Editing Dimensional analysis (section)

== Properties ==

=== 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group]]: The [[identity element|identity]] is written as 1;{{citation needed|reason=Both the new SI and the as-yet unpublished VIM4 make no such statement.|date=May 2021}} {{nowrap|1=L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.

=== Mechanics ===
The dimension of physical quantities of interest in [[mechanics]] can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a [[change of basis]]. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a [[Basis (linear algebra)|basis]]: they must [[Linear span|span]] the space, and be [[linearly independent]].

For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T<sup>2</sup>], L, M, while the latter can be expressed as [T = (LM/F)<sup>1/2</sup>], L, M.

On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons:
* There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not ''span the space'').
* Velocity, being expressible in terms of length and time ({{nowrap|1=V = L/T}}), is redundant (the set is not ''linearly independent'').

=== Other fields of physics and chemistry ===
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of [[electric charge]].  In [[thermodynamics]], the base set of dimensions is often extended to include a dimension for temperature, Θ.  In chemistry, the [[amount of substance]] (the number of molecules divided by the [[Avogadro constant]], ≈ {{val|6.02|e=23|u=mol-1}}) is also defined as a base dimension, N.
In the interaction of [[relativistic plasma]] with strong laser pulses, a dimensionless [[relativistic similarity parameter]], connected with the symmetry properties of the collisionless [[Vlasov equation]], is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features.

=== Polynomials and transcendental functions ===
Bridgman's theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor.<ref>{{harvnb|Bridgman|1922|loc=2. Dimensional Formulas pp. 17–27}}</ref><ref>{{cite journal |first1=Mário N. |last1=Berberan-Santos |first2=Lionello |last2=Pogliani |title=Two alternative derivations of Bridgman's theorem |journal=Journal of Mathematical Chemistry |volume=26 |pages=255–261, See §5 General Results p. 259 |date=1999 |doi=10.1023/A:1019102415633 |s2cid=14833238 |url=https://core.ac.uk/download/pdf/22873054.pdf}}</ref>  This excludes polynomials of more than one term or transcendental functions not of that form.

[[Scalar (physics)|Scalar]] arguments to [[transcendental function]]s such as [[Exponential function|exponential]], [[Trigonometric function|trigonometric]] and [[logarithm]]ic functions, or to [[inhomogeneous polynomial]]s, must be [[dimensionless quantities]].  (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.)

<!--see discussion page/transcendental functions This requirement is clear when one observes the [[Taylor expansion]]s for these functions (a sum of various powers of the function argument). For example, the logarithm of 3&nbsp;kg is undefined even though the logarithm of&nbsp;3 is nearly&nbsp;0.477. An attempt to compute ln&nbsp;3&nbsp;kg would produce, if one naively took ln&nbsp;3&nbsp;kg to mean the dimensionally meaningless "ln(1&nbsp;+&nbsp;2&nbsp;kg)",
: <math>\mathrm{2\,kg} - \frac{\mathrm{4\,kg}^2}{2} + \cdots ,</math>
which is dimensionally incompatible – the sum has no meaningful dimension – requiring the argument of transcendental functions to be dimensionless.

Another way to understand this problem is that the different coefficients ''scale'' differently under change of unit – were one to reconsider this in grams as "ln&thinsp;3000&nbsp;g" instead of "ln&thinsp;3&nbsp;kg", one could compute ln&thinsp;3000, but in terms of the [[Taylor series]], the degree 1 term would scale by 1000, the degree-2 term would scale by 1000<sup>2</sup>, and so forth – the overall output would not scale as a particular dimension.
-->
While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity {{math|1=log(''a''/''b'') = log&thinsp;''a'' − log&thinsp;''b''}}, where the logarithm is taken in any base, holds for dimensionless numbers {{math|''a''}} and {{math|''b''}}, but it does ''not'' hold if {{math|''a''}} and {{math|''b''}} are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.<ref>{{harnvb|Berberan-Santos|Pogliani|1999|page=256}}</ref>

Similarly, while one can evaluate [[monomials]] ({{math|''x''<sup>''n''</sup>}}) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for {{math|''x''<sup>2</sup>}}, the expression {{nowrap|1=(3 m)<sup>2</sup> = 9 m<sup>2</sup>}} makes sense (as an area), while for {{math|''x''<sup>2</sup> + ''x''}}, the expression {{nowrap|1=(3 m)<sup>2</sup> + 3 m = 9 m<sup>2</sup> + 3 m}} does not make sense.

However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless.  For example,
: <math> \tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot t^2 + (\mathrm{500~m/s}) \cdot t. </math>

This is the height to which an object rises in time&nbsp;{{math|''t''}} if the acceleration of [[gravity]] is 9.8 {{nowrap|metres per second per second}} and the initial upward speed is 500 {{nowrap|metres per second}}.  It is not necessary for {{math|''t''}} to be in ''seconds''.  For example, suppose {{math|''t''}}&nbsp;=&nbsp;0.01&nbsp;minutes.  Then the first term would be
: <math>\begin{align}
      &\tfrac{1}{2} \cdot (\mathrm{-9.8~m/s^2}) \cdot (\mathrm{0.01~min})^2 \\[10pt]
  ={} &\tfrac{1}{2} \cdot -9.8 \cdot \left(0.01^2\right) (\mathrm{min/s})^2 \cdot \mathrm{m} \\[10pt]
  ={} &\tfrac{1}{2} \cdot -9.8 \cdot \left(0.01^2\right) \cdot 60^2 \cdot \mathrm{m}.
\end{align}</math>

=== Combining units and numerical values ===
{{main|Physical quantity#Components}}

The value of a dimensional physical quantity {{math|''Z''}} is written as the product of a [[Unit of measurement|unit]] [{{math|''Z''}}] within the dimension and a dimensionless numerical value or numerical factor, {{math|''n''}}.<ref name=Pisanty13>For a review of the different conventions in use see: {{cite web |url=http://physics.stackexchange.com/q/77690 |title=Square bracket notation for dimensions and units: usage and conventions |last1=Pisanty |first1= E|date=17 September 2013 |website=Physics Stack Exchange  |access-date=15 July 2014}}</ref>
: <math>Z = n \times [Z] = n [Z]</math>

When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A [[conversion factor]], which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed:
: <math> \mathrm{1\,ft} = \mathrm{0.3048\,m}</math> is identical to <math> 1 = \frac{\mathrm{0.3048\,m}}{\mathrm{1\,ft}}.</math>

The factor 0.3048&nbsp;m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted.

Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units.

=== Quantity equations ===
{{distinguish|Quantity theory of money}}

A '''quantity equation''', also sometimes called a '''complete equation''', is an equation that remains valid independently of the [[unit of measurement]] used when expressing the [[physical quantities]].<ref name ="nist" />

In contrast, in a ''numerical-value equation'', just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit.

For example, a quantity equation for [[displacement (geometry)|displacement]] {{math|''d''}} as [[speed]] {{math|''s''}} multiplied by time difference {{math|''t''}} would be:
: {{math|1=''d'' = ''s'' ''t''}}
for {{math|''s''}} = 5&nbsp;m/s, where {{math|''t''}} and {{math|''d''}} may be expressed in any units, [[conversion of units|converted]] if necessary.
In contrast, a corresponding numerical-value equation would be:
: {{math|1=''D'' = 5 ''T''}}
where {{math|''T''}} is the numeric value of {{math|''t''}} when expressed in seconds and {{math|''D''}} is the numeric value of {{math|''d''}} when expressed in metres.

Generally, the use of numerical-value equations is discouraged.<ref name ="nist">{{Cite book|url=https://physics.nist.gov/cuu/pdf/sp811.pdf|title=Guide for the Use of the International System of Units (SI): The Metric System|last=Thompson|first=Ambler|date=November 2009 |publisher=DIANE Publishing|isbn=9781437915594|language=en}}</ref>