Guide for the Use of the International System of Units (SI)

4 The Three Classes of SI Units and the SI Prefixes

SI units are currently divided into three classes: which together form what is called "the coherent system of SI units." (Footnote 2) The SI also includes 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 (including the radian and steradian which are the two supplementary units - see Sec. 4.3). 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 Sec. 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
wave number reciprocal meter m-1
mass density (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
amount-of-substance concentration (concentration)   mole per cubic meter mol/m3
luminance candela per square meter cd/m2

4.2.1 SI derived units with special names and symbols

Certain SI derived units have special names and symbols; these are given in Table 3a and Table 3b. As discussed in Sec. Sec. 4.3, the radian and steradian, which are the two supplementary units, are included in Table 3a.

Table 3a. SI derived units with special names and symbols, including the radian and steradian
  SI derived unit
Derived quantity Special
Name		 Special
Symbol		 Expression
in terms of
other SI units		 Expression
in terms of
SI base units
plane angle radian rad   - m · m-1 = 1
solid angle steradian sr   - m2 · m-2 = 1
frequency hertz Hz   - s-1
force newton N   - m · kg · s-2
pressure, stress pascal Pa N/m2 m-1 · kg · s-2
energy, work, quantity of heat   joule J N · m m2 · kg · s-2
power, radiant flux watt W J/s m2 · kg · s-3
electric charge, quantity of electricity coulomb C   - s · A
electric potential, potential difference, 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 (a) degree Celsius °C   - K
luminous flux lumen lm cd · sr cd · sr (b)
illuminance lux lx lm/m2 m-2 · cd · sr (b)
(a) See Sec. 4.2.1.1, Sec. 6.2.8, and Sec. 7.2.

(b) The steradian (sr) is not an SI base unit. However, in photometry the steradian (sr) is maintained in expressions for units (see Sec. 4.3).

Table 3b. SI derived units with special names and symbols admitted for reasons of safeguarding human health

  SI derived unit
Derived quantity Special
Name (a)		 Special
Symbol (a)		 Expression
in terms of
other SI units		 Expression
in terms of
SI base units
activity (of a radionuclide) becquerel 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, equivalent dose, sievert  Sv  J/kg  m2 · s-2
(a) The derived quantities to be expressed in the gray and the sievert have been revised in accordance with the recommendations of the International Commission on Radiation Units and Measurements (ICRU); see Ref. [19].

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 can, however, 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 (including the radian and steradian) are given in Table 4.

Table 4. Examples of SI derived units expressed with the aid of SI derived units having special names and symbols
  SI derived unit
Derived quantity Name Symbol Expression
in terms of
SI base units
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
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
heat flux density, irradiance watt per square meter W/m2 kg · s-3
radiant intensity watt per steradian W/sr m2 · kg · s-3 · sr-1 (a)
radiance watt per square meter steradian   W/(m2 · sr)   kg · s-3 · sr-1 (a)
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 · A-1
electric charge density coulomb per cubic meter C/m3 m-3 · s · A
electric flux density coulomb per square meter C/m2 m-2 · s · A
permittivity farad per meter F/m m-3 · kg-1 · s4 · A2
permeability henry per meter H/m m · kg · s-2 · A-2
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 (x and &;gamma; rays) coulomb per kilogram C/kg kg-1 · s · A
absorbed dose rate gray per second Gy/s m2 · s-3
(a) The steradian (sr) is not an SI base unit. However, in radiometry the steradian (sr) is maintained in expressions for units (see Sec. 4.3).

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 3a) is expressed in terms of the volt second (V · s) instead of the weber (Wb).

Table 3a, Table 3b, and Table 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 SI supplementary units

[See Preface NOTE TO SECOND PRINTING]

As previously stated, there are two units in this class: the radian, symbol rad, the SI unit of the quantity plane angle; and the steradian, symbol sr, the SI unit of the quantity solid angle. Definitions of these units are given in Appendix A.

The SI supplementary units are now interpreted as so-called dimensionless derived units (see Sec. 7.14) for which the CGPM allows the freedom of using or not using them in expressions for SI derived units. (Footnote 3) Thus the radian and steradian are not given in a separate table but have been included in Table 3a together with other derived units with special names and symbols (see Sec. 4.2.1). This interpretation of the supplementary units implies that plane angle and solid angle are considered derived quantities of dimension one (so-called dimensionless quantities - see Sec. 7.14), each of which has the unit one, symbol 1, as its coherent SI unit. However, in practice, when one expresses the values of derived quantities involving plane angle or solid angle, it often aids understanding if the special names (or symbols) "radian" (rad) or "steradian" (sr) are used in place of the number 1. For example, although values of the derived quantity angular velocity (plane angle divided by time) may be expressed in the unit s-1, such values are usually expressed in the unit rad/s.

Because the radian and steradian are now viewed as so-called dimensionless derived units, the Consultative Committee for Units (CCU, Comité Consultatif des Unités) of the CIPM (footnote **), as a result of a 1993 request it received from ISO/TC 12 (see Ref. [22]), recommended to the CIPM that it request the CGPM to abolish the class of supplementary units as a separate class in the SI. The CIPM accepted the CCU recommendation, and if the abolishment is approved by the CGPM as is likely (the question will be on the agenda of the 20th CGPM, October 1995), the SI will consist of only two classes of units: base units and derived units, with the radian and steradian subsumed into the class of derived units of the SI. (The option of using or not using them in expressions for SI derived units, as is convenient, would remain unchanged.)

4.4 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 SI units. They allow very large or very small numerical values (see Sec. 7.1) to be avoided. A prefix attaches directly to the name of a unit, and a prefix symbol attaches directly to the symbol for a unit. For example, one kilometer, symbol 1 km, is equal to one thousand meters, symbol 1000 m or 103 m. When prefixes are attached to SI units, the units so formed are called "multiples and submultiples of SI units" in order to distinguish them from the coherent system of SI units. (See footnote 2 for a brief discussion of coherence. The rules and style conventions for printing and using SI prefixes are given in Sec. 6.2.1 to Sec. 6.2.8. The special rule for forming decimal multiples and submultiples of the unit of mass is given in Sec. 6.2.7.)

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.

Table 5.  SI prefixes
Factor Prefix   Symbol                 Factor Prefix   Symbol
1024 = (103)8   yotta Y   10-1 deci d
1021 = (103)7   zetta Z   10-2 centi c
1018 = (103)6   exa E   10-3 = (103)-1   milli m
1015 = (103)5   peta P   10-6 = (103)-2   micro µ
1012 = (103)4   tera T   10-9 = (103)-3   nano n
109 = (103)3   giga G   10-12 = (103)-4   pico p
106 = (103)2   mega M   10-15 = (103)-5   femto f
103 = (103)1   kilo k   10-18 = (103)-6   atto a
102 hecto h   10-21 = (103)-7   zepto z
101 deka da   10-24 = (103)-8   yocto y
Table of Contents
Previous Section Next Section