




Coherent units, derived units with special names, and the SI prefixes

SI Brochure, Section 1.4

Derived units are defined as products of powers of the base units. When the product of powers includes no numerical factor other than one, the derived units are called coherent derived units. The base and coherent derived units of the SI form a coherent set, designated the set of coherent SI units. The word coherent is used here in the following sense: when coherent units are used, equations between the numerical values of quantities take exactly the same form as the equations between the quantities themselves. Thus if only units from a coherent set are used, conversion factors between units are never required.
The expression for the coherent unit of a derived quantity may be obtained from the dimensional product of that quantity by replacing the symbol for each dimension by the symbol of the corresponding base unit.
Some of the coherent derived units in the SI are given special names,* to simplify their expression (see Section 2.2.2). It is important to emphasize that each physical quantity has only one coherent SI unit, even if this unit can be expressed in different forms by using some of the special names and symbols. The inverse, however, is not true: in some cases the same SI unit can be used to express the values of several different quantities (see Section 2.2.2).


*. As an example of a special name, the particular combination of base units
m^{2} kg s^{–2} for energy is given the special name joule, symbol J, where by definition J = m^{2} kg s^{–2}.


The CGPM has, in addition, adopted a series of prefixes for use in forming the decimal multiples and submultiples of the coherent SI units (see 3.1, where the prefix names and symbols are listed). These are convenient for expressing the values of quantities that are much larger than or much smaller than the coherent unit.† Following the CIPM Recommendation 1 (1969) these are given the name SI prefixes. (These prefixes are also sometimes used with other nonSI units, as described in Chapter 4.) However when prefixes are used with SI units, the resulting units are no longer coherent, because a prefix on a derived unit effectively introduces a numerical factor in the expression for the derived unit in terms of the base units.
As an exception, the name of the kilogram, which is the base unit of mass, includes the prefix kilo, for historical reasons. It is nonetheless taken to be a base unit of the SI. The multiples and submultiples of the kilogram are formed by attaching prefix names to the unit name "gram", and prefix symbols to the unit symbol "g" (see Section 3.2). Thus 10^{–6} kg is written as a milligram, mg, not as a microkilogram, µkg.
The complete set of SI units, including both the coherent set and the multiples and submultiples of these units formed by combining them with the SI prefixes, are designated as the complete set of SI units, or simply the SI units, or the units of the SI. Note, however, that the decimal multiples and submultiples of the SI units do not form a coherent set.‡

†. The length of a chemical bond is more conveniently given in nanometres, nm, than in metres, m; and the distance from London to Paris is more conveniently given in kilometres, km, than in metres, m.
‡. The metre per second, symbol m/s, is the coherent SI unit of speed. The kilometre per second, km/s, the centimetre per second, cm/s, and the millimetre per second, mm/s, are also SI units, but they are not coherent SI units.



We are pleased to present the updated (2014) 8th edition of the SI Brochure, which defines and presents the Système International d'Unités, the SI (known in English as the International System of Units).
 SI prefixes

Factor 
Name 
Symbol 

Factor 
Name 
Symbol 

10^{1} 
deca 
da 

10^{–1} 
deci 
d 
10^{2} 
hecto 
h 
10^{–2} 
centi 
c 
10^{3} 
kilo 
k 
10^{–3} 
milli 
m 
10^{6} 
mega 
M 
10^{–6} 
micro 
µ 
10^{9} 
giga 
G 
10^{–9} 
nano 
n 
10^{12} 
tera 
T 
10^{–12} 
pico 
p 
10^{15} 
peta 
P 
10^{–15} 
femto 
f 
10^{18} 
exa 
E 
10^{–18} 
atto 
a 
10^{21} 
zetta 
Z 
10^{–21} 
zepto 
z 
10^{24} 
yotta 
Y 
10^{–24} 
yocto 
y 

 The kilogram
General principles for the writing of unit symbols and numbers were first given by the 9th CGPM (1948, Resolution 7). These were subsequently elaborated by ISO, IEC, and other international bodies. As a consequence, there now exists a general consensus on how unit symbols and names, including prefix symbols and names, as well as quantity symbols should be written and used, and how the values of quantities should be expressed. Compliance with these rules and style conventions, the most important of which are presented in this chapter, supports the readability of scientific and technical papers.
This appendix lists those decisions of the CGPM and the CIPM that bear directly upon definitions of the units of the SI, prefixes defined for use as part of the SI, and conventions for the writing of unit symbols and numbers. It is not a complete list of CGPM and CIPM decisions. For a complete list, reference must be made to the BIPM website, successive volumes of the Comptes Rendus des Séances de la Conférence Générale des Poids et Mesures (CR) and ProcèsVerbaux des Séances du Comité International des Poids et Mesures (PV) or, for recent decisions, to Metrologia.
Since the SI is not a static convention, but evolves following developments in the science of measurement, some decisions have been abrogated or modified; others have been clarified by additions. In the SI Brochure, a number of notes have been added by the BIPM to make the text more understandable; they do not form part of the original text.
In the printed brochure, the decisions of the CGPM and CIPM are listed in strict chronological order in order to preserve the continuity with which they were taken. However in order to make it easy to locate decisions related to particular topics a table of contents is also provided, ordered by subject:
Optical radiation is able to cause chemical changes in certain living or nonliving materials: this property is called actinism, and radiation capable of causing such changes is referred to as actinic radiation. Actinic radiation has the fundamental characteristic that, at the molecular level, one photon interacts with one molecule to alter or break the molecule into new molecular species. It is therefore possible to define specific photochemical or photobiological quantities in terms of the result of optical radiation on the associated chemical or biological receptors.
In the field of metrology, the only photobiological quantity which has been formally defined for measurement in the SI is for the interaction of light with the human eye in vision. An SI base unit, the candela, has been defined for this important photobiological quantity. Several other photometric quantities with units derived from the candela have also been defined (such as the lumen and the lux, see Table 3 in Chapter 2).





