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).
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 non-SI 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 106 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.
*. As an example of a special name, the particular combination of base units
m2 kg s2 for energy is given the special name joule, symbol J, where by definition J = m2 kg s2.
. 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.