BIPM - quantities and units

SI brochure, Section 1.1

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 $v ital$ of a particle may be expressed in the form $v ital$ = 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, $v ital$ , may be expressed in terms of the quantities distance, x, and time, t, by the equation $v ital$ = 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 $v ital$ 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 $v ital$ to the electric field strength, E: u = $v ital$ /E. The derived unit of electric mobility is then given as (m/s)/(V/m) = m² 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).