NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixes | NIST
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Physical Measurement Laboratory

NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixes

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Since the 1995 edition of this Guide, the 20th CGPM, which met October 9 − 12, 1995, decided to eliminate the class of supplementary units as a separate unit class in the SI. The SI now consists of only two classes of units: base units and derived units. The radian and steradian, which were the two supplementary units, are now subsumed into the class of SI derived units. Thus the SI units are currently divided into base units and derived units, which together form what is called "the coherent system of SI units."2 The SI also includes the prefixes to form decimal multiples and submultiples of SI units.

4.1 SI base units

Table 1 gives the seven base quantities, assumed to be mutually independent, on which the SI is founded, and the names and symbols of their respective units, called "SI base units." Definitions of the SI base units are given in Appendix A. The kelvin and its symbol K are also used to express the value of a temperature interval or a temperature difference (see Sec. 8.5).

Table 1.  SI base units
  SI base unit
Base quantity Name Symbol
length meter m
mass kilogram       kg
time second s
electric current ampere A
thermodynamic temperature      kelvin K
amount of substance mole mol
luminous intensity candela cd

4.2 SI derived units

Derived units are expressed algebraically in terms of base units or other derived units. The symbols for derived units are obtained by means of the mathematical operations of multiplication and division. For example, the derived unit for the derived quantity molar mass (mass divided by amount of substance) is the kilogram per mole, symbol kg/mol. Additional examples of derived units expressed in terms of SI base units are given in Table 2. (The rules and style conventions for printing and using SI unit symbols are given in Sec. 6.1.1 to 6.1.8.)

Table 2. Examples of SI derived units expressed in terms of SI base units
  SI derived unit
Derived quantity Name Symbol
area square meter m2
volume cubic meter m3
speed, velocity meter per second m/s
acceleration meter per second squared m/s2
wavenumber reciprocal meter m−1
density, mass density kilogram per cubic meter kg/m3
specific volume cubic meter per kilogram m3/kg
current density ampere per square meter A/m2
magnetic field strength ampere per meter A/m
luminance candela per square meter cd/m2
amount−of−substance concentration    
amount concentration , concentration mole per cubic meter mol/m3

4.2.1 SI coherent derived units with special names and symbols

Certain SI coherent derived units have special names and symbols; these are given in Table 3. Consistent with the discussion in Sec. 4, the radian and steradian, which are the two former supplementary units, are included in Table 3. The last four units in Table 3 were introduced into the SI for reasons of safeguarding human health.

Table 3. The 22 SI coherent derived units with special names and symbols.
  SI coherent derived unit (a)
  Special Name Special
symbol		 Expression in
terms of other
SI units		 Expression 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   m · kg · s−2
pressure, stress pascal Pa N/m2 m−1 · kg · s−2
energy, work, amount of heat joule J N · m m2 · kg · s−2
power, radiant flux watt W J/s m2 · kg · s−3
electric charge, amount of electricity coulomb C   s · A
electric potential difference(e), electromotive force volt V W/A m2 · kg · s−3 · A−1
capacitance farad F C/V m−2 · kg−1 · s4 · A2
electric resistance ohm Ω V/A m2 · kg · s−3 · A−2
electric conductance siemens S A/V m−2 · kg−1 · s3 · A2
magnetic flux weber Wb V · s m2 · kg · s−2 · A−1
magnetic flux density tesla T Wb/m2 kg · s−2 · A−1
inductance henry H Wb/A m2 · kg· s−2 · A−2
Celsius temperature degree Celsius (f) °C   K
luminous flux lumen lm cd · sr(c) Cd
illuminance lux lx lm/m2 m−2· cd
activity referred to a radionuclide(g) 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 (h) Sv J/kg m2· s−2
catalytic activity katal kat   s−1· mol

(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. (See Sec. 6.2.8.)
(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. (See Sec 7.10)
(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) Electric potential difference is also called "voltage" in the United States.
(f) 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. (See Secs. 4.2.1.1 and 8.5.)
(g) Activity referred to a radionuclide is sometimes incorrectly called radioactivity.
(h) See Refs. [1, 2], on the use of the sievert.

4.2.1.1 Degree Celsius

In addition to the quantity thermodynamic temperature (symbol T), expressed in the unit kelvin, use is also made of the quantity Celsius temperature (symbol t) defined by the equation t = T − T0 , where T0 = 273.15 K by definition. To express Celsius temperature, the unit degree Celsius, symbol °C, which is equal in magnitude to the unit kelvin, is used; in this case, "degree Celsius" is a special name used in place of "kelvin." An interval or difference of Celsius temperature, however, can be expressed in the unit kelvin as well as in the unit degree Celsius (see Sec. 8.5). (Note that the thermodynamic temperature T0 is exactly 0.01 K below the thermodynamic temperature of the triple point of water (see Sec. A.6).)

4.2.2 Use of SI derived units with special names and symbols

Examples of SI derived units that can be expressed with the aid of SI derived units having special names and symbols are given in Table 4.

Table 4. Examples of SI coherent derived units expressed with the aid of SI derived units having special names and symbols.
  SI coherent derived unit
Derived quantity Name Symbol Expression in terms of SI base units
dynamic viscosity pascal second Pa · s m−1 · kg · s−1
moment of force newton meter N · m m2 · kg · s−2
surface tension newton per meter 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 meter W/m2 kg · s−3
heat capacity, entropy joule per kelvin J/K m2 · kg · 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 meter kelvin W/(m · K) m · kg · s−3 · K−1
energy density joule per cubic meter J/m3 m−1 · kg · s−2
electric field strength volt per meter V/m m · kg · s−3 s· A−1
electric charge density coulomb per cubic meter C/m3 m−3 · s · A
surface charge density coulomb per square meter C/m2 m−2 · s · A
electric flux density, electric displacement coulomb per square meter C/m2 m−2 · s · A
permittivity farad per meter F/m m−3 · kg−1 · s4 · A−2
permeability henry per meter H/m m · kg · s−2 · A2
molar energy joule per mole J/mol m2 · kg · s−2 · mol−1
molar entropy, molar heat capacity joule per mole kelvin J/(mol · K) m2 · kg · s−2 · K−1 · mol−1
exposure (χ and γ rays) coulomb per kilogram C/kg kg−1 · s · A
absorbed dose rate gray per second Gy/s m2 · s−3
radiant intensity watt per steradian W/sr m4 · m−2 · kg · s−3 =m2 · kg · s−3
radiance watt per square meter steradian W/(m2 · sr) m2 · m−2 · kg · s−3 = kg · s−3
catalytic activity concentration katal per cubic meter kat/m3 m−3 · s−1 · mol

The advantages of using the special names and symbols of SI derived units are apparent in Table 4. Consider, for example, the quantity molar entropy: the unit J/ (mol · K) is obviously more easily understood than its SI base−unit equivalent, m2 · kg · s−2 · K−1 · mol−1. Nevertheless, it should always be recognized that the special names and symbols exist for convenience;either the form in which special names or symbols are used for certain combinations of units or the form in which they are not used is correct. For example, because of the descriptive value implicit in the compound−unit form, communication is sometimes facilitated if magnetic flux (see Table 3) is expressed in terms of the volt second (V · s) instead of the weber (Wb) or the combination of SI base units, m2 · kg · s−2 · A−1.

Tables 3 and 4 also show that the values of several different quantities are expressed in the same SI unit. For example, the joule per kelvin (J/K) is the SI unit for heat capacity as well as for entropy. Thus the name of the unit is not sufficient to define the quantity measured.

A derived unit can often be expressed in several different ways through the use of base units and derived units with special names. In practice, with certain quantities, preference is given to using certain units with special names, or combinations of units, to facilitate the distinction between quantities whose values have identical expressions in terms of SI base units. For example, the SI unit of frequency is specified as the hertz (Hz) rather than the reciprocal second (s−1), and the SI unit of moment of force is specified as the newton meter (N · m) rather than the joule (J).

Similarly, in the field of ionizing radiation, the SI unit of activity is designated as the becquerel (Bq) rather than the reciprocal second (s−1), and the SI units of absorbed dose and dose equivalent are designated as the gray (Gy) and the sievert (Sv), respectively, rather than the joule per kilogram (J/kg).

4.3 Decimal multiples and submultiples of SI units: SI prefixes

Table 5 gives the SI prefixes that are used to form decimal multiples and submultiples of units. They allow very large or very small numerical values (see Sec. 7.1) to be avoided. A prefix name attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit. For example, one kilometer, 1 km, is equal to one thousand meters, 1000 m or 103 m. When prefixes are used to form multiples and submultiples of SI base and derived units, the resulting units are no longer coherent. (See footnote 2 for a brief discussion of coherence.) The rules and style conventions for printing and using SI prefixes are given in Secs. 6.2.1 to 6.2.8. The special rule for forming decimal multiples and submultiples of the unit of mass is given in Sec. 6.2.7.

Table 5. SI prefixes*
Factor Prefix Symbol Factor Prefix Symbol
1030=(103)10 quetta Q 10−1 deci d
1027=(103)9 ronna R 10−2 centi c
1024=(103)8 yotta Y 10−3=(103)−1 milli m
1021=(103)7 zetta Z 10−6=(103)−2 micro μ
1018=(103)6 exa E 10−9=(103)−3 nano n
1015=(103)5 peta P 10−12=(103)−4 pico p
1012=(103)4 tera T 10−15=(103)−5 femto f
109=(103)3 giga G 10−18=(103)−6 atto a
106=(103)2 mega M 10−21=(103)−7 zepto z
103=(103)1 kilo k 10−24=(103)−8 yocto y
102 hecto h 10−27=(103)−9 ronto r
101 deka da 10−30=(103)−10 quecto q

Note: Alternative definitions of the SI prefixes and their symbols are not permitted. For example, it is unacceptable to use kilo (k) to represent 210 = 1024, mega (M) to represent 220 = 1 048 576, or giga (G) to represent 230 = 1 073 741 824. See the note to Ref. [5] on page 74 for the prefixes for binary powers adopted by the IEC.

* Note: Four new SI prefixes are added to Table 5 to align with the BIPM SI Brochure, 9th edition, version 2.01 (2022).

Metrology
Created January 28, 2016, Updated September 22, 2023