[solid angle]]
| style="text-align:center;" | m<sup>2</sup>/m<sup>2</sup>
| style="text-align:center;" | 1
|-
! scope="row" | [[hertz]]
| style="text-align:center;" | Hz
| [[frequency]]
| style="text-align:center;" | s<sup>−1</sup>
| <!-- intentionally left blank -->
|-
! scope="row" | [[newton (unit)|newton]]
| style="text-align:center;" | N
| [[force]] 
| style="text-align:center;" | kg⋅m⋅s<sup>−2</sup>
| <!-- intentionally left blank -->
|-
! scope="row" | [[pascal (unit)|pascal]]
| style="text-align:center;" | Pa
| [[pressure]], [[stress (physics)|stress]]
| style="text-align:center;" | kg⋅m<sup>−1</sup>⋅s<sup>−2</sup>
| style="text-align:center;" | N/m<sup>2</sup> = J/m<sup>3</sup>
|-
! scope="row" | [[joule]]
| style="text-align:center;" | J
| [[energy]], [[mechanical work|work]], amount of [[heat]]
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−2</sup>
| style="text-align:center;" | N⋅m = Pa⋅m<sup>3</sup>
|-
! scope="row" | [[watt]]
| style="text-align:center;" | W
| [[Power (physics)|power]], [[radiant flux]]
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−3</sup>
| style="text-align:center;" | J/s
|-
! scope="row" | [[coulomb]]
| style="text-align:center;" | C
| [[electric charge]]
| style="text-align:center;" | s⋅A
| <!-- intentionally left blank -->
|-
! scope="row" | [[volt]]
| style="text-align:center;" | V
| [[electric potential difference]]{{efn|Electric potential difference is also called "voltage" in many countries, as well as "electric tension" or simply "tension" in some countries.}}
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−1</sup>
| style="text-align:center;" | W/A = J/C
|-
! scope="row" | [[farad]]
| style="text-align:center;" | F
| [[capacitance]]
| style="text-align:center;" | kg<sup>−1</sup>⋅m<sup>−2</sup>⋅s<sup>4</sup>⋅A<sup>2</sup>
| style="text-align:center;" | C/V = C<sup>2</sup>/J
|-
! scope="row" | [[ohm (unit)|ohm]]
| style="text-align:center;" | Ω
| [[electrical resistance]]
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−2</sup>
| style="text-align:center;" | V/A = J⋅s/C<sup>2</sup>
|-
! scope="row" | [[siemens (unit)|siemens]]
| style="text-align:center;" | S
| [[electrical conductance]]
| style="text-align:center;" | kg<sup>−1</sup>⋅m<sup>−2</sup>⋅s<sup>3</sup>⋅A<sup>2</sup>
| style="text-align:center;" | Ω<sup>−1</sup>
|-
! scope="row" | [[weber (unit)|weber]]
| style="text-align:center;" | Wb
| [[magnetic flux]]
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅A<sup>−1</sup>
| style="text-align:center;" | V⋅s
|-
! scope="row" | [[tesla (unit)|tesla]]
| style="text-align:center;" | T
| [[magnetic flux density]]
| style="text-align:center;" | kg⋅s<sup>−2</sup>⋅A<sup>−1</sup>
| style="text-align:center;" | Wb/m<sup>2</sup>
|-
! scope="row" | [[henry (unit)|henry]]
| style="text-align:center;" | H
| [[inductance]]
| style="text-align:center;" | kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅A<sup>−2</sup>
| style="text-align:center;" | Wb/A
|-
! scope="row" | [[degree Celsius]]
| style="text-align:center;" | °C
| [[Celsius temperature]]
| style="text-align:center;" | K
| <!-- intentionally left blank -->
|-
! scope="row" | [[lumen (unit)|lumen]]
| style="text-align:center;" | lm
| [[luminous flux]]
| style="text-align:center;" | cd⋅m<sup>2</sup>/m<sup>2</sup>
| style="text-align:center;" | cd⋅sr
|-
! scope="row" | [[lux]]
| style="text-align:center;" | lx
| [[illuminance]]
| style="text-align:center;" | cd⋅m<sup>2</sup>/m<sup>4</sup>
| style="text-align:center;" | lm/m<sup>2</sup> = cd⋅sr⋅m<sup>−2</sup>
|-
! scope="row" | [[becquerel]]
| style="text-align:center;" | Bq
| [[Radioactive decay|activity referred to a radionuclide]]
| style="text-align:center;" | s<sup>−1</sup>
| <!-- intentionally left blank -->
|-
! scope="row" | [[gray (unit)|gray]]
| style="text-align:center;" | Gy
| [[absorbed dose]], [[kerma (physics)|kerma]]
| style="text-align:center;" | m<sup>2</sup>⋅s<sup>−2</sup>
| style="text-align:center;" | J/kg
|-
! scope="row" | [[sievert]]
| style="text-align:center;" | Sv
| [[dose equivalent]]
| style="text-align:center;" | m<sup>2</sup>⋅s<sup>−2</sup>
| style="text-align:center;" | J/kg
|-
! scope="row" | [[katal]]
| style="text-align:center;" | kat <!-- Note: this is the last row, there are exactly 22 in the ref -->
| [[catalytic activity]]
| style="text-align:center;" | mol⋅s<sup>−1</sup>
| <!-- intentionally left blank -->
|-
<!-- Note: there are exactly 22 in the ref -->
| colspan="5" | '''Notes'''
<references group="N" />
|}
{{Clear}}

The derived units in the SI are formed by powers, products, or quotients of the base units and are unlimited in number.<ref name="SIBrochure">{{SIbrochure8th}}</ref>{{rp|page=103}}<ref name="NIST330"/>{{rp|pages=14,16}}

[[File:Physics measurements SI units.png|thumb|Arrangement of the principal measurements in physics based on the mathematical manipulation of length, time, and mass]]

[[Derived unit]]s apply to some [[derived quantity|derived quantities]], which may by definition be expressed in terms of [[base quantity|base quantities]], and thus are not independent; for example, [[electrical conductance]] is the inverse of [[electrical resistance]], with the consequence that the siemens is the inverse of the ohm, and similarly, the ohm and siemens can be replaced with a ratio of an ampere and a volt, because those quantities bear a defined relationship to each other.{{efn|Ohm's law: {{nowrap|1=1 Ω = 1 V/A}} from the relationship {{nowrap|1=''E'' = ''I'' × ''R''}}, where ''E'' is electromotive force or voltage (unit: volt), ''I'' is current (unit: ampere), and ''R'' is resistance (unit: ohm).}} Other useful derived quantities can be specified in terms of the SI base and derived units that have no named units in the SI, such as acceleration, which has the SI unit m/s<sup>2</sup>.<ref name="SIBrochure9thEd" />{{rp|page=139}}

A combination of base and derived units may be used to express a derived unit. For example, the SI unit of [[force]] is the [[newton (unit)|newton]] (N), the SI unit of [[pressure]] is the [[Pascal (unit)|pascal]] (Pa) – and the pascal can be defined as one newton per square metre (N/m<sup>2</sup>).<ref>{{cite web |title=Units & Symbols for Electrical & Electronic Engineers |url=http://www.theiet.org/students/resources/units-symbols.cfm |publisher=Institution of Engineering and Technology |date=1996 |pages=8–11 |access-date=19 August 2013 |archive-url=https://web.archive.org/web/20130628212624/http://www.theiet.org/students/resources/units-symbols.cfm |archive-date=28 June 2013}}</ref>

=== Prefixes ===

{{Main|Metric prefix}}

Like all metric systems, the SI uses [[metric prefix]]es to systematically construct, for the same physical quantity, a set of units that are decimal multiples of each other over a wide range. For example, driving distances are normally given in [[kilometre]]s (symbol {{val|u=km}}) rather than in metres. Here the metric prefix '[[kilo-]]' (symbol 'k') stands for a factor of 1000; thus, {{val|1|u=km}} = {{val|1000|u=m}}.

The SI provides twenty-four metric prefixes that signify decimal powers ranging from 10<sup>−30</sup> to 10<sup>30</sup>, the most recent being adopted in 2022.<ref name="SIBrochure9thEd" />{{rp|pages=143–144}}<ref>{{cite web |first1=Daniel|last1=Lawler|title=Earth now weighs six ronnagrams: New metric prefixes voted in |url=https://phys.org/news/2022-11-earth-ronnagrams-metric-prefixes-voted.html |publisher=phys.org |date=18 November 2022}}</ref><ref>{{cite web |date=18 November 2022 |title=List of Resolutions for the 27th meeting of the General Conference on Weights and Measures |url=https://www.bipm.org/documents/20126/64811223/Resolutions-2022.pdf |publisher=BIPM|url-status=dead |access-date=18 November 2022 |archive-date=18 November 2022 |archive-url=https://web.archive.org/web/20221118153958/https://www.bipm.org/documents/20126/64811223/Resolutions-2022.pdf}}</ref><ref>{{cite web |title=New prefixes for the SI adopted by the General Conference on Weights and Measures |url=https://www.bipm.org/en/-/2022-12-19-si-prefixes |access-date=11 January 2023 |website=BIPM }}</ref> Most prefixes correspond to integer powers of 1000; the only ones that do not are those for 10, 1/10, 100, and 1/100.
The conversion between different SI units for one and the same physical quantity is always through a power of ten. This is why the SI (and metric systems more generally) are called ''decimal systems of measurement units''.<ref name="DecimalSystem">{{cite web |url=https://usma.org/decimal-nature-of-the-metric-system |title=Decimal Nature of the Metric System |publisher=[[US Metric Association]] | date=2015 |access-date=15 April 2020 |archive-url=https://web.archive.org/web/20200415225727/https://usma.org/decimal-nature-of-the-metric-system/ |archive-date=15 April 2020 |url-status=live}}</ref>

{{Anchor|Compound unit}}The grouping formed by a prefix symbol attached to a unit symbol (e.g. '{{val|u=km}}', '{{val|u=cm}}') constitutes a new inseparable unit symbol. This new symbol can be raised to a positive or negative power. It can also be combined with other unit symbols to form ''compound unit'' symbols.<ref name="SIBrochure9thEd" />{{rp|page=143}} For example, {{val|u=g/cm3}} is an SI unit of [[density]], where {{val|u=cm3}} is to be interpreted as ({{val|u=cm}}){{sup|3}}.

Prefixes are added to unit names to produce multiples and [[submultiple]]s of the original unit. All of these are integer powers of ten, and above a hundred or below a hundredth all are integer powers of a thousand. For example, ''kilo-'' denotes a multiple of a thousand and ''milli-'' denotes a multiple of a thousandth, so there are one thousand millimetres to the metre and one thousand metres to the kilometre. The prefixes are never combined, so for example a millionth of a metre is a ''micrometre'', not a ''millimillimetre''. Multiples of the kilogram are named as if the gram were the base unit, so a millionth of a kilogram is a ''milligram'', not a ''microkilogram''.<ref name="SIBrochure"/>{{rp|page=122}}<ref name="NIST811"/>{{rp|page=14}}

The BIPM specifies 24 prefixes for the International System of Units (SI):

{{SI prefixes (infobox)}}

=== Coherent and non-coherent SI units <span class="anchor" id="Coherent SI units"></span> ===

{{Further|Coherent unit}}

The base units and the derived units formed as the product of powers of the base units with a numerical factor of one form a [[coherent system of units]]. Every physical quantity has exactly one coherent SI unit. For example, {{nowrap|1=1 m/s = (1 m) / (1 s)}} is the coherent derived unit for velocity.<ref name="SIBrochure9thEd" />{{rp|page=139}} With the exception of the kilogram (for which the prefix kilo- is required for a coherent unit), when prefixes are used with the coherent SI units, the resulting units are no longer coherent, because the prefix introduces a numerical factor other than one.<ref name="SIBrochure9thEd" />{{rp|page=137}} For example, the metre, kilometre, centimetre, nanometre, etc. are all SI units of length, though only the metre is a {{em|coherent}} SI unit. The complete set of SI units consists of both the coherent set and the multiples and sub-multiples of coherent units formed by using the SI prefixes.<ref name="SIBrochure9thEd" />{{rp|page=138}}

The kilogram is the only coherent SI unit whose name and symbol include a prefix. For historical reasons, the names and symbols for multiples and sub-multiples of the unit of mass are formed as if the [[gram]] were the base unit. Prefix names and symbols are attached to the unit name ''gram'' and the unit symbol g respectively. For example, {{val|e=-6|u=kg}} is written ''milligram'' and {{val|u=mg}}, not ''microkilogram'' and {{val|u=μkg}}.<ref name="SIBrochure9thEd" />{{rp|page=144}}

Several different quantities may share the same coherent SI unit. For example, the joule per kelvin (symbol {{val|u=J/K}}) is the coherent SI unit for two distinct quantities: [[heat capacity]] and [[entropy]]; another example is the ampere, which is the coherent SI unit for both [[electric current]] and [[magnetomotive force]]. This illustrates why it is important not to use the unit alone to specify the quantity. As the ''SI Brochure'' states,<ref name="SIBrochure9thEd" />{{rp|page=140}} "this applies not only to technical texts, but also, for example, to measuring instruments (i.e. the instrument read-out needs to indicate both the unit and the quantity measured)".

Furthermore, the same coherent SI unit may be a base unit in one context, but a coherent derived unit in another. For example, the ampere is a base unit when it is a unit of electric current, but a coherent derived unit when it is a unit of magnetomotive force.<ref name="SIBrochure9thEd" />{{rp|page=140}}

{| class="wikitable floatleft" style="margin:1em auto 1em auto;line-height:1.4"
|+ <big>Examples of coherent derived units in terms of base units</big><ref name="NIST330" />{{rp|page=17}}
|-
! scope="col" | Name
! scope="col" | Symbol
! scope="col" | Derived quantity
! scope="col" | Typical symbol
|-
! scope="row" | [[square metre]]
| style="text-align:center;" | {{val|u=m2}}
| [[area]]
| style="text-align:center;" | {{math|''A''}}
|-
! scope="row" | [[cubic metre]]
| style="text-align:center;" | {{val|u=m3}}
| [[volume]]
| style="text-align:center;" | {{math|''V''}}
|-
! scope="row" | [[metre per second]]
| style="text-align:center;" | {{val|u=m/s}}
| [[speed]], [[velocity]]
| style="text-align:center;" | {{math|''v''}}
|-
! scope="row" | [[metre per second squared]]
| style="text-align:center;" | {{val|u=m/s2}}
| [[acceleration]]
| style="text-align:center;" | {{math|''a''}}
|-
! scope="row" rowspan="2" | [[reciprocal metre]]
| rowspan="2" style="text-align:center;" | {{val|u=m-1}}
| [[wavenumber]]
| style="text-align:center;" | {{math|''σ''}}, {{math|''ṽ''}}
|-
| [[vergence (optics)]]
| style="text-align:center;" | {{math|''V''}}, 1/{{math|''f''}}
|-
! scope="row" | [[kilogram per cubic metre]]
| style="text-align:center;" | {{val|u=kg/m3}}
| [[density]]
| style="text-align:center;" | {{math|''ρ''}}
|-
! scope="row" | kilogram per square metre
| style="text-align:center;" | {{val|u=kg/m2}}
| [[surface density]]
| style="text-align:center;" | {{math|''ρ''{{smallsub|A}}}}
|-
! scope="row" | cubic metre per kilogram
| style="text-align:center;" | {{val|u=m3/kg}}
| [[specific volume]]
| style="text-align:center;" | {{math|''v''}}
|-
! scope="row" | ampere per square metre
| style="text-align:center;" | {{val|u=A/m<sup>2</sup>}}
| [[current density]]
| style="text-align:center;" | {{math|''j''}}
|-
! scope="row" | [[amperes per metre|ampere per metre]]
| style="text-align:center;" | {{val|u=A/m}}
| [[magnetic field strength]]
| style="text-align:center;" | {{math|''H''}}
|-
! scope="row" | mole per cubic metre
| style="text-align:center;" | {{val|u=mol/m3}}
| [[concentration]]
| style="text-align:center;" | {{math|''c''}}
|-
! scope="row" | [[kilogram per cubic metre]]
| style="text-align:center;" | {{val|u=kg/m3}}
| [[Mass concentration (chemistry)|mass concentration]]
| style="text-align:center;" | {{math|''ρ''}}, {{math|''γ''}}
|-
! scope="row" | [[candela per square metre]]
| style="text-align:center;" | {{val|u=cd/m<sup>2</sup>}}
| [[luminance]]
| style="text-align:center;" | {{math|''L''<sub>v</sub>}}
|}

{| class="wikitable floatleft" style="margin:1em auto 1em auto;line-height:1.4"
|+ <big>Examples of derived units that include units with special names</big><ref name="NIST330" />{{rp|page=18}}
|-
! scope="col" | Name
! scope="col" | Symbol
! scope="col" | Quantity
! scope="col" | In SI base units
|-
! scope="row" | [[pascal-second]]
| style="text-align:center;" | Pa⋅s
| [[dynamic viscosity]]
| style="text-align:center;" | m<sup>−1</sup>⋅kg⋅s<sup>−1</sup>
|-
! scope="row" | [[newton-metre]]
| style="text-align:center;" | N⋅m
| [[moment of force]]
| style="text-align:center;" | m<sup>2</sup>⋅kg⋅s<sup>−2</sup>
|-
! scope="row" | newton per metre
| style="text-align:center;" | N/m
| [[surface tension]]
| style="text-align:center;" | kg⋅s<sup>−2</sup>
|-
! scope="row" | [[radian per second]]
| style="text-align:center;" | rad/s
| [[angular velocity]], [[angular frequency]]
| style="text-align:center;" | s<sup>−1</sup>
|-
! scope="row" | [[radian per second squared]]
| style="text-align:center;" | rad/s<sup>2</sup>
| [[angular acceleration]]
| style="text-align:center;" | s<sup>−2</sup>
|-
! scope="row" | [[watt per square metre]]
| style="text-align:center;" | W/m<sup>2</sup>
| heat flux density, [[irradiance]]
| style="text-align:center;" | kg⋅s<sup>−3</sup>
|-
! scope="row" | [[joule per kelvin]]
| style="text-align:center;" | J/K
| [[entropy]], [[heat capacity]]
| style="text-align:center;" | m<sup>2</sup>⋅kg⋅s<sup>−2</sup>⋅K<sup>−1</sup>
|-
! scope="row" | joule per kilogram-kelvin
| style="text-align:center;" | J/(kg⋅K)
| [[specific heat capacity]], [[specific entropy]]
| style="text-align:center;" | m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup>
|-
! scope="row" | joule per kilogram
| style="text-align:center;" | J/kg
| [[specific energy]]
| style="text-align:center;" | m<sup>2</sup>⋅s<sup>−2</sup>
|-
! scope="row" | watt per metre-kelvin
| style="text-align:center;" | W/(m⋅K)
| [[thermal conductivity]]
| style="text-align:center;" | m⋅kg⋅s<sup>−3</sup>⋅K<sup>−1</sup>
|-
! scope="row" | joule per cubic metre
| style="text-align:center;" | J/m<sup>3</sup>
| [[energy density]]
| style="text-align:center;" | m<sup>−1</sup>⋅kg⋅s<sup>−2</sup>
|-
! scope="row" | volt per metre
| style="text-align:center;" | V/m
| [[electric field strength]]
| style="text-align:center;" | m⋅kg⋅s<sup>−3</sup>⋅A<sup>−1</sup>
|-
! scope="row" | coulomb per cubic metre
| style="text-align:center;" | C/m<sup>3</sup>
| [[electric charge density]]
| style="text-align:center;" | m<sup>−3</sup>⋅s⋅A
|-
! scope="row" | coulomb per square metre
| style="text-align:center;" | C/m<sup>2</sup>
| [[surface charge density]], [[electric flux density]], [[electric displacement]]
| style="text-align:center;" | m<sup>−2</sup>⋅s⋅A
|-
! scope="row" | farad per metre
| style="text-align:center;" | F/m
| [[permittivity]]
| style="text-align:center;" | m<sup>−3</sup>⋅kg<sup>−1</sup>⋅s<sup>4</sup>⋅A<sup>2</sup>
|-
! scope="row" | henry per metre
| style="text-align:center;" | H/m
| [[permeability (electromagnetism)|permeability]]
| style="text-align:center;" | m⋅kg⋅s<sup>−2</sup>⋅A<sup>−2</sup>
|-
! scope="row" | joule per mole
| style="text-align:center;" | J/mol
| [[joule per mole|molar energy]]
| style="text-align:center;" | m<sup>2</sup>⋅kg⋅s<sup>−2</sup>⋅mol<sup>−1</sup>
|-
! scope="row" | joule per mole-kelvin
| style="text-align:center;" | J/(mol⋅K)
| [[molar entropy]], [[molar heat capacity]]
| style="text-align:center;" | m<sup>2</sup>⋅kg⋅s<sup>−2</sup>⋅K<sup>−1</sup>⋅mol<sup>−1</sup>
|-
! scope="row" | coulomb per kilogram
| style="text-align:center;" | C/kg
| [[radiation exposure|exposure]] (x- and γ-rays)
| style="text-align:center;" | kg<sup>−1</sup>⋅s⋅A
|-
! scope="row" | gray per second
| style="text-align:center;" | Gy/s
| [[absorbed dose rate]]
| style="text-align:center;" | m<sup>2</sup>⋅s<sup>−3</sup>
|-
! scope="row" | watt per steradian
| style="text-align:center;" | W/sr
| [[radiant intensity]]
| style="text-align:center;" | m<sup>2</sup>⋅kg⋅s<sup>−3</sup>
|-
! scope="row" | watt per square metre-steradian
| style="text-align:center;" | W/(m<sup>2</sup>⋅sr)
| [[radiance]]
| style="text-align:center;" | kg⋅s<sup>−3</sup>
|-
! scope="row" | katal per cubic metre
| style="text-align:center;" | kat/m<sup>3</sup>
| [[catalytic activity concentration]]
| style="text-align:center;" | m<sup>−3</sup>⋅s<sup>−1</sup>⋅mol
|}
{{Clear}}

== Lexicographic conventions <span class="anchor" id="SI_writing_style"></span> ==

{{See also|ISO 31-0#Typographic conventions|Space (punctuation)#Unit symbols and numbers}}
[[File:981ms2.png|thumb|Example of lexical conventions. In the expression of acceleration due to gravity, a space separates the value and the units, both the 'm' and the 's' are lowercase because neither the metre nor the second are named after people, and exponentiation is represented with a [[superscript]] '2'.]]

=== Unit names ===

According to the SI Brochure,<ref name="SIBrochure9thEd" />{{rp|page=148}} unit names should be treated as [[common noun]]s of the context language. This means that they should be typeset in the same character set as other common nouns (e.g. [[Latin alphabet]] in English, [[Cyrillic script]] in Russian, etc.), following the usual grammatical and [[orthography|orthographical]] rules of the context language. For example, in English and French, even when the unit is named after a person and its symbol begins with a capital letter, the unit name in running text should start with a lowercase letter (e.g., newton, hertz, pascal) and is [[capitalised]] only at the beginning of a sentence and in [[title capitalization|headings and publication titles]]. As a nontrivial application of this rule, the SI Brochure notes<ref name="SIBrochure9thEd" />{{rp|page=148}} that the name of the unit with the symbol {{val|u=degC}} is correctly spelled as 'degree [[Celsius]]': the first letter of the name of the unit, 'd', is in lowercase, while the modifier 'Celsius' is capitalised because it is a proper name.<ref name="SIBrochure9thEd" />{{rp|page=148}}

The English spelling and even names for certain SI units and metric prefixes depend on the variety of English used. [[US English]] uses the spelling ''deka-'', ''meter'', and ''liter'', and [[International English]] uses ''deca-'', ''metre'', and ''litre''. The name of the unit whose symbol is t and which is defined according to {{val|1|u=t}} {{=}} {{val|e=3|u=kg}} is 'metric ton' in US English and 'tonne' in International English.<ref name="NIST330"/>{{rp|page=iii}}

=== Unit symbols and the values of quantities ===

Symbols of SI units are intended to be unique and universal, independent of the context language.<ref name= "SIBrochure" />{{rp|pages=130–135}} The SI Brochure has specific rules for writing them.<ref name= "SIBrochure" />{{rp|pages=130–135}}

In addition, the SI Brochure provides style conventions for among other aspects of displaying quantities units: the quantity symbols, formatting of numbers and the decimal marker, expressing measurement uncertainty, multiplication and division of quantity symbols, and the use of pure numbers and various angles.<ref name="SIBrochure9thEd"/>{{rp|page=147}}

In the United States, the guideline produced by the [[National Institute of Standards and Technology]] (NIST)<ref name= "NIST811">{{cite report |url=https://www.nist.gov/pml/special-publication-811 |title=Guide for the Use of the International System of Units (SI) |last1=Thompson |first1=Ambler |last2=Taylor |first2=Barry N. |date=March 2008 |publisher=[[National Institute of Standards and Technology]] | access-date=21 January 2022 |at=§10.5.3}}</ref>{{rp|page=37}} clarifies language-specific details for American English that were left unclear by the SI Brochure, but is otherwise identical to the SI Brochure.<ref name="NISTInterprepation">{{cite journal |date=9 May 2008 |url=http://edocket.access.gpo.gov/2008/pdf/E8-11058.pdf |title=Interpretation of the International System of Units (the Metric System of Measurement) for the United States |journal=Federal Register |volume=73 |issue=96 |pages=28432–28433 |id=FR Doc number E8-11058 |access-date=28 October 2009}}</ref> For example, since 1979, the [[litre]] may exceptionally be written using either an uppercase&nbsp;"L" or a lowercase&nbsp;"l", a decision prompted by the similarity of the lowercase letter&nbsp;"l" to the numeral&nbsp;"1", especially with certain typefaces or English-style handwriting. The American NIST recommends that within the United States "L" be used rather than&nbsp;"l".<ref name="NIST811" />

== Realisation of units <span class="anchor" id="Realisation"></span> ==

{{Main|Realisation (metrology)}}
[[File:Silicon sphere for Avogadro project.jpg|thumb|upright|Silicon sphere for the [[Avogadro project]] used for measuring the Avogadro constant to a relative [[standard uncertainty]] of {{val|2|e=−8}} or less, held by [[Achim Leistner]]<ref>{{cite web |title=Avogadro Project |url=http://www.npl.co.uk/science-technology/mass-and-force/research/avogadro-project |publisher=National Physical Laboratory |access-date=19 August 2010}}</ref>]]
Metrologists carefully distinguish between the definition of a unit and its realisation. The SI units are defined by declaring that seven ''defining constants''<ref name="SIBrochure9thEd" />{{rp|pages=125–129}} have certain exact numerical values when expressed in terms of their SI units. The realisation of the definition of a unit is the procedure by which the definition may be used to establish the value and associated uncertainty of a quantity of the same kind as the unit.<ref name="SIBrochure9thEd"/>{{rp|page=135}}

For each base unit the BIPM publishes a {{lang|fr|mises en pratique}}, ([[French language|French]] for 'putting into practice; implementation',<ref name="NIST mise en pratique">{{cite journal |url=https://www.nist.gov/programs-projects/nist-mise-en-pratique-new-kilogram-definition |title=NIST Mise en Pratique of the New Kilogram Definition |journal=[[NIST]] | date=2013 |access-date=9 May 2020 |archive-url=https://web.archive.org/web/20170714202843/https://www.nist.gov/programs-projects/nist-mise-en-pratique-new-kilogram-definition |archive-date=14 July 2017 |url-status=live}}</ref>) describing the current best practical realisations of the unit.<ref name="MisesEnPratique">{{cite web |url=https://www.bipm.org/en/publications/mises-en-pratique/ |title=Practical realizations of the definitions of some important units |publisher=[[BIPM]] | date=2019 |access-date=11 April 2020 |archive-url=https://web.archive.org/web/20200409115245/http://www.bipm.org/en/publications/mises-en-pratique/ |archive-date=9 April 2020 |url-status=live}}</ref> The separation of the defining constants from the definitions of units means that improved measurements can be developed leading to changes in the {{lang|fr|mises en pratique}} as science and technology develop, without having to revise the definitions.

The published {{lang|fr|mise en pratique}} is not the only way in which a base unit can be determined: the SI Brochure states that "any method consistent with the laws of physics could be used to realise any SI unit".<ref name="SIBrochure"/>{{rp|page=111}} Various consultative committees of the [[CIPM]] decided in 2016 that more than one {{lang|fr|mise en pratique}} would be developed for determining the value of each unit.<ref>{{cite web|title=International Committee for Weights and Measures – Proceedings of the 106th meeting|url=https://www.bipm.org/utils/en/pdf/CIPM/CIPM2017-EN.pdf}}</ref> These methods include the following:
* At least three separate experiments be carried out yielding values having a relative [[standard uncertainty]] in the determination of the [[kilogram]] of no more than {{val|5|e=-8}} and at least one of these values should be better than {{val|2|e=-8}}. Both the [[Kibble balance]] and the [[Avogadro project]] should be included in the experiments and any differences between these be reconciled.<ref>{{cite web |url=http://www.bipm.org/utils/common/pdf/CCM12.pdf#page=23 |title=Recommendations of the Consultative Committee for Mass and Related Quantities to the International Committee for Weights and Measures |website=12th Meeting of the CCM |date=26 March 2010 |publisher=Bureau International des Poids et Mesures |location=Sèvres |access-date=27 June 2012 |archive-url=https://web.archive.org/web/20130514081750/http://www.bipm.org/utils/common/pdf/CCM12.pdf#page=23 |archive-date=14 May 2013 |url-status=dead}}</ref><ref>{{cite web |url=http://www.bipm.org/utils/common/pdf/CCQM16.pdf#page=40 |title=Recommendations of the Consultative Committee for Amount of Substance – Metrology in Chemistry to the International Committee for Weights and Measures |website=16th Meeting of the CCQM |date=15–16 April 2010 |publisher=Bureau International des Poids et Mesures |location=Sèvres |access-date=27 June 2012 |archive-url=https://web.archive.org/web/20130514072057/http://www.bipm.org/utils/common/pdf/CCQM16.pdf#page=40 |archive-date=14 May 2013 |url-status=dead}}</ref>
* The definition of the [[kelvin]] measured with a relative uncertainty of the [[Boltzmann constant]] derived from two fundamentally different methods such as acoustic gas [[thermometry]] and dielectric constant gas thermometry be better than one part in {{val|e=-6}} and that these values be corroborated by other measurements.<ref>{{cite web |url=http://www.bipm.org/utils/common/pdf/CCT25.pdf#page=53 |title=Recommendations of the Consultative Committee for Thermometry to the International Committee for Weights and Measures |website=25th Meeting of the CCT |date=6–7 May 2010 |publisher=Bureau International des Poids et Mesures |location=Sèvres |access-date=27 June 2012 |archive-url=https://web.archive.org/web/20130514064646/http://www.bipm.org/utils/common/pdf/CCT25.pdf#page=53 |archive-date=14 May 2013 |url-status=dead}}</ref>

== Organizational status ==

[[File:Metric and imperial systems (2019).svg|thumb|upright=1.3|Countries using the [[Metric system|metric]] (SI), [[Imperial units|imperial]], and [[US customary]] systems as of 2019]]
The International System of Units, or SI,<ref name="SIBrochure9thEd">{{Citation |last=International Bureau of Weights and Measures |title=The International System of Units (SI) |date=Dec 2022 |url=https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf |volume=2 |issue=1 |archive-url=https://web.archive.org/web/20211018184555/https://www.bipm.org/documents/20126/41483022/SI-Brochure-9.pdf/fcf090b2-04e6-88cc-1149-c3e029ad8232 |url-status=live |edition=9th |isbn=978-92-822-2272-0 |archive-date=18 October 2021 |author-link=New SI}}</ref>{{rp|page=123|quote=... since its establishment in 1960, the International System of Units has always been referred to as "the SI" in its shortened form.}} is a [[decimal]] and [[metric system|metric]] [[system of units]] established in 1960 and periodically updated since then. The SI has an [[metrication|official status]] in most countries, including [[metrication in the United States|the United States]], [[metrication in Canada|Canada]], and [[metrication in the United Kingdom|the United Kingdom]], although these three countries are among the handful of nations that, to various degrees, also continue to use their customary systems. Nevertheless, with this nearly universal level of acceptance, the SI "has been used around the world as the preferred system of units, the basic language for science, technology, industry, and trade."<ref name="SIBrochure9thEd" />{{rp|pages=123,126}}

The only other types of measurement system that still have widespread use across the world are the [[imperial and US customary measurement systems]]. The [[international yard and pound]] are defined in terms of the SI.<ref name="Standards1959">{{cite book|author=United States. National Bureau of Standards|author-link=National Institute of Standards and Technology|title=Research Highlights of the National Bureau of Standards|url=https://books.google.com/books?id=4aWN-VRV1AoC&pg=PA13|access-date = 31 July 2019|year=1959|publisher=U.S. Department of Commerce, National Bureau of Standards|page=13}}</ref>

=== International System of Quantities ===

{{Main|International System of Quantities}}
The quantities and equations that provide the context in which the SI units are defined are now referred to as the ''[[International System of Quantities]]'' (ISQ).
The ISQ is based on the [[quantity#Quantity in physical science|quantities]] underlying each of the [[SI base units|seven base units of the SI]]. Other quantities, such as [[area]], [[pressure]], and [[electrical resistance]], are derived from these base quantities by clear, non-contradictory equations. The ISQ defines the quantities that are measured with the SI units.<ref>{{cite book |title=International vocabulary of metrology – Basic and general concepts and associated terms (VIM) |date=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> The ISQ is formalised, in part, in the international standard [[ISO/IEC 80000]], which was completed in 2009 with the publication of [[ISO 80000-1]],<ref>S. V. Gupta, ''Units of Measurement: Past, Present and Future. International System of Units'', p.&nbsp;16, Springer, 2009. {{ISBN|3642007384}}.</ref> and has largely been revised in 2019–2020.<ref>{{Cite web |title=ISO 80000-1:2022 Quantities and units Part 1: General |url=https://www.iso.org/standard/76921.html}}</ref>

=== Controlling authority ===

{{main | General Conference on Weights and Measures | International Bureau of Weights and Measures }}
The SI is regulated and continually developed by three international organisations that were established in 1875 under the terms of the [[Metre Convention]]. They are the [[General Conference on Weights and Measures]] (CGPM{{efn|name="French CGPM"|From [[French language|French]]: {{lang|fr|Conférence générale des poids et mesures.}}}}),<ref name="FR Interpretation of the SI">{{cite journal |date=16 May 2008 |title=Interpretation of the International System of Units (the Metric System of Measurement) for the United States |url=https://www.federalregister.gov/documents/2008/05/16/E8-11058/interpretation-of-the-international-system-of-units-the-metric-system-of-measurement-for-the-united |url-status=live |journal=[[Federal Register]] | publisher=[[National Institute of Standards and Technology]] | volume=73 |page=28432 |archive-url=https://web.archive.org/web/20170816003324/https://www.federalregister.gov/documents/2008/05/16/E8-11058/interpretation-of-the-international-system-of-units-the-metric-system-of-measurement-for-the-united |archive-date=16 August 2017 |access-date=6 December 2022}}</ref> the International Committee for Weights and Measures (CIPM{{efn|name="French CIPM"|from {{langx|fr|Comité international des poids et mesures}}}}), and the [[International Bureau of Weights and Measures]] (BIPM{{efn|name="BIPM from French"|from {{langx|fr|Bureau international des poids et mesures}}}}).
{{Anchor|SI Brochure}}All the decisions and recommendations concerning units are collected in a brochure called ''The International System of Units (SI)'',<ref name="SIBrochure9thEd" /> which is published in French and English by the BIPM and periodically updated. The writing and maintenance of the brochure is carried out by one of the committees of the CIPM. The definitions of the terms "quantity", "unit", "dimension", etc. that are used in the ''SI Brochure'' are those given in the [[international vocabulary of metrology]].<ref>{{cite web|url=http://www.bipm.org/en/publications/guides/vim.html|title= VIM3: International Vocabulary of Metrology |website=BIPM |url-status=dead |archive-url= https://web.archive.org/web/20201031042511/http://www.bipm.org/en/publications/guides/vim.html |archive-date= 31 October 2020 }}</ref> The brochure leaves some scope for local variations, particularly regarding unit names and terms in different languages. For example, the United States' [[National Institute of Standards and Technology]] (NIST) has produced a version of the CGPM document (NIST SP 330) which clarifies usage for English-language publications that use [[American English]].<ref name="NIST330">{{cite book |editor1=David B. Newell |editor2=Eite Tiesinga |title=The International System of Units (SI) |url=https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.330-2019.pdf |access-date=30 November 2019 |publisher=[[National Institute of Standards and Technology|NIST]] | location=Gaithersburg, MD |date=2019|edition=NIST Special publication 330, 2019}}</ref>

== History ==

[[File:Alter Grenzstein Pontebba 01.jpg|thumb|upright|[[Boundary marker|Stone marking]] the [[Austria-Hungary|Austro-Hungarian]]/Italian border at [[Pontebba]] displaying [[myriametre]]s, a unit of 10&nbsp;km used in [[Central Europe]] in the 19th century (but since [[deprecation|deprecated]])<ref name="Europa1842">{{cite web |url=http://www.spasslernen.de/geschichte/groessen/mas1.htm |date= 1 May 2009 |website=spasslernen |title=Amtliche Maßeinheiten in Europa 1842 |language=de |trans-title=Official units of measure in Europe 1842 |access-date=26 March 2011 |url-status=dead |archive-url=https://web.archive.org/web/20120925070621/http://www.spasslernen.de/geschichte/groessen/mas1.htm |archive-date= 25 September 2012 }} Text version of Malaisé's book: {{cite book |url=https://archive.org/details/bub_gb_TQgHAAAAcAAJ |title=Theoretisch-practischer Unterricht im Rechnen |publisher=Verlag des Verf. |language=de |trans-title=Theoretical and practical instruction in arithmetic |author-first= Ferdinand von |author-last=Malaisé |place=München |date=1842 |pages=307–322 |access-date=7 January 2013}}</ref>]]
{{broader|History of the metric system}}

=== CGS and MKS systems ===

{{See also|CGS system of units}}
[[File:US National Length Meter.JPG|thumb|right|Closeup of the National Prototype Metre, serial number 27, allocated to the United States]]

The concept of a system of units emerged a hundred years before the SI.
In the 1860s, [[James Clerk Maxwell]], [[William Thomson, 1st Baron Kelvin|William Thomson]] (later Lord Kelvin), and others working under the auspices of the [[British Association for the Advancement of Science]], building on previous work of [[Carl Gauss]], developed the [[centimetre–gram–second system of units]] or cgs system in 1874. The systems formalised the concept of a collection of related units called a ''coherent'' system of units. In a coherent system, ''base units'' combine to define ''derived units'' without extra factors.<ref name=NIST330/>{{rp|page=2}} For example, using metre per second is coherent in a system that uses metre for length and second for time, but kilometre per hour is not coherent. The principle of coherence was successfully used to define a number of units of measure based on the CGS, including the [[erg]] for [[energy]], the [[dyne]] for [[force]], the [[barye]] for [[pressure]], the [[poise (unit)|poise]] for [[dynamic viscosity]] and the [[stokes (unit)|stokes]] for [[kinematic viscosity]].<ref name="BIPMCentenary">{{cite book |url=https://archive.org/details/internationalbur420page |page=[https://archive.org/details/internationalbur420page/page/12 12] |title=The International Bureau of Weights and Measures 1875–1975: NBS Special Publication 420 |date=20 May 1975 |editor-last1=Page
| editor-first1=Chester H. |editor-last2=Vigoureux |editor-first2=Paul |publisher=[[National Bureau of Standards]] | location=Washington, D.C.}}</ref>

=== Metre Convention ===

{{Main|Metre Convention |MKS system of units}}
A French-inspired initiative for international cooperation in [[metrology]] led to the signing in 1875 of the [[Metre Convention]], also called Treaty of the Metre, by 17 nations.{{efn|Argentina, Austria-Hungary, Belgium, Brazil, Denmark, France, German Empire, Italy, Peru, Portugal, Russia, Spain, Sweden and Norway, Switzerland, Ottoman Empire, United States, and Venezuela.}}<ref name="Alder">{{cite book |title=The Measure of all Things – The Seven-Year-Odyssey that Transformed the World |author-last=Alder |author-first=Ken |date=2002 |publisher=Abacus |location=London |isbn=978-0-349-11507-8}}</ref>{{rp|pages=353–354}} The [[General Conference on Weights and Measures]] (French: {{lang|fr|Conférence générale des poids et mesures}} – CGPM), which was established by the Metre Convention,<ref name="BIPMCentenary"/> brought together many international organisations to establish the definitions and standards of a new system and to standardise the rules for writing and presenting measurements.<ref name=Giunta/>{{rp|page=37}}<ref>{{Cite book |last=Quinn |first=Terry J. |title=From artefacts to atoms: the BIPM and the search for ultimate measurement standards |date=2012 |publisher=Oxford University Press |isbn=978-0-19-530786-3 |location=New York Oxford}}</ref> Initially the convention only covered standards for the metre and the kilogram. This became the foundation of the MKS system of units.<ref name=NIST330/>{{rp|page=2}}

=== Giovanni Giorgi and the problem of electrical units ===

At the close of the 19th century three different systems of units of measure existed for electrical measurements: a [[CGS-based system for electrostatic units]], also known as the Gaussian or ESU system, a [[CGS-based system for electromechanical units]] (EMU), and an International system based on units defined by the Metre Convention<ref>{{cite book |title=Weights, Measures and Units |url=https://archive.org/details/dictionaryofweig0000fenn |url-access=registration |author-first=Donald |author-last=Fenna |at=International unit |isbn=978-0-19-860522-5 |publisher=[[Oxford University Press]] | date=2002}}</ref> for electrical distribution systems. Attempts to resolve the electrical units in terms of length, mass, and time using [[dimensional analysis]] was beset with difficulties – the dimensions depended on whether one used the ESU or EMU systems.<ref name="Maxwell2">{{cite book |title=A treatise on electricity and magnetism |volume=2 |author-first=J. C. |author-last=Maxwell |date=1873 |publisher=Clarendon Press |location=Oxford |url=https://archive.org/stream/electricandmag02maxwrich |pages=242–245 |access-date=12 May 2011}}</ref> This anomaly was resolved in 1901 when [[Giovanni Giorgi]] published a paper in which he advocated using a fourth base unit alongside the existing three base units. The fourth unit could be chosen to be [[electric current]], [[voltage]], or [[electrical resistance]].<ref name="IECGiorgi">{{cite web |url=http://www.iec.ch/about/history/beginning/giovanni_giorgi.htm |title=In the beginning...: Giovanni Giorgi |publisher=[[International Electrotechnical Commission]] | access-date=5 April 2011 |archive-date=15 May 2011 |archive-url=https://web.archive.org/web/20110515134553/http://www.iec.ch/about/history/beginning/giovanni_giorgi.htm |url-status=dead}}</ref>

Electric current with named unit 'ampere' was chosen as the base unit, and the other electrical quantities derived from it according to the laws of physics. When combined with the MKS the new system, known as MKSA, was approved in 1946.<ref name=NIST330/>

=== 9th CGPM, the precursor to SI ===

In 1948, the 9th CGPM commissioned a study to assess the measurement needs of the scientific, technical, and educational communities and "to make recommendations for a single practical system of units of measurement, suitable for adoption by all countries adhering to the Metre Convention".<ref>{{cite web|url=http://www.bipm.org/en/CGPM/db/9/6/|title= Resolution 6 of the 9th CGPM|website=BIPM |access-date=22 August 2017 |date=1948 |url-status=dead |archive-url=https://web.archive.org/web/20170822153835/http://www.bipm.org/en/CGPM/db/9/6/ |archive-date= 22 August 2017  }}</ref> This working document was ''Practical system of units of measurement''. Based on this study, the 10th CGPM in 1954 defined an international system derived six base units: the metre, kilogram, second, ampere, degree Kelvin, and candela.

The 9th CGPM also approved the first formal recommendation for the writing of symbols in the metric system when the basis of the rules as they are now known was laid down.<ref>{{cite web |url=http://www.bipm.org/en/CGPM/db/9/7/ |title=Resolution 7 of the 9th meeting of the CGPM (1948): Writing and printing of unit symbols and of numbers |date=1948 |access-date=6 November 2012 |publisher=[[International Bureau of Weights and Measures]] |url-status=dead |archive-url=https://web.archive.org/web/20130514091113/http://www.bipm.org/en/CGPM/db/9/7/ |archive-date= May 14, 2013 }}</ref> These rules were subsequently extended and now cover unit symbols and names, prefix symbols and names, how quantity symbols should be written and used, and how the values of quantities should be expressed.<ref name="SIBrochure"/>{{rp|pages=104,130}}

=== Birth of the SI ===

The 10th CGPM in 1954 resolved to create an international system of units<ref name=Giunta>{{Cite book |last=Giunta |first=Carmen J. |url=https://link.springer.com/10.1007/978-3-031-28436-6 |title=A Brief History of the Metric System: From Revolutionary France to the Constant-Based SI |date=2023 |publisher=Springer International Publishing |isbn=978-3-031-28435-9 |series=SpringerBriefs in Molecular Science |location=Cham |language=en |doi=10.1007/978-3-031-28436-6|bibcode=2023bhms.book.....G |s2cid=258172637 }}</ref>{{rp|page=41}} and in 1960, the 11th CGPM adopted the ''International System of Units'', abbreviated SI from the French name {{lang|fr|Le Système international d'unités}}, which included a specification for units of measurement.<ref name="SIBrochure"/>{{rp|page=110}}

The [[International Bureau of Weights and Measures]] (BIPM) has described SI as "the modern form of metric system".<ref name="SIBrochure"/>{{rp|page=95}} In 1971 the [[mole (unit)|mole]] became the seventh base unit of the SI.<ref name=NIST330/>{{rp|page=2}}

=== 2019 redefinition ===

[[File:Unit relations in the new SI.svg|thumb |right |Reverse dependencies of the SI base units on seven [[physical constant]]s, which are assigned exact numerical values in the [[2019 revision of the SI|2019 redefinition]]. Unlike in the previous definitions, the base units are all derived exclusively from constants of nature. Here, <math>a \rightarrow b</math> means that <math>a</math> is used to define <math>b</math>.]]
{{Main|2019 revision of the SI}}

After the [[history of the metre|metre was redefined]] in 1960, the [[International Prototype of the Kilogram]] (IPK) was the only physical artefact upon which base units (directly the kilogram and indirectly the ampere, mole and candela) depended for their definition, making these units subject to periodic comparisons of national standard kilograms with the IPK.<ref name="NPL kg">{{cite web |title=Redefining the kilogram |url=http://www.npl.co.uk/educate-explore/redefining-the-kilogram/|publisher=UK National Physical Laboratory |access-date=30 November 2014 |url-status=live |archive-url= https://web.archive.org/web/20141227083141/http://www.npl.co.uk/educate-explore/redefining-the-kilogram/ |archive-date= Dec 27, 2014 }}</ref> During the 2nd and 3rd Periodic Verification of National Prototypes of the Kilogram, a significant divergence had occurred between the mass of the IPK and all of its official copies stored around the world: the copies had all noticeably increased in mass with respect to the IPK. During ''extraordinary verifications'' carried out in 2014 preparatory to redefinition of metric standards, continuing divergence was not confirmed. Nonetheless, the residual and irreducible instability of a physical IPK undermined the reliability of the entire metric system to precision measurement from small (atomic) to large (astrophysical) scales.<ref>{{Cite web |date=12 May 2018 |title=A Turning Point for Humanity: Redefining the World's Measurement System |url=https://www.nist.gov/si-redefinition/turning-point-humanity-redefining-worlds-measurement-system |access-date=16 January 2024 |website=NIST |language=en}}</ref>
By avoiding the use of an artefact to define units, all issues with the loss, damage, and change of the artefact are avoided.<ref name=SIBrochure9thEd/>{{rp|page=125}}

A proposal was made that:<ref>{{cite web |title=Appendix 1. Decisions of the CGPM and the CIPM |url=https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-App4-EN.pdf |page=188 |publisher=[[BIPM]] | access-date=27 April 2021}}</ref>
* In addition to the speed of light, four constants of nature – the [[Planck constant]], an [[elementary charge]], the [[Boltzmann constant]], and the [[Avogadro constant]] – be defined to have exact values
* The [[International Prototype of the Kilogram]] be retired
* The current definitions of the kilogram, ampere, kelvin, and mole be revised
* The wording of base unit definitions should change emphasis from explicit unit to explicit constant definitions.

The new definitions were adopted at the 26th CGPM on 16 November 2018, and came into effect on 20 May 2019.<ref>{{cite web |url=http://www.bipm.org/cc/TGFC/Allowed/Minutes/CODATA_Minutes_14-BIPM-public.pdf |title=Report on the Meeting of the CODATA Task Group on Fundamental Constants |date=3–4 November 2014 |publisher=[[BIPM]] | author-first=B. |author-last=Wood |page=7 |quote=[BIPM director Martin] Milton responded to a question about what would happen if ... the CIPM or the CGPM voted not to move forward with the redefinition of the SI. He responded that he felt that by that time the decision to move forward should be seen as a foregone conclusion.}}</ref> The change was adopted by the European Union through Directive (EU) 2019/1258.<ref>{{cite web |url=https://eur-lex.europa.eu/eli/dir/2019/1258/oj |title=Commission Directive (EU) 2019/1258 of 23 July 2019 amending, for the purpose of its adaptation to technical progress, the Annex to Council Directive 80/181/EEC as regards the definitions of SI base units |author=<!--Not stated--> |date=23 July 2019 |website=[[Eur-Lex]] | access-date=28 August 2019}}</ref>

Prior to its redefinition in 2019, the SI was defined through the seven base units from which the derived units were constructed as products of powers of the base units. After the redefinition, the SI is defined by fixing the numerical values of seven defining constants. This has the effect that the distinction between the base units and derived units is, in principle, not needed, since all units, base as well as derived, may be constructed directly from the defining constants. Nevertheless, the distinction is retained because "it is useful and historically well established", and also because the [[ISO/IEC 80000]] series of standards, which define the [[International System of Quantities]] (ISQ), specifies base and derived quantities that necessarily have the corresponding SI units.<ref name="SIBrochure9thEd" />{{rp|page=129}}

== Related units ==

=== Non-SI units accepted for use with SI ===

{{Main|Non-SI units mentioned in the SI}}

[[File:CubeLitre.svg|right|thumb|upright=1.1|While not an SI-unit, the litre may be used with SI units. It is equivalent to {{nowrap|1=({{val|10|u=cm}}){{sup|3}} = ({{val|1|u=dm}}){{sup|3}} = {{val|e=-3|u=m3}}}}.]]
Many non-SI units continue to be used in the scientific, technical, and commercial literature. Some units are deeply embedded in history and culture, and their use has not been entirely replaced by their SI alternatives. The CIPM recognised and acknowledged such traditions by compiling a list of [[Non-SI units mentioned in the SI|non-SI units accepted for use with SI]],<ref name="SIBrochure"/> including the hour, minute, degree of angle, litre, and decibel.

=== Metric units not recognised by SI ===

{{Main|List of metric units}}
Although the term ''metric system'' is often used as an informal alternative name for the International System of Units,<ref name="MetricSystemAsNameForSI">{{cite journal |last=Olthoff|first=Jim|url=https://www.nist.gov/blogs/taking-measure/all-times-all-peoples-how-replacing-kilogram-empowers-industry |title=For All Times, For All Peoples: How Replacing the Kilogram Empowers Industry |journal=[[NIST]] | date=2018 |access-date=14 April 2020 |archive-url=https://web.archive.org/web/20200316195625/https://www.nist.gov/blogs/taking-measure/all-times-all-peoples-how-replacing-kilogram-empowers-industry |archive-date=16 March 2020 |url-status=live |quote=... the International System of Units (SI), popularly known as the metric system.}}</ref> other metric systems exist, some of which were in widespread use in the past or are even still used in particular areas. There are also individual [[metric units]] such as the [[sverdrup]] and the [[darcy (unit)|darcy]] that exist outside of any system of units. Most of the units of the other metric systems are not recognised by the SI.

=== Unacceptable uses ===

Sometimes, SI unit name variations are introduced, mixing information about the corresponding physical quantity or the conditions of its measurement; however, this practice is unacceptable with the SI. "Unacceptability of mixing information with units: When one gives the value of a quantity, any information concerning the quantity or its conditions of measurement must be presented in such a way as not to be associated with the unit."<ref name="SIBrochure"/>
Instances include: "[[watt-peak]]" and "[[watt RMS]]"; "[[geopotential metre]]" and "[[vertical metre]]"; "[[standard cubic metre]]"; "[[atomic second]]", "[[ephemeris second]]", and "[[sidereal second]]".

== See also ==

{{Columns-list|colwidth=30em|
* {{annotated link|Conversion of units}}
* {{annotated link|List of international common standards}}
* [[Metrication]]
* {{annotated link|Outline of the metric system}}

{{br}}
'''Organisations'''
* {{annotated link|International Bureau of Weights and Measures}}
* {{annotated link|Institute for Reference Materials and Measurements}} (EU)
* {{annotated link|National Institute of Standards and Technology}} (US)
'''Standards and conventions'''
* {{annotated link|Conventional electrical unit}}
* {{annotated link|Coordinated Universal Time|abbreviation=UTC}}
* {{annotated link|Unified Code for Units of Measure}}
}}

== Notes ==

{{notelist}}

; Attribution
<ref name="SIBrochure9thEd" />{{Creative Commons text attribution notice|cc=by3|url=https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf|authors=Bureau International des Poids et Mesures|from this source=yes}}

== References ==

{{reflist}}

== Further reading ==

{{refbegin}}
* {{GreenBook2nd}}
* [http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/#rms Unit Systems in Electromagnetism] {{Webarchive|url=https://web.archive.org/web/20201030081637/http://info.ee.surrey.ac.uk/Workshop/advice/coils/unit_systems/#rms |date=30 October 2020 }}
* [https://www.nist.gov/customcf/get_pdf.cfm?pub_id=32943 MW Keller ''et al.''] (PDF) Metrology Triangle Using a Watt Balance, a Calculable Capacitor, and a Single-Electron Tunnelling Device
* [http://nvlpubs.nist.gov/nistpubs/jres/116/6/V116.N06.A01.pdf "The Current SI Seen From the Perspective of the Proposed New SI"] (PDF). Barry N. Taylor. Journal of Research of the National Institute of Standards and Technology, Vol. 116, No. 6, Pgs. 797–807, Nov–Dec 2011.
* B. N. Taylor, Ambler Thompson, ''International System of Units (SI)'', [[National Institute of Standards and Technology]] 2008 edition, {{ISBN|1437915582}}.
{{refend}}

== External links ==

{{Commons category|International System of Units}}
{{Refbegin}}
* [https://www.bipm.org/ BIPM (International Bureau of Weights and Measures) official web site]
{{Refend}}

{{SI units}}
{{Systems of measurement}}
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[[Category:International standards]]
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