# Physical quantity

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A physical quantity is either a quantity within physics that can be measured (e.g. mass, volume, etc.), or the result of a measurement. A physical quantity Q is usually expressed as the product of a numerical value {Q} and a physical unit [Q].

Q = {Q} x [Q]

(SI units are usually preferred today). The notion of physical dimension of a physical quantity was introduced by Fourier (1822).

## Example

If a certain value of power is written as

P = 42.3 x 103 W = 42.3 kW,

then

P represents the physical quantity of power
42.3 x 103 is the numerical value {P}
k is the SI prefix kilo, representing 103
W is the symbol for the unit of power [P], the watt

## Symbols for physical quantities

Usually, the symbols for physical quantities are chosen to be a single lower case or capital letter of the Latin or Greek alphabet written in italic type. Often, the symbols are modified by subscripts and superscripts, in order to specify what they pertain to - for instance Ep is usually used to denote potential energy and cp heat capacity at constant pressure.

Symbols for quantities should be chosen according to the international recommendations from ISO 31, the IUPAC red book and the IUPAC green book. For example, the recommended symbol for a physical quantity of mass is m, and the recommended symbol for a quantity of charge is Q.

## Units of physical quantities

Most physical quantites Q include a unit [Q] (where [Q] means "unit of Q"). Neither the name of a physical quantity, nor the symbol used to denote it, implies a particular choice of unit. For example, a quantity of mass might be represented by the symbol m, and could be expressed in the units kilogram (kg), pounds (lbs), or dalton (Da).

## Base quantities, derived quantities and dimensions

By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension. In the SI system of units, there are seven base units, but other conventions may have a different number of fundamental units. The base quantities according to the International System of Quantities (ISQ) and their dimensions are listed in the following table:

ISQ base quantities
Name Symbol for quantity Symbol for dimension SI base unit
Length l L meter
Time t T second
Mass m M kilogram
Electrical current I I ampere
Thermodynamic temperature T θ kelvin
Amount of substance n N mole
Luminous intensity Iv J candela

All other quantities are derived quantities since their dimensions are derived from those of base quantities by multiplication and division. For example, a physical quantity velocity is derived from base quantities length and time and has dimension L/T. See Dimensional Analysis for further information.

Some derived physical quantities have no dimension and are said to be dimensionless quantities.

## Extensive and intensive quantities

A quantity is called:

• extensive when its magnitude is additive for subsystems (volume, mass, etc.)
• intensive when the magnitude is independent of the extent of the system (temperature, pressure, etc.)

Some extensive physical quantities may be prefixed in order to further qualify their meaning:

• specific is added to refer to the quantity divided by its mass (such as specific volume)
• molar is added to refer to the quantity divided by the amount of substance (such as molar volume)

There are also physical quantities that can be classified as neither extensive nor intensive, for example angular momentum, area, force, length, and time.

## Physical quantities as coordinates over spaces of physical qualities

The meaning of the term physical quantity is generally well understood (everyone understands what it is meant by the frequency of a periodic phenomenon, or the resistance of an electric wire). It is clear that behind a set of quantities like temperature − inverse temperature − logarithmic temperature, there is a qualitative notion: the cold−hot quality. Over this one-dimensional quality space, we may choose different coordinates: the temperature, the inverse temperature, etc. Other quality spaces are multidimensional. For instance, to represent the properties of an ideal elastic medium we need 21 coefficients, that can be the 21 components of the elastic stiffness tensor $c_{ijkl}$ , or the 21 components of the elastic compliance tensor (inverse of the stiffness tensor), or the proper elements (six eigenvalues and 15 angles) of any of the two tensors, etc. Again, we are selecting coordinates over a 21-dimensional quality space. On this space, each point represents a particular elastic medium.

It is always possible to define the distance between two points of any quality space, and this distance is —inside a given theoretical context— uniquely defined. For instance, two periodic phenomena can be characterized by their periods, $T_1$ and $T_2$, or by their frequencies, $\nu_1$ and $\nu_2$ . The only definition of distance that respects some clearly defined invariances is $D = |$log$(T_2/T_1 ) | = |$log$(\nu_2/\nu_1 ) |$ .

These notions have implications in physics. As soon as we accept that behind the usual physical quantities there are quality spaces, that usual quantities are only special coordinates over these quality spaces, and that there is a metric in each space, the following question arises: Can we do physics intrinsically, i.e., can we develop physics using directly the notion of physical quality, and of metric, and without using particular coordinates (i.e., without any particular choice of physical quantities)? In fact, physics can (and must?) be developed independently of any particular choice of coordinates over the quality spaces, i.e., independently of any particular choice of physical quantities to represent the measurable physical qualities. This point of view has recently been developed (Tarantola, 2006 [1] ).