Editing International System of Units (section)

=== 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=SIBrochure9thEd/>{{rp|page=137}}
|-
! 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="nc">The radian and steradian are defined as dimensionless derived units.</ref>
| style="text-align:center;" | rad
| [[angle|plane angle]]
| style="text-align:center;" | <!-- intentionally left blank to reflect version 3.01 (2024) of the 9th SI brochure -->
| style="text-align:center;" | 1
|-
! scope="row" | [[steradian]]<ref name=":0" group="nc" />
| style="text-align:center;" | sr
| [[solid angle]]
| style="text-align:center;" | <!-- intentionally left blank to reflect version 3.01 (2024) of the 9th SI brochure -->
| 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⋅sr<ref name="keep-sr" group="nc">In photometry, the steradian is usually retained in expressions for units.</ref>
| style="text-align:center;" | cd⋅sr
|-
! scope="row" | [[lux]]
| style="text-align:center;" | lx
| [[illuminance]]
| style="text-align:center;" | cd⋅sr⋅m<sup>−2</sup><ref name="keep-sr" group="nc"/>
| style="text-align:center;" | lm/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
| [[catalytic activity]]
| style="text-align:center;" | mol⋅s<sup>−1</sup>
| <!-- intentionally left blank -->
|-
<!-- Note: there are exactly 22 entries in the ref -->
| colspan="5" | '''Notes'''
<references group="nc" />
|}
{{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=SIBrochure9thEd/>{{rp|page=138}}<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>