Editing Dimensional analysis (section)

=== Mathematical properties ===
{{further|Buckingham π theorem}}

The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an [[abelian group]]: The [[identity element|identity]] is written as 1;{{citation needed|reason=Both the new SI and the as-yet unpublished VIM4 make no such statement.|date=May 2021}} {{nowrap|1=L<sup>0</sup> = 1}}, and the inverse of L is 1/L or L<sup>−1</sup>. L raised to any integer power {{math|''p''}} is a member of the group, having an inverse of L<sup>{{math|−''p''}}</sup> or 1/L<sup>{{math|''p''}}</sup>.  The operation of the group is multiplication, having the usual rules for handling exponents ({{nowrap|1=L<sup>{{math|''n''}}</sup> × L<sup>{{math|''m''}}</sup> = L<sup>{{math|''n''+''m''}}</sup>}}). Physically, 1/L can be interpreted as [[reciprocal length]], and 1/T as reciprocal time (see [[reciprocal second]]).

An abelian group is equivalent to a [[module (mathematics)|module]] over the integers, with the dimensional symbol {{gaps|T<sup>{{math|''i''}}</sup>|L<sup>{{math|''j''}}</sup>|M<sup>{{math|''k''}}</sup>}} corresponding to the tuple {{math|(''i'', ''j'', ''k'')}}. When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to [[scalar multiplication]] in the module.

A basis for such a module of dimensional symbols is called a set of [[base quantities]], and all other vectors are called derived units. As in any module, one may choose different [[Basis (linear algebra)|bases]], which yields different systems of units (e.g., [[ampere#Proposed future definition|choosing]] whether the unit for charge is derived from the unit for current, or vice versa).

The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module, {{math|(0, 0, 0)}}.

In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like {{math|''V''{{isup|''L''{{sup|1/2}}}}}}.{{sfn|Tao|2012|loc="With a bit of additional effort (and taking full advantage of the one-dimensionality of the vector spaces), one can also define spaces with fractional exponents&nbsp;..."}} However, it is not possible to take arbitrary fractional powers of units, due to [[representation theory|representation-theoretic]] obstructions.{{sfn|Tao|2012|loc="However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units&nbsp;..."}}

One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions {{math|''M''}} and {{math|''L''}}, one has the vector spaces {{math|''V''{{isup|''M''}}}} and {{math|''V''{{isup|''L''}}}}, and can define {{math|1=''V''{{isup|''ML''}} := ''V''{{isup|''M''}} ⊗ ''V''{{isup|''L''}}}} as the [[tensor product]]. Similarly, the dual space can be interpreted as having "negative" dimensions.<ref>{{harvnb|Tao|2012}} "Similarly, one can define {{math|''V''{{isup|''T''{{isup|−1}}}}}} as the dual space to {{math|''V''{{isup|''T''}}}} ..."</ref> This corresponds to the fact that under the [[natural pairing]] between a vector space and its dual, the dimensions cancel, leaving a [[dimensionless]] scalar.

The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The [[Kernel (linear algebra)#nullity|nullity]] describes some number (e.g., {{math|''m''}}) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {{math|{{mset|π<sub>1</sub>, ..., π<sub>''m''</sub>}}}}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and [[exponentiating]]) together the measured quantities to produce something with the same unit as some derived quantity {{math|''X''}} can be expressed in the general form
: <math>X = \prod_{i=1}^m (\pi_i)^{k_i}\,.</math>

Consequently, every possible [[#Commensurability|commensurate]] equation for the physics of the system can be rewritten in the form
: <math>f(\pi_1,\pi_2, ..., \pi_m)=0\,.</math>

Knowing this restriction can be a powerful tool for obtaining new insight into the system.