




Dimensions of quantities

SI Brochure, Section 1.3

By convention physical quantities are organized in a system of dimensions. Each of the seven base quantities used in the SI is regarded as having its own dimension, which is symbolically represented by a single sans serif roman capital letter. The symbols used for the base quantities, and the symbols used to denote their dimension, are given as follows.
Base quantities and dimensions used in the SI

Base quantity 
Symbol for quantity 
Symbol for dimension 

length 
l, x, r, etc. 
L 
mass 
m 
M 
time, duration 
t 
T 
electric current 
I, i 
l 
thermodynamic temperature 
T 

amount of substance 
n 
N 
luminous intensity 
I_{ v} 
J 

All other quantities are derived quantities, which may be written in terms of the base quantities by the equations of physics. The dimensions of the derived quantities are written as products of powers of the dimensions of the base quantities using the equations that relate the derived quantities to the base quantities. In general the dimension of any quantity Q is written in the form of a dimensional product,
dim Q = L^{} M T l^{} ^{} N^{} J^{}
where the exponents , _{}, _{}, , , _{}, and _{}, which are generally small integers which can be positive, negative or zero, are called the dimensional exponents. The dimension of a derived quantity provides the same information about the relation of that quantity to the base quantities as is provided by the SI unit of the derived quantity as a product of powers of the SI base units.


Quantity symbols are always written in an italic font, and symbols for dimensions in sansserif roman capitals.
For some quantities a variety of alternative symbols may be used, as indicated for length and electric current.
Note that symbols for quantities are only recommendations, in contrast to symbols for units that appear elsewhere in this brochure whose style and form is mandatory (see Chapter 5).
Dimensional symbols and exponents are manipulated using the ordinary rules of algebra. For example, the dimension of area is written as L^{2}; the dimension of velocity as LT^{–1}; the dimension of force as LMT^{–2}; and the dimension of energy is written as L^{2}MT^{–2}.


There are some derived quantities Q for which the defining equation is such that all of the dimensional exponents in the expression for the dimension of Q are zero. This is true, in particular, for any quantity that is defined as the ratio of two quantities of the same kind. Such quantities are described as being dimensionless, or alternatively as being of dimension one. The coherent derived unit for such dimensionless quantities is always the number one, 1, since it is the ratio of two identical units for two quantities of the same kind.
There are also some quantities that cannot be described in terms of the seven base quantities of the SI at all, but have the nature of a count. Examples are number of molecules, degeneracy in quantum mechanics (the number of independent states of the same energy), and the partition function in statistical thermodynamics (the number of thermally accessible states). Such counting quantities are also usually regarded as dimensionless quantities, or quantities of dimension one, with the unit one, 1. 
For example, refractive index is defined as the ratio of the speed of light in vacuum to that in the medium, and is thus a ratio of two quantities of the same kind. It is therefore a dimensionless quantity.
Other examples of dimensionless quantities are plane angle, mass fraction, relative permittivity, relative permeability, and finesse of a FabryPerot cavity. 


We are pleased to present the updated (2014) 8th edition of the SI Brochure, which defines and presents the Système International d'Unités, the SI (known in English as the International System of Units).
Chapter 1: Introduction
Chapter 2: SI units
Chapter 3: Decimal multiples and submultiples of SI units
 SI prefixes

Factor 
Name 
Symbol 

Factor 
Name 
Symbol 

10^{1} 
deca 
da 

10^{–1} 
deci 
d 
10^{2} 
hecto 
h 
10^{–2} 
centi 
c 
10^{3} 
kilo 
k 
10^{–3} 
milli 
m 
10^{6} 
mega 
M 
10^{–6} 
micro 
µ 
10^{9} 
giga 
G 
10^{–9} 
nano 
n 
10^{12} 
tera 
T 
10^{–12} 
pico 
p 
10^{15} 
peta 
P 
10^{–15} 
femto 
f 
10^{18} 
exa 
E 
10^{–18} 
atto 
a 
10^{21} 
zetta 
Z 
10^{–21} 
zepto 
z 
10^{24} 
yotta 
Y 
10^{–24} 
yocto 
y 

 The kilogram
Chapter 4: Units outside the SI
Chapter 5: Writing unit symbols and names, and expressing the values of quantities
General principles for the writing of unit symbols and numbers were first given by the 9th CGPM (1948, Resolution 7). These were subsequently elaborated by ISO, IEC, and other international bodies. As a consequence, there now exists a general consensus on how unit symbols and names, including prefix symbols and names, as well as quantity symbols should be written and used, and how the values of quantities should be expressed. Compliance with these rules and style conventions, the most important of which are presented in this chapter, supports the readability of scientific and technical papers.
This appendix lists those decisions of the CGPM and the CIPM that bear directly upon definitions of the units of the SI, prefixes defined for use as part of the SI, and conventions for the writing of unit symbols and numbers. It is not a complete list of CGPM and CIPM decisions. For a complete list, reference must be made to the BIPM website, successive volumes of the Comptes Rendus des Séances de la Conférence Générale des Poids et Mesures (CR) and ProcèsVerbaux des Séances du Comité International des Poids et Mesures (PV) or, for recent decisions, to Metrologia.
Since the SI is not a static convention, but evolves following developments in the science of measurement, some decisions have been abrogated or modified; others have been clarified by additions. In the SI Brochure, a number of notes have been added by the BIPM to make the text more understandable; they do not form part of the original text.
In the printed brochure, the decisions of the CGPM and CIPM are listed in strict chronological order in order to preserve the continuity with which they were taken. However in order to make it easy to locate decisions related to particular topics a table of contents is also provided, ordered by subject:
Optical radiation is able to cause chemical changes in certain living or nonliving materials: this property is called actinism, and radiation capable of causing such changes is referred to as actinic radiation. Actinic radiation has the fundamental characteristic that, at the molecular level, one photon interacts with one molecule to alter or break the molecule into new molecular species. It is therefore possible to define specific photochemical or photobiological quantities in terms of the result of optical radiation on the associated chemical or biological receptors.
In the field of metrology, the only photobiological quantity which has been formally defined for measurement in the SI is for the interaction of light with the human eye in vision. An SI base unit, the candela, has been defined for this important photobiological quantity. Several other photometric quantities with units derived from the candela have also been defined (such as the lumen and the lux, see Table 3 in Chapter 2).





