Editing International System of Units (section)

=== 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}}