Physical quantity

A physical quantity is a physical property that can be quantified.

Formally, the International Vocabulary of Metrology', 3rd edition (VIM) defines quantity as:

property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference^[1]

Hence the value of a physical quantity Q is expressed as the product of a numerical value {Q} and a unit of measurement [Q].

Q = {Q} x [Q]

Quantity calculus describes how to do math with quantities.

1 Examples
2 Symbols for physical quantities
3 Units of physical quantities
4 Base quantities, derived quantities and dimensions
5 Extensive and intensive quantities
6 Physical quantities as coordinates over spaces of physical qualities
7 See also
8 Notes
9 References

Examples

If the temperature T of a body is quantified (measured) as 25 degrees Celsius this is written as:

T = 25 x °C = 25 °C

where T is the symbol of the physical quantity "temperature", 25 is the numerical factor and °C is the unit.

If a person weighs 120 pounds, then "120" is the numerical value and "pound" is the unit. This physical quantity mass would be written as "120 lbs", or

m = 120 lbs

An example employing SI units and scientific notation for the number, might be a measurement of power written as

P = 42.3 x 10³ W,

Here, P represents the physical quantity of power, 42.3 x 10³ is the numerical value {P}, and 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 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 E_k is usually used to denote kinetic energy and c_p heat capacity at constant pressure. (Note the difference in the style of the subscripts: “k” is the abbreviation of the word “kinetic”, whereas “p” is the symbol of the physical quantity “pressure” rather than the abbreviation of the word “pressure”.)

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

Symbols for physical quantities that are vectors are bold italic type. If, e.g., u is the speed of a particle, then the straightforward notation for its velocity is u.

Note that concrete numbers, even those denoted by letters, are always roman (upright) type, e.g.: 1, 2, e (for the base of natural logarithm), i (for the imaginary unit) or π (for 3.14...). Symbols of concrete functions such as sin α must be roman type too. Although not followed by Wikipedia, operators like d in dx should also be roman type.

Units of physical quantities

Most physical quantities 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 kilograms (kg), pounds (lb), or Daltons (Da). SI units are usually preferred today.

Base quantities, derived quantities and dimensions

The notion of physical dimension of a physical quantity was introduced by Fourier in 1822.^[2] By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension. The seven base quantities of the International System of Quantities (ISQ) and their corresponding SI units are listed in the following table. Other conventions may have a different number of fundamental units (e.g. the CGS and MKS systems of units).

International System of Units base quantities
Name	Symbol for quantity	Symbol for dimension	SI base unit	Symbol for unit
Length	l, x, r, etc.	L	meter	m
Time	t	T	second	s
Mass	m	M	kilogram	kg
Electric current	I, i	I	ampere	A
Thermodynamic temperature	T	θ	kelvin	K
Amount of substance	n	N	mole	mol
Luminous intensity	I_v	J	candela	cd

All other quantities are derived quantities since their dimensions are derived from those of base quantities by multiplication and division. For example, the physical quantity velocity is derived from base quantities length and time and has dimension L/T. Some derived physical quantities have dimension 1 and are said to be dimensionless quantities.

Further information: dimensional analysis

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 physical quantities are prefixed in order to further qualify their meaning:

specific is added to refer to a quantity which is expressed per unit mass (such as specific heat capacity)
molar is added to refer to a quantity which is expressed per unit 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 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.^[3]

Notes

↑ Joint Committee for Guides in Metrology (JCGM), International Vocabulary of Metrology, Basic and General Concepts and Associated Terms (VIM), III ed., Pavillon de Breteuil : JCGM 200:2008, 1.1 (on-line)
↑ Fourier, Joseph. Théorie analytique de la chaleur, Firmin Didot, Paris, 1822. (In this book, Fourier introduces the concept of physical dimensions for the physical quantities.)
↑ Tarantola, Albert. Elements for physics - Quantities, qualities and intrinsic theories, Springer, 2006. ISBN 3-540-25302-5. [1]

References

Cook, Alan H. The observational foundations of physics, Cambridge, 1994. ISBN 0-521-45597-