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Dimensions of quantities
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Quantities and units
The International System of Units (SI) and the corresponding system of quantities
Dimensions of quantities
Coherent units, derived units with special names, and the SI prefixes
SI units in the framework of general relativity
Units for quantities that describe biological effects
Legislation on units
Historical note

Search the SI brochure
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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 capital theta
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 = Lalpha Mbeta Tgamma ldelta capital thetaepsilon Nzeta Jeta

where the exponents alpha, beta, gamma, delta, epsilon, zeta, and eta, 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.

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.

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Note: For the official text, please refer to the PDF files available at:
  • (in English) and
  • (in French).