




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).
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



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 m^{0}, kg^{0}, 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 readout 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 2 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.





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)} 
m^{2}/m^{2} 
frequency 
hertz ^{(d)} 
Hz 

s^{–1} 
force 
newton 
N 

kg m s^{–2} 
pressure, stress 
pascal 
Pa 
N/m^{2} 
kg m^{–1} s^{–2} 
energy, work, amount of heat 
joule 
J 
N m 
kg m^{2} s^{–2} 
power, radiant flux 
watt 
W 
J/s 
kg m^{2} s^{–3} 
electric charge, amount of electricity 
coulomb 
C 

A s 
electric potential difference, electromotive force 
volt 
V 
W/A 
kg m^{2} s^{–3} A^{–1} 
capacitance 
farad 
F 
C/V 
kg^{–1} m^{–2} s^{4} A^{2} 
electric resistance 
ohm 

V/A 
kg m^{2} s^{–3} A^{–2} 
electric conductance 
siemens 
S 
A/V 
kg^{–1} m^{–2} s^{3} A^{2} 
magnetic flux 
weber 
Wb 
V s 
kg m^{2} s^{–2} A^{–1} 
magnetic flux density 
tesla 
T 
Wb/m^{2} 
kg s^{–2} A^{–1} 
inductance 
henry 
H 
Wb/A 
kg m^{2} s^{–2} A^{–2} 
Celsius temperature 
degree Celsius ^{(e)} 
°C 

K 
luminous flux 
lumen 
lm 
cd sr ^{(c)} 
cd sr 
illuminance 
lux 
lx 
lm/m^{2} 
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 
m^{2} s^{–2} 
dose equivalent, ambient dose equivalent, directional dose equivalent, personal dose equivalent 
sievert ^{(g)} 
Sv 
J/kg 
m^{2} 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 (CI2002) on the use of the sievert. 


[ updated 2014 ] 



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 = _{}L/_{}, where _{} is mass density, _{} is dynamic viscosity, 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.









