In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, and weight, and units measure them. Quantities are sometimes also referred to as dimensions although the term is only strictly true of time and space measurements; see dimensional analysis.
A system of fundamental quantities (or sometimes fundamental dimensions) is such that every other quantity can be generated from them. Traditionally, the accepted fundamental quantities are mass, length, time, and temperature, but in principle, other fundamental quantities could be used.
Correspondingly, a system of fundamental units is such that every other unit can be generated from them. The kilogram, metre, second, ampere, kelvin, mole and candela are the fundamental units of the SI system of units, termed SI base units; 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 quantity velocity, for example, is length divided by time, so its unit m/s can be generated from the fundamental units metre and second.
It is an important basic fact of dimensional analysis 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 important values have units: dimensionless numbers occur in many fields of science.
See also
In algebraic number theory, a fundamental unit is a generator for the torsion-free unit group of the ring of integers of a number field, when that group is infinite cyclic. See also Dirichlet's unit theorem.