




Chapter 1: Introduction
The value of a quantity is generally expressed as the product of a number and a unit. The unit is simply a particular example of the quantity concerned which is used as a reference, and the number is the ratio of the value of the quantity to the unit. For a particular quantity, many different units may be used. For example, the speed of a particle may be expressed in the form = 25 m/s = 90 km/h, where metre per second and kilometre per hour are alternative units for expressing the same value of the quantity speed. However, because of the importance of a set of well defined and easily accessible units universally agreed for the multitude of measurements that support today's complex society, units should be chosen so that they are readily available to all, are constant throughout time and space, and are easy to realize with high accuracy.
In order to establish a system of units, such as the International System of Units, the SI, it is necessary first to establish a system of quantities, including a set of equations defining the relations between those quantities. This is necessary because the equations between the quantities determine the equations relating the units, as described below. It is also convenient to choose definitions for a small number of units that we call base units, and then to define units for all other quantities as products of powers of the base units that we call derived units. In a similar way the corresponding quantities are described as base quantities and derived quantities, and the equations giving the derived quantities in terms of the base quantities are used to determine the expression for the derived units in terms of the base units, as discussed further in section 1.4. Thus in a logical development of this subject, the choice of quantities and the equations relating the quantities comes first, and the choice of units comes second.*
From a scientific point of view, the division of quantities into base quantities and derived quantities is a matter of convention, and is not essential to the physics of the subject. However for the corresponding units, it is important that the definition of each base unit is made with particular care, to satisfy the requirements outlined in the first paragraph above, since they provide the foundation for the entire system of units. The definitions of the derived units in terms of the base units then follow from the equations defining the derived quantities in terms of the base quantities. Thus the establishment of a system of units, which is the subject of this brochure, is intimately connected with the algebraic equations relating the corresponding quantities.
The number of derived quantities of interest in science and technology can, of course, be extended without limit. As new fields of science develop, new quantities are devised by researchers to represent the interests of the field, and with these new quantities come new equations relating them to those quantities that were previously familiar, and hence ultimately to the base quantities. In this way the derived units to be used with the new quantities may always be defined as products of powers of the previously chosen base units.


The terms quantity and unit are defined in the International Vocabulary of Basic and General Terms in Metrology, the VIM.
The quantity speed, , may be expressed in terms of the quantities distance, x, and time, t, by the equation
= dx/dt.
In most systems of quantities and units, distance x and time t are regarded as base quantities, for which the metre, m, and the second, s, may be chosen as base units. Speed is then taken as a derived quantity, with the derived unit metre per second, m/s.
*. For example, in electrochemistry, the electric mobility of an ion, u, is defined as the ratio of its velocity to the electric field strength, E: u = /E. The derived unit of electric mobility is then given as (m/s)/(V/m) = m^{2} V^{–1} s^{–1},
in units which may be easily related to the chosen base units (V is the symbol for the SI derived unit volt).



This Brochure is concerned with presenting the information necessary to define and use the International System of Units, universally known as the SI (from the French Système International d'Unités). The SI was established by and is defined by the General Conference on Weights and Measures, the CGPM, as described in the Historical note in Section 1.8*.
The system of quantities, including the equations relating the quantities, to be used with the SI, is in fact just the quantities and equations of physics that are familiar to all scientists, technologists, and engineers. They are listed in many textbooks and in many references, but any such list can only be a selection of the possible quantities and equations, which is without limit. Many of the quantities, their recommended names and symbols, and the equations relating them, are listed in the International Standard 80000 of ISO and IEC, Quantities and units, composed of 14 parts and produced by Technical Committee 12 of the International Organization for Standardization, ISO/TC 12, and by Technical Committee 25 of the International Electrotechnical Commission, IEC/TC 25. In the ISO and IEC 80000 series the quantities and equations used with the SI are known as the International System of Quantities.
The base quantities used in the SI are length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The base quantities are by convention assumed to be independent. The corresponding base units of the SI were chosen by the CGPM to be the metre, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. The definitions of these base units are presented in Section 2.1.1 in the following chapter. The derived units of the SI are then formed as products of powers of the base units, according to the algebraic relations that define the corresponding derived quantities in terms of the base quantities, see Section 1.4.
On rare occasions a choice may arise between different forms of the relations between the quantities. An important example occurs in defining the electromagnetic quantities. In this case the rationalized fourquantity electromagnetic equations used with the SI are based on length, mass, time, and electric current. In these equations, the electric constant _{0} (the permittivity of vacuum) and the magnetic constant _{}_{0} (the permeability of vacuum) have dimensions and values such that _{0}_{}_{0} = 1/c_{0}^{2}, where c_{0} is the speed of light in vacuum. The Coulomb law of electrostatic force between two particles with charges q_{1} and q_{2} separated by a distance r is written**
F = 
q_{1}q_{2} r 

4_{0} r^{3} 
and the corresponding equation for the magnetic force between two thin wire elements carrying electric currents, i_{1}dl_{1} and i_{2}dl_{2}, is written
d^{2}F = 
_{}_{0} 
i_{1}dl_{1} x (i_{2}dl_{2} x r) 


4 
r^{3} 
where d^{2}F is the double differential of the force F. These equations, on which the SI is based, are different from those used in the CGSESU, CGSEMU, and CGSGaussian systems, where _{0} and _{}_{0} are dimensionless quantities, chosen to be equal to one, and where the rationalizing factors of 4 are omitted.
* 
Acronyms used in this Brochure are listed with their meaning here. 
** 
Symbols in bold print are used to denote vectors. 


The name Système International d'Unités, and the abbreviation SI, were established by the 11th CGPM in 1960.
Examples of the equations relating quantities used in the SI are the Newtonian inertial equation relating force, F, to mass, m, and acceleration, a,
for a particle: F = m a, and the equation giving the kinetic energy, T, of a particle moving with velocity, :
T = m^{2}/2.



By convention physical quantities are organized in a system of dimensions. Each of the seven base quantities used in the SI is regarded as having its own dimension, which is symbolically represented by a single sans serif roman capital letter. The symbols used for the base quantities, and the symbols used to denote their dimension, are given as follows.
Base quantities and dimensions used in the SI

Base quantity 
Symbol for quantity 
Symbol for dimension 

length 
l, x, r, etc. 
L 
mass 
m 
M 
time, duration 
t 
T 
electric current 
I, i 
l 
thermodynamic temperature 
T 

amount of substance 
n 
N 
luminous intensity 
I_{ v} 
J 

All other quantities are derived quantities, which may be written in terms of the base quantities by the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product,
dim Q = L^{} M T l^{} ^{} N^{} J^{}
where the exponents , _{}, _{}, , , _{}, and _{}, which are generally small integers which can be positive, negative or zero, are called the dimensional exponents. The dimension of a derived quantity provides the same information about the relation of that quantity to the base quantities as is provided by the SI unit of the derived quantity as a product of powers of the SI base units.


Quantity symbols are always written in an italic font, and symbols for dimensions in sansserif roman capitals.
For some quantities a variety of alternative symbols may be used, as indicated for length and electric current.
Note that symbols for quantities are only recommendations, in contrast to symbols for units that appear elsewhere in this brochure whose style and form is mandatory (see Chapter 5).
Dimensional symbols and exponents are manipulated using the ordinary rules of algebra. For example, the dimension of area is written as L^{2}; the dimension of velocity as LT^{–1}; the dimension of force as LMT^{–2}; and the dimension of energy is written as L^{2}MT^{–2}.


There are some derived quantities Q for which the defining equation is such that all of the dimensional exponents in the expression for the dimension of Q are zero. This is true, in particular, for any quantity that is defined as the ratio of two quantities of the same kind. Such quantities are described as being dimensionless, or alternatively as being of dimension one. The coherent derived unit for such dimensionless quantities is always the number one, 1, since it is the ratio of two identical units for two quantities of the same kind.
There are also some quantities that cannot be described in terms of the seven base quantities of the SI at all, but have the nature of a count. Examples are number of molecules, degeneracy in quantum mechanics (the number of independent states of the same energy), and the partition function in statistical thermodynamics (the number of thermally accessible states). Such counting quantities are also usually regarded as dimensionless quantities, or quantities of dimension one, with the unit one, 1. 
For example, refractive index is defined as the ratio of the speed of light in vacuum to that in the medium, and is thus a ratio of two quantities of the same kind. It is therefore a dimensionless quantity.
Other examples of dimensionless quantities are plane angle, mass fraction, relative permittivity, relative permeability, and finesse of a FabryPerot cavity. 


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.



The definitions of the base units of the SI were adopted in a context that takes no account of relativistic effects. When such account is taken, it is clear that the definitions apply only in a small spatial domain sharing the motion of the standards that realize them. These units are known as proper units; they are realized from local experiments in which the relativistic effects that need to be taken into account are those of special relativity. The constants of physics are local quantities with their values expressed in proper units.
Physical realizations of the definition of a unit are usually compared locally. For frequency standards, however, it is possible to make such comparisons at a distance by means of electromagnetic signals. To interpret the results the theory of general relativity is required since it predicts, among other things, a relative frequency shift between standards of about 1 part in 10^{16} per metre of altitude difference at the surface of the Earth. Effects of this magnitude cannot be neglected when comparing the best frequency standards.


The question of proper units is addressed in Resolution A4 adopted by the
XXIst General Assembly
of the International Astronomical Union (IAU) in 1991 and by the report of the CCDS Working Group on the Application of General Relativity to Metrology (Metrologia, 1997, 34, 261290).



Units for quantities that describe biological effects are often difficult to relate to units of the SI because they typically involve weighting factors that may not be precisely known or defined, and which may be both energy and frequency dependent. These units, which are not SI units, are described briefly in this section.
Optical radiation may cause chemical changes in living or nonliving materials: this property is called actinism and radiation capable of causing such changes is referred to as actinic radiation. In some cases, the results of measurements of photochemical and photobiological quantities of this kind can be expressed in terms of SI units. This is discussed briefly in Appendix 3.
Sound causes small pressure fluctuations in the air, superimposed on the normal atmospheric pressure, that are sensed by the human ear. The sensitivity of the ear depends on the frequency of the sound, but is not a simple function of either the pressure changes or the frequency. Therefore frequencyweighted quantities are used in acoustics to approximate the way in which sound is perceived. Such frequencyweighted quantities are employed, for example, in work to protect against hearing damage. The effects of ultrasonic acoustic waves pose similar concerns in medical diagnosis and therapy.
Ionizing radiation deposits energy in irradiated matter. The ratio of deposited energy to mass is termed absorbed dose. High doses of ionizing radiation kill cells, and this is used in radiation therapy. Appropriate biological weighting functions are used to compare therapeutic effects of different radiation treatments. Low sublethal doses can cause damage to living organisms, for instance by inducing cancer. Appropriate riskweighted functions are used at low doses as the basis of radiation protection regulations.
There is a class of units for quantifying the biological activity of certain substances used in medical diagnosis and therapy that cannot yet be defined in terms of the units of the SI. This is because the mechanism of the specific biological effect that gives these substances their medical use is not yet sufficiently well understood for it to be quantifiable in terms of physicochemical parameters. In view of their importance for human health and safety, the World Health Organization (WHO) has taken responsibility for defining WHO International Units (IU) for the biological activity of such substances.




By legislation, individual countries have established rules concerning the use of units on a national basis, either for general use or for specific areas such as commerce, health, public safety, and education. In almost all countries this legislation is based on the International System of Units.
The International Organization of Legal Metrology (OIML), founded in 1955, is charged with the international harmonization of this legislation.



The previous paragraphs of this chapter give a brief overview of the way in which a system of units, and the International System of Units in particular, is established. This note gives a brief account of the historical development of the International System.
The 9th CGPM (1948, Resolution 6) instructed the CIPM:
 to study the establishment of a complete set of rules for units of measurement;
 to find out for this purpose, by official enquiry, the opinion prevailing in scientific, technical and educational circles in all countries;
 to make recommendations on the establishment of a practical system of units of measurement suitable for adoption by all signatories to the Convention du Mètre.
The same CGPM also laid down, in Resolution 7, general principles for the writing of unit symbols, and listed some coherent derived units which were assigned special names.
The 10th CGPM (1954, Resolution 6) and the 14th CGPM (1971, Resolution 3) adopted as base units of this practical system of units the units of the following seven quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
The 11th CGPM (1960, Resolution 12) adopted the name Système International d'Unités, with the international abbreviation SI, for this practical system of units and laid down rules for prefixes, derived units, and the former supplementary units, and other matters; it thus established a comprehensive specification for units of measurement. Subsequent meetings of the CGPM and CIPM have added to, and modified as necessary, the original structure of the SI to take account of advances in science and of the needs of users.
The historical sequence that led to these important CGPM decisions is summarized here: www.bipm.org/en/measurementunits/historysi/.









