Fundamental unit

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For the use in number theory, see Fundamental unit (number theory).

A set of fundamental units is a set of units for physical quantities from which every other unit can be generated.

In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, momentum, energy, and weight, and units are used to describe their measure. Quantities are sometimes also referred to as dimensions (i.e. "The dimensions of the package to be mailed are 12" x 8" x 4".") although the term is normally meant to apply to the class of physical quantity (length, time, mass, momentum, energy, etc.) being measured (see dimensional analysis).

A system of fundamental quantities (or sometimes fundamental dimensions) is such that every other physical quantity (or dimension of physical quantity) can be generated from them. The traditional fundamental dimensions of physical quantity are mass, length, time, and temperature, but in principle, other fundamental quantities could be used. Some physicists have not recognized temperature as a fundamental dimension of physical quantity since it simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). In addition, some physicists recognize electric charge as a separate fundamental dimension of physical quantity, even if it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic cgs system. There are also physicists who have cast doubt on the very existence of incompatible fundamental quantities^[1].

The fundamental units of the SI system are kilogram, second, ampere, kelvin, and mole. The metre and candela were formerly fundamental units but have been redefined in terms of the other units. They are still considered SI base units, however, mainly for historical reasons. Other units such as the newton, joule, and volt can be derived from the SI base units and are therefore termed SI derived units.

The dimension speed, for example, is length divided by time, so its unit m/s can be generated from the fundamental unit of seconds (the metre being also defined in terms of the second).

An important basic fact of dimensional analysis is that the fundamental units can be regarded as the basis of a special kind of vector space, the space of all units. This is a vector space over the field of rational numbers where the vector addition is given by the multiplication of units and the scalar multiplication is exponentiation of units.

Not all physically salient values have units: dimensionless numbers occur in many fields of science. In fact, dimensionless quantities are fundamentally what we measure, even when we are measuring dimensionful quantities. We always measure a physical quantity against a like dimensioned standard. When one measures a length with a ruler or tape-measure, that person is actually counting tick marks on the ruler (a standard used to measure length) and the net result is a dimensionless number. But when that quantity is expressed as the dimensionless number attached to (multiplying) a dimensionful unit, it becomes, conceptually, a dimensionful quantity.