Fundamental unit |
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.
In the language of measurement, physical quantity are quantifiable aspects of the world, such as time, distance, velocity, mass, momentum, energy, and weight, and Physical unit 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.
Correspondingly, a system of fundamental units is such that every other unit can be generated from them. The kilogram, metre, second, ampere, kelvin, mole (unit) and candela are the fundamental units of the SI system of units, termed SI base unit; other units such as the newton, joule, and volt can be derived from the SI base units and are therefore termed SI derived unit.
The dimension velocity, for example, is length divided by time, so its unit m/s can be generated from the fundamental units metre and second.
An important basic fact of dimensional analysis is that the fundamental units can be regarded as the basis (linear algebra) 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.
=References=
# M. J. Duff, L. B. Okun and G. Veneziano, Trialogue on the number of fundamental constants , JHEP 0203, 023 (2002) [http://arxiv.org/abs/physics/0110060 preprint].
=See also=
*SI base unit *SI *Natural units *dimensional analysis|
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