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SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014]

Section 2.2: SI derived units

    Derived units are products of powers of base units. Coherent derived units are products of powers of base units that include no numerical factor other than 1. The base and coherent derived units of the SI form a coherent set, designated the set of coherent SI units (see section 1.4).

Section 2.2.1: Derived units expressed in terms of base units

The number of quantities in science is without limit, and it is not possible to provide a complete list of derived quantities and derived units. However, Table 2 lists some examples of derived quantities, and the corresponding coherent derived units expressed directly in terms of base units.

Table 2. Examples of coherent derived units in the SI expressed in terms of base units

Derived quantity SI coherent derived unit
Name Symbol Name Symbol
area A square metre m2
volume V cubic metre m3
speed, velocity v ital metre per second m/s
acceleration a metre per second squared m/s2
wavenumber sigma, nu tilde reciprocal metre m–1
density, mass density rho kilogram per cubic metre kg/m3
surface density rhoA kilogram per square metre kg/m2
specific volume v cubic metre per kilogram m3/kg
current density j ampere per square metre A/m2
magnetic field strength H ampere per metre A/m
amount concentration (a), concentration c mole per cubic metre mol/m3
mass concentration rho, gamma kilogram per cubic metre kg/m3
luminance Lv candela per square metre cd/m2
refractive index (b) n one 1
relative permeability (b) r one 1

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Section 2.2.2: Units with special names and symbols; units that incorporate special names and symbols

For convenience, certain coherent derived units have been given special names and symbols. There are 22 such units, as listed in Table 3. These special names and symbols may themselves be used in combination with the names and symbols for base units and for other derived units to express the units of other derived quantities. Some examples are given in Table 4. The special names and symbols are simply a compact form for the expression of combinations of base units that are used frequently, but in many cases they also serve to remind the reader of the quantity involved. The SI prefixes may be used with any of the special names and symbols, but when this is done the resulting unit will no longer be coherent.

Among these names and symbols the last four entries in Table 3 are of particular note since they were adopted by the 15th CGPM (1975, Resolutions 8 and 9), the 16th CGPM (1979, Resolution 5) and the 21st CGPM (1999, Resolution 12) specifically with a view to safeguarding human health.

In both Tables 3 and 4 the final column shows how the SI units concerned may be expressed in terms of SI base units. In this column factors such as m0, kg0, etc., which are all equal to 1, are not shown explicitly.

The values of several different quantities may be expressed using the same name and symbol for the SI unit. Thus for the quantity heat capacity as well as the quantity entropy, the SI unit is the joule per kelvin. Similarly for the base quantity electric current as well as the derived quantity magnetomotive force, the SI unit is the ampere. It is therefore important not to use the unit alone to specify the quantity. This applies not only to scientific and technical texts, but also, for example, to measuring instruments (i.e. an instrument read-out should indicate both the unit and the quantity measured).

A derived unit can often be expressed in different ways by combining base units with derived units having special names. Joule, for example, may formally be written newton metre, or kilogram metre squared per second squared. This, however, is an algebraic freedom to be governed by common sense physical considerations; in a given situation some forms may be more helpful than others.

In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension. When using this freedom, one may recall the process by which the quantity is defined. For example, the quantity torque may be thought of as the cross product of force and distance, suggesting the unit newton metre, or it may be thought of as energy per angle, suggesting the unit joule per radian. The SI unit of frequency is given as the hertz, implying the unit cycles per second; the SI unit of angular velocity is given as the radian per second; and the SI unit of activity is designated the becquerel, implying the unit counts per second. Although it would be formally correct to write all three of these units as the reciprocal second, the use of the different names emphasises the different nature of the quantities concerned. Using the unit radian per second for angular velocity, and hertz for frequency, also emphasizes that the numerical value of the angular velocity in radian per second is 2pi times the numerical value of the corresponding frequency in hertz.

In the field of ionizing radiation, the SI unit of activity is designated the becquerel rather than the reciprocal second, and the SI units of absorbed dose and dose equivalent are designated the gray and the sievert, respectively, rather than the joule per kilogram. The special names becquerel, gray, and sievert were specifically introduced because of the dangers to human health that might arise from mistakes involving the units reciprocal second and joule per kilogram, in case the latter units were incorrectly taken to identify the different quantities involved.

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    Table 3. Coherent derived units in the SI with special names and symbols

Derived quantity SI coherent derived unit (a)
Name Symbol Expressed
in terms of
other SI units
Expressed
in terms of
SI base units
plane angle radian (b) rad 1 (b) m/m
solid angle steradian (b) sr (c) 1 (b) m2/m2
frequency hertz (d) Hz   s–1
force newton N   kg m s–2
pressure, stress pascal Pa N/m2 kg m–1 s–2
energy, work, amount of heat joule J N m kg m2 s–2
power, radiant flux watt W J/s kg m2 s–3
electric charge, amount of electricity coulomb C   A s
electric potential difference, electromotive force volt V W/A kg m2 s–3 A–1
capacitance farad F C/V kg–1 m–2 s4 A2
electric resistance ohm capital omega V/A kg m2 s–3 A–2
electric conductance siemens S A/V kg–1 m–2 s3 A2
magnetic flux weber Wb V s kg m2 s–2 A–1
magnetic flux density tesla T Wb/m2 kg s–2 A–1
inductance henry H Wb/A kg m2 s–2 A–2
Celsius temperature degree Celsius (e) °C   K
luminous flux lumen lm cd sr (c) cd sr
illuminance lux lx lm/m2 cd sr m–2
activity referred to a radionuclide (f) becquerel (d) Bq   s–1
absorbed dose, specific energy (imparted), kerma gray Gy J/kg m2 s–2
dose equivalent,
ambient dose equivalent,
directional dose equivalent,
personal dose equivalent
sievert (g) Sv J/kg m2 s–2
catalytic activity katal kat   mol s–1

(a) The SI prefixes may be used with any of the special names and symbols, but when this is done the resulting unit will no longer be coherent.
(b) The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities.
(c) In photometry the name steradian and the symbol sr are usually retained in expressions for units.
(d) The hertz is used only for periodic phenomena, and the becquerel is used only for stochastic processes in activity referred to a radionuclide.
(e) The degree Celsius is the special name for the kelvin used to express Celsius temperatures. The degree Celsius and the kelvin are equal in size, so that the numerical value of a temperature difference or temperature interval is the same when expressed in either degrees Celsius or in kelvins.
(f) Activity referred to a radionuclide is sometimes incorrectly called radioactivity.
(g) See CIPM Recommendation 2 (CI-2002) on the use of the sievert.

[ updated 2014 ]
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    Table 4. Examples of SI coherent derived units whose names and symbols include SI coherent derived units with special names and symbols

Derived quantity SI coherent derived unit
Name Symbol Expressed in terms of
SI base units
dynamic viscosity pascal second Pa s kg m–1 s–1
moment of force newton metre N m kg m2 s–2
surface tension newton per metre N/m kg s–2
angular velocity radian per second rad/s m m–1 s–1 = s–1
angular acceleration radian per second squared rad/s2 m m–1 s–2 = s–2
heat flux density,
irradiance
watt per square metre W/m2 kg s–3
heat capacity, entropy joule per kelvin J/K kg m2 s–2 K–1
specific heat capacity,
specific entropy
joule per kilogram kelvin J/(kg K) m2 s–2 K–1
specific energy joule per kilogram J/kg m2 s–2
thermal conductivity watt per metre kelvin W/(m K) kg m s–3 K–1
energy density joule per cubic metre J/m3 kg m–1 s–2
electric field strength volt per metre V/m kg m s–3 A–1
electric charge density coulomb per cubic metre C/m3 A s m–3
surface charge density coulomb per square metre C/m2 A s m–2
electric flux density,
electric displacement
coulomb per square metre C/m2 A s m–2
permittivity farad per metre F/m kg–1 m–3 s4 A2
permeability henry per metre H/m kg m s–2 A–2
molar energy joule per mole J/mol kg m2 s–2 mol–1
molar entropy,
molar heat capacity
joule per mole kelvin J/(mol K) kg m2 s–2 mol–1 K–1
exposure
(x- and gamma-rays)
coulomb per kilogram C/kg A s kg–1
absorbed dose rate gray per second Gy/s m2 s–3
radiant intensity watt per steradian W/sr kg m2 s–3 sr–1
radiance watt per square metre steradian W/(m2 sr) kg s–3 sr–1
catalytic activity
concentration
katal per cubic metre kat/m3 mol s–1 m–3

[ updated 2014 ]
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Section 2.2.3: Units for dimensionless quantities, also called quantities of dimension one

Certain quantities are defined as the ratio of two quantities of the same kind, and are thus dimensionless, or have a dimension that may be expressed by the number one. The coherent SI unit of all such dimensionless quantities, or quantities of dimension one, is the number one, since the unit must be the ratio of two identical SI units. The values of all such quantities are simply expressed as numbers, and the unit one is not explicitly shown. Examples of such quantities are refractive index, relative permeability, and friction factor. There are also some quantities that are defined as a more complex product of simpler quantities in such a way that the product is dimensionless. Examples include the "characteristic numbers" like the Reynolds number Re = rho v italL/eta, where rho is mass density, eta is dynamic viscosity,  v ital is speed, and L is length. For all these cases the unit may be considered as the number one, which is a dimensionless derived unit.

Another class of dimensionless quantities are numbers that represent a count, such as a number of molecules, degeneracy (number of energy levels), and partition function in statistical thermodynamics (number of thermally accessible states). All of these counting quantities are also described as being dimensionless, or of dimension one, and are taken to have the SI unit one, although the unit of counting quantities cannot be described as a derived unit expressed in terms of the base units of the SI. For such quantities, the unit one may instead be regarded as a further base unit.

In a few cases, however, a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian and the steradian. The radian and steradian have been identified by the CGPM as special names for the coherent derived unit one, to be used to express values of plane angle and solid angle, respectively, and are therefore included in Table 3.

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