View source for Variable (mathematics)

{{Short description|Symbol representing a mathematical object}}
{{distinguish|Variable (computer science)}}
{{Use mdy dates|date=November 2021}}
In [[mathematics]], a '''variable''' (from [[Latin language|Latin]] ''[[wikt:variabilis|variabilis]]'', "changeable") is a [[Mathematical symbol|symbol]], typically a letter, that refers to an unspecified [[mathematical object]].<ref>{{cite book |last=Sobolev |first=S.K. (originator) |title=Encyclopedia of Mathematics |title-link=Encyclopedia of Mathematics |publisher=[[Springer Publishing|Springer]] |isbn=1402006098 |chapter=Individual variable |quote=A symbol of a formal language used to denote an arbitrary element (individual) in the structure described by this language. |access-date=2024-09-05 |chapter-url=http://encyclopediaofmath.org/index.php?title=Individual_variable&oldid=17515}}</ref><ref>{{cite book |last=Beckenbach |first=Edwin F |url=https://archive.org/details/collegealgebra0000beck_d0s1/mode/2up |title=College algebra |publisher=Wadsworth |year=1982 |isbn=0-534-01007-5 |edition=5th |quote=A variable is a symbol representing an unspecified element of a given set.}}</ref><ref>{{cite book |last=Landin |first=Joseph |url=https://archive.org/details/introductiontoal00land/page/2/mode/2up?q=variable |title=An Introduction to Algebraic Structures |publisher=[[Dover Publications]] |year=1989 |isbn=0-486-65940-2 |location=New York |page=204 |quote=A variable is a symbol that holds a place for constants.}}</ref> One says colloquially that the variable ''represents'' or ''denotes'' the object, and that any valid candidate for the object is the [[value (mathematics)|value]] of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a [[Set (mathematics)|set]], such as the set of [[real numbers]].

The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables {{mvar|p}} and {{mvar|q}} and require that the value of the square of {{mvar|p}} is twice the square of {{mvar|q}}, which in algebraic notation can be written {{math|1=''p''<sup>2</sup> = 2 ''q''<sup>2</sup>}}. A definitive proof that this relationship is impossible to satisfy when {{mvar|p}} and {{mvar|q}} are restricted to integer numbers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.

Originally, the term "variable" was used primarily for the [[argument of a function]], in which case its value can ''vary'' in the [[domain of a function|domain of the function]]. This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as {{math|''y''}} in {{math|1=''y'' = ''f''(''x'')}}.

A variable may represent an unspecified number that remains fixed during the resolution of a problem; in which case, it is often called a ''parameter''. A variable may denote an unknown<!-- Please, do not link this occurrence of "unknown". This is an adjective used in its common-language meaning --> number that has to be determined; in which case, it is called an [[unknown (mathematics)|unknown]]; for example, in the [[quadratic equation]] {{math|1=''ax''<sup>2</sup> + ''bx'' + ''c'' = 0}}, the variables {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are parameters, and {{math|''x''}} is the unknown.

Sometimes the same symbol can be used to denote both a variable and a [[Constant (mathematics)|constant]], that is a well defined mathematical object. For example, the [[Greek letter]] {{math|''π''}} generally represents the number [[Pi|{{math|''π''}}]], but has also been used to denote a [[Projection (mathematics)|projection]]. Similarly, the letter {{math|''e''}} often denotes [[E (mathematical constant)|Euler's number]], but has been used to denote an unassigned [[coefficient]] for [[quartic function]] and higher [[Degree of a polynomial|degree polynomials]]. Even the symbol {{math|1}} has been used to denote an [[identity element]] of an arbitrary [[Field (mathematics)|field]]. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant.<ref name="80000-2:2019">{{cite web |title=ISO 80000-2:2019 |url=https://www.iso.org/standard/64973.html |archive-url=https://cdn.standards.iteh.ai/samples/64973/329519100abd447ea0d49747258d1094/ISO-80000-2-2019.pdf |archive-date=2019-09-15 |access-date=2019-09-15 |website=Quantities and units, Part 2: Mathematics |publisher=[[International Organization for Standardization]]}}</ref>

Variables are often used for representing [[matrix (mathematics)|matrices]], [[Function (mathematics)|functions]], their arguments, [[Set (mathematics)|set]]s and their [[Element (mathematics)|element]]s, [[vector (mathematics and physics)|vector]]s, [[space (mathematics)|space]]s, etc.<ref>[[#SW|Stover & Weisstein]].</ref>

In [[mathematical logic]], a ''variable'' is a symbol that either represents an unspecified constant of the theory, or is being [[Quantifier (logic)|quantified]] over.<ref>{{cite journal |last=van Dalen |first=Dirk |date=2008 |title=Logic and Structure |url=https://www.cin.ufpe.br/~mlogica/livros/Logic%20and%20Structure%20-%20Van%20Dalen.pdf |journal=[[Springer-Verlag]] |edition=4th |page=57 |doi=10.1007/978-3-540-85108-0 |isbn=978-3-540-20879-2}}</ref><ref>{{cite book |last1=Feys |first1=Robert |author-link=Robert Feys |title=Dictionary of symbols of mathematical logic |last2=Fitch |first2=Frederic Brenton |author-link2=Frederic Fitch |publisher=[[North-Holland Publishing Company|North-Holland Pub. Co]] |year=1969 |location=Amsterdam |lccn=67030883}}</ref><ref>{{citation |last1=Shapiro |first1=Stewart |title=Classical Logic |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/logic-classical/#BuilBloc |access-date=2024-09-01 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |last2=Kouri Kissel |first2=Teresa |editor2-last=Nodelman |editor2-first=Uri}}</ref>

== History ==
{{Broader|History of algebra|History of mathematical notation}}

=== Early history ===
[[File:Rhind Mathematical Papyrus.jpg|thumb|284x284px|[[Rhind Mathematical Papyrus]]]]
The earliest uses of an "unknown quantity" date back to at least the [[Ancient Egyptian mathematics|Ancient Egyptians]] with the [[Moscow Mathematical Papyrus]] (c. 1500 BC) which described problems with unknowns rhetorically, called the "Aha problems". The "Aha problems" involve finding unknown quantities (referred to as ''aha'', "stack") if the sum of the quantity and part(s) of it are given (The [[Rhind Mathematical Papyrus]] also contains four of these type of problems). For example, problem 19 asks one to calculate a quantity taken {{frac|1|1|2}} times and added to 4 to make 10.<ref name="Clagett">Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. {{ISBN|0-87169-232-5}}</ref> In modern mathematical notation: {{math|1={{sfrac|3|2}}''x'' + 4 = 10}}. Around the same time in Mesopotamia, [[Babylonian mathematics|mathematics of the Old Babylonian period]] (c. 2000 BC – 1500 BC) was more advanced, also studying quadratic and [[Cubic equation|cubic equations]].<ref name=":0" />

[[File:Euclid-proof-white-background.svg|thumb|A page from [[Euclid's Elements|Euclid's ''Elements'']]]]
In works of [[Greek mathematics|ancient greece]] such as [[Euclid's Elements|Euclid's ''Elements'']] (c. 300 BC), mathematics was described [[Geometry|gemoetrically]]. For example, ''The Elements'', proposition 1 of Book II, [[Euclid]] includes the proposition:

''"If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments."''

This corresponds to the algebraic identity {{math|1=''a''(''b'' + ''c'') = ''ab'' + ''ac''}} ([[distributivity]]), but is described entirely geometrically. Euclid, and other greek geometers, also used single letters refer to geometric points and shapes. This kind of algebra is now sometimes called [[Greek geometric algebra]].<ref name=":0">{{cite book |last=Boyer |first=Carl B. (Carl Benjamin) |author-link=Carl Benjamin |url=https://archive.org/details/historyofmathema00boye |title=A History of Mathematics |date=1991 |publisher=[[Wiley (publisher)|Wiley]] |isbn=978-0-471-54397-8 |location=New York}}</ref>

[[Diophantus]] of [[History of Alexandria|Alexandria]],<ref>[http://www.ms.uky.edu/~carl/ma330/projects/diophanfin1.html Diophantine Equations]. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999.</ref> pioneered a form of [[syncopated algebra]] in his ''[[Arithmetica]]'' (c. 200 AD), which introduced symbolic manipulation of expressions with unknowns and powers, but without modern symbols for [[Relation (mathematics)|relations]] (such as [[Equality (mathematics)|equality]] or [[Inequality (mathematics)|inequality]]) or [[Exponentiation|exponents]].<ref name="Boyer2">Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."</ref> An unknown number was called <math>\zeta</math>.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub_gb_7DDQAAAAMAAJ/page/n472|456]]</ref> The square of <math>\zeta</math> was <math>\Delta^v</math>; the cube was <math>K^v</math>; the fourth power was <math>\Delta^v\Delta</math>; and the fifth power was <math>\Delta K^v</math>.<ref>A History of Greek Mathematics: From Aristarchus to Diophantus.  By Sir Thomas Little Heath. Pg [[iarchive:bub_gb_7DDQAAAAMAAJ/page/n474|458]]</ref> So for example, what would be written in modern notation as:
<math display="block">x^3 - 2x^2 + 10x -1,</math>
would be written in Diophantus's syncopated notation as:
: <math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>

In the 7th century BC, [[Brahmagupta]] used different colours to represent the unknowns in algebraic equations in the ''[[Brāhmasphuṭasiddhānta]]''. One section of this book is called "Equations of Several Colours".<ref>{{Harvnb|Tabak|2014|page=[https://books.google.com/books?id=h-zRieb7VbwC&pg=PA40 40]}}.</ref> Greek and other ancient mathematical advances, were often trapped in long periods of stagnation, and so there were few revolutions in notation, but this began to change by the [[early modern period]].

=== Early modern period ===
At the end of the 16th century, [[François Viète]] introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns.<ref>{{Harvnb|Fraleigh|1989|page=[https://archive.org/details/firstcourseinabs0000fral/page/276/mode/2up 276]}}.</ref>

In 1637, [[René Descartes]] "invented the convention of representing unknowns in equations by {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}, and knowns by {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}".<ref>{{Harvnb|Sorell|2000|page=19}}.</ref> Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887 [[Scientific American]] article.<ref>{{cite book|url=https://books.google.com/books?id=moM9AQAAIAAJ|title=Scientific American|date=1887-09-03|publisher=Munn & Company|pages=148|language=en}}</ref>

Starting in the 1660s, [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] independently developed the [[infinitesimal calculus]], which essentially consists of studying how an [[infinitesimal]] variation of a ''time-varying quantity,'' called a [[Fluent (mathematics)|Fluent]], induces a corresponding variation of another quantity which is a ''[[function (mathematics)|function]]'' of the first variable. Almost a century later, [[Leonhard Euler]] fixed the terminology of infinitesimal calculus, and introduced the notation {{math|1=''y'' = ''f''(''x'')}} for a function {{math|''f''}}, its '''variable''' {{math|''x''}} and its value {{math|''y''}}. Until the end of the 19th century, the word ''variable'' referred almost exclusively to the [[argument of a function|arguments]] and the [[value (mathematics)|values]] of functions.

In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere [[differentiable function|differentiable]] [[continuous function]]. To solve this problem, [[Karl Weierstrass]] introduced a new formalism consisting of replacing the intuitive notion of [[limit (mathematics)|limit]] by a formal definition. The older notion of limit was "when the ''variable'' {{math|''x''}} varies and tends toward {{math|''a''}}, then {{math|''f''(''x'')}} tends toward {{math|''L''}}", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
: <math>(\forall \epsilon >0) (\exists \eta >0) (\forall x) \;|x-a|<\eta </math><math> \; \Rightarrow |L-f(x)|<\epsilon,</math>
in which none of the five variables is considered as varying.

This static formulation led to the modern notion of variable, which is simply a symbol representing a [[mathematical object]] that either is unknown, or may be replaced by any element of a given [[set (mathematics)|set]] (e.g., the set of [[real number]]s).

== Notation ==
Variables are generally denoted by a single letter, most often from the [[Latin alphabet]] and less often from the [[Greek alphabet|Greek]], which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in {{math|''x''<sub>2</sub>}}), another variable ({{math|''x''<sub>''i''</sub>}}), a word or abbreviation of a word as a label ({{math|''x''<sub>total</sub>}}) or a [[mathematical expression]] ({{math|''x''<sub>2''i''+1</sub>}}). Under the influence of [[Variable (computer science)|computer science]], some variable names in pure mathematics consist of several letters and digits. Following [[René Descartes]] (1596–1650), letters at the beginning of the alphabet such as {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are commonly used for known values and parameters, and letters at the end of the alphabet such as {{math|''x''}}, {{math|''y''}}, {{math|''z''}} are commonly used for unknowns and variables of functions.<ref name=E004>Edwards Art. 4</ref> In printed mathematics, the norm is to set variables and constants in an italic typeface.<ref>{{Harvnb|Hosch|2010|page=[https://books.google.com/books?id=ad0P0elU1_0C&pg=PA71 71].}}</ref>

For example, a general [[quadratic function]] is conventionally written as {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}}, where {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are parameters (also called [[Constant (mathematics)|constants]], because they are [[constant function]]s), while {{math|''x''}} is the variable of the function. A more explicit way to denote this function is {{math|''x'' ↦ ''ax''<sup>2</sup> + ''bx'' + ''c''}}, which clarifies the function-argument status of {{math|''x''}} and the constant status of {{math|''a''}}, {{math|''b''}} and {{math|''c''}}. Since {{math|''c''}} occurs in a term that is a constant function of {{math|''x''}}, it is called the [[constant term]].<ref>{{Harvnb|Foerster|2006|page=[https://archive.org/details/algebratrigonome00paul_0/page/18/mode/2up 18]}}.</ref>

Specific branches and applications of mathematics have specific [[naming conventions]] for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D [[coordinate space]] are conventionally called {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}. In physics, the names of variables are largely determined by the [[physical quantity]] they describe, but various naming conventions exist. A convention often followed in [[probability]] and [[statistics]] is to use {{math|''X''}}, {{math|''Y''}}, {{math|''Z''}} for the names of [[random variable]]s, keeping {{math|''x''}}, {{math|''y''}}, {{math|''z''}} for variables representing corresponding better-defined values.

=== Conventional variable names ===
<!-- Not a comprehensive list -->
* {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''d''}} (sometimes extended to {{math|''e''}}, {{math|''f''}}) for parameters or [[coefficient]]s
* {{math|''a''<sub>0</sub>}}, {{math|''a''<sub>1</sub>}}, {{math|''a''<sub>2</sub>}}, ... for situations where distinct letters are inconvenient
* {{math|''a<sub>i</sub>''}} or {{math|''u<sub>i</sub>''}} for the {{math|''i''}}th term of a [[sequence]] or the {{math|''i''}}th coefficient of a [[Series (mathematics)|series]]
* {{math|''f''}}, {{math|''g''}}, {{math|''h''}} for [[Function (mathematics)|functions]] (as in {{math|''f''(''x'')}})
* {{math|''i''}}, {{math|''j''}}, {{math|''k''}} (sometimes {{math|''l''}} or {{math|''h''}}) for varying [[integers]] or indices in an [[indexed family]], or [[unit vectors]]
* {{math|''l''}} and {{math|''w''}} for the length and width of a figure
* {{math|''l''}}  also for a line, or in number theory for a prime number not equal to {{math|''p''}}
* {{math|''n''}} (with {{math|''m''}} as a second choice) for a fixed integer, such as a count of objects or the [[degree of a polynomial]]
* {{math|''p''}} for a [[prime number]] or a [[probability]]
* {{math|''q''}} for a [[prime power]] or a [[quotient]]
* {{math|''r''}} for a [[radius]], a [[remainder]] or a [[correlation coefficient]]
* {{math|''t''}} for [[time]]
* {{math|''x''}}, {{math|''y''}}, {{math|''z''}} for the three [[Cartesian coordinates]] of a point in [[Euclidean geometry]] or the corresponding [[Cartesian coordinate system|axes]]
* {{math|''z''}} for a [[complex number]], or in statistics a [[normal distribution|normal random variable]]
* {{math|''α''}}, {{math|''β''}}, {{math|''γ''}}, {{math|''θ''}}, {{math|''φ''}} for [[angle measure]]s
* {{math|''ε''}} (with {{math|''δ''}} as a second choice) for an arbitrarily small positive number
* {{math|''λ''}} for an [[eigenvalue]]
* {{math|Σ}} (capital sigma) for a sum, or {{math|''σ''}} (lowercase sigma) in statistics for the [[standard deviation]]<ref>{{cite web |last=Weisstein |first=Eric W. |title=Sum |url=https://mathworld.wolfram.com/ |access-date=2022-02-14 |website=mathworld.wolfram.com |language=en}}</ref>
* {{math|''μ''}} for a [[mean]]

== Specific kinds of variables ==
It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general [[cubic equation]]
: <math>ax^3+bx^2+cx+d=0,</math>
is interpreted as having five variables: four, {{math|''a'', ''b'', ''c'', ''d''}}, which are taken to be given numbers and the fifth variable, {{math|''x'',}} is understood to be an ''unknown'' number. To distinguish them, the variable {{math|''x''}} is called ''an unknown'', and the other variables are called ''parameters'' or ''[[coefficient]]s'', or sometimes ''constants'', although this last terminology is incorrect for an equation, and should be reserved for the [[function (mathematics)|function]] defined by the left-hand side of this equation.

In the context of functions, the term ''variable'' refers commonly to the arguments of the functions. This is typically the case in sentences like "[[function of a real variable]]", "{{math|''x''}} is the variable of the function {{math|1=''f'' : ''x'' ↦ ''f''(''x'')}}", "{{math|''f''}} is a function of the variable {{math|''x''}}" (meaning that the argument of the function is referred to by the variable {{math|''x''}}).

In the same context, variables that are independent of {{math|''x''}} define [[constant function]]s and are therefore called ''constant''. For example, a ''[[constant of integration]]'' is an arbitrary constant function that is added to a particular [[antiderivative]] to obtain the other antiderivatives. Because of the strong relationship between [[polynomial]]s and [[polynomial function]]s, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.

Other specific names for variables are:
* An '''unknown''' is a variable in an [[equation]] which has to be solved for.
* An '''[[indeterminate (variable)|indeterminate]]''' is a symbol, commonly called variable, that appears in a [[polynomial]] or a [[formal power series]]. Formally speaking, an indeterminate is not a variable, but a [[constant (mathematics)|constant]] in the [[polynomial ring]] or the ring of [[formal power series]]. However, because of the strong relationship between polynomials or [[power series]] and the [[function (mathematics)|functions]] that they define, many authors consider indeterminates as a special kind of variables.
* A '''[[parameter]]''' is a quantity (usually a number) which is a part of the input of a problem, and remains constant during the whole solution of this problem. For example, in [[mechanics]] the mass and the size of a solid body are ''parameters'' for the study of its movement. In [[computer science]], ''parameter'' has a different meaning and denotes an argument of a function.
* '''[[Free variables and bound variables]]'''
* A '''[[random variable]]''' is a kind of variable that is used in [[probability theory]] and its applications.

All these denominations of variables are of [[semantics|semantic]] nature, and the way of computing with them ([[syntax (logic)|syntax]]) is the same for all.

=== Dependent and independent variables ===
{{main|Dependent and independent variables}}
In [[calculus]] and its application to [[physics]] and other sciences, it is rather common to consider a variable, say {{math|''y''}}, whose possible values depend on the value of another variable, say {{math|''x''}}. In mathematical terms, the ''dependent'' variable {{math|''y''}} represents the value of a [[function (mathematics)|function]] of {{math|''x''}}. To simplify formulas, it is often useful to use the same symbol for the dependent variable {{math|''y''}} and the function mapping {{math|''x''}} onto {{math|''y''}}. For example, the state of a physical system depends on measurable quantities such as the [[pressure]], the [[temperature]], the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.

Therefore, in a formula, a '''dependent variable''' is a variable that is implicitly a function of another (or several other) variables. An '''independent variable''' is a variable that is not dependent.<ref>Edwards Art. 5</ref>

The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation {{math|''f''(''x'', ''y'', ''z'')}}, the three variables may be all independent and the notation represents a function of three variables. On the other hand, if {{math|''y''}} and {{math|''z''}} depend on {{math|''x''}} (are ''dependent variables'') then the notation represents a function of the single ''independent variable'' {{math|''x''}}.<ref>Edwards Art. 6</ref>

=== Examples ===
If one defines a function {{math|''f''}} from the [[real number]]s to the real numbers by
: <math>f(x) = x^2+\sin(x+4)</math>
then ''x'' is a variable standing for the [[argument of a function|argument]] of the function being defined, which can be any real number.

In the identity
: <math>\sum_{i=1}^n i = \frac{n^2+n}2</math>
the variable {{math|''i''}} is a summation variable which designates in turn each of the integers {{math|1, 2, ...,  ''n''}} (it is also called '''index''' because its variation is over a discrete set of values) while {{math|''n''}} is a parameter (it does not vary within the formula).

In the theory of [[polynomials]], a polynomial of degree 2 is generally denoted as {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}}, where {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are called [[coefficient]]s (they are assumed to be fixed, i.e., parameters of the problem considered) while {{math|''x''}} is called a variable. When studying this polynomial for its [[polynomial function]] this {{math|''x''}} stands for the function argument. When studying the polynomial as an object in itself, {{math|''x''}} is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

==== Example: the ideal gas law ====
Consider the equation describing the ideal gas law, 
<math display=block>PV = Nk_\text{B}T.</math>
This equation would generally be interpreted to have four variables, and one constant. The constant is {{math|''k''<sub>B</sub>}}}, the [[Boltzmann constant]]. One of the variables, {{math|''N''}}, the number of particles, is a positive integer (and therefore a discrete variable), while the other three, {{math|''P''}}, {{math|''V''}} and {{math|''T''}}, for pressure, volume and temperature, are continuous variables.

One could rearrange this equation to obtain {{math|''P''}} as a function of the other variables,
<math display=block>P(V, N, T) = \frac{Nk_\text{B}T}{V}.</math>
Then {{math|''P''}}, as a function of the other variables, is the dependent variable, while its arguments, {{math|''V''}}, {{math|''N''}} and {{math|''T''}}, are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here {{math|''P''}} is a function <math>P: \mathbb{R}_{>0} \times \mathbb{N} \times \mathbb{R}_{>0} \rightarrow \mathbb{R}</math>.

However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say {{math|''T''}}. This gives a function
<math display = block>P(T) = \frac{Nk_\text{B}T}{V},</math>
where now {{math|''N''}} and {{math|''V''}} are also regarded as constants. Mathematically, this constitutes a [[partial application]] of the earlier function {{math|''P''}}.

This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard {{math|''k''<sub>B</sub>}} as a variable to obtain a function 
<math display=block>P(V, N, T, k_\text{B}) = \frac{Nk_\text{B}T}{V}.</math>

== Moduli spaces ==

{{See also |moduli spaces}}

Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a [[parabola]], 
<math display = block>y = ax^2 + bx + c,</math>
where {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, {{math|''x''}} and {{math|''y''}} are all considered to be real. The set of points {{math|(''x'', ''y'')}} in the 2D plane satisfying this equation trace out the graph of a parabola. Here, {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are regarded as constants, which specify the parabola, while {{math|''x''}} and {{math|''y''}} are variables.

Then instead regarding {{math|''a''}}, {{math|''b''}} and {{math|''c''}} as variables, we observe that each set of 3-tuples {{math|(''a'', ''b'', ''c'')}} corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a '''moduli space of parabolas'''.

== See also ==
* [[Lambda calculus]]
* [[Observable variable]]
* [[Physical constant]]
* [[Propositional variable]]

== References ==
{{reflist}}

== Bibliography ==
{{refbegin}}
* {{cite book |last=Edwards |first=Joseph |title=An Elementary Treatise on the Differential Calculus  |publisher= MacMillan and Co. |location=London | year=1892 |edition=2nd |url=https://archive.org/details/in.ernet.dli.2015.109607}}
* {{cite book |last=Foerster |first=Paul A. |title=Algebra and Trigonometry: Functions and Applications |edition=classics |year=2006 |isbn=    978-0-13-165711-3 |publisher=Prentice Hall |location=Upper Saddle River, NJ |url-access=registration |url=https://archive.org/details/algebratrigonome00paul_0}}
* {{cite book |last=Fraleigh |first=John B. |title=A First Course in Abstract Algebra |publisher=Addison-Wesley |edition=4th |year=1989 |location=United States |isbn=978-0-201-52821-3 |url=https://archive.org/details/firstcourseinabs0000fral |url-access=registration}}
* {{cite book |editor-last=Hosch |editor-first=William L. |title=The Britannica Guide to Algebra and Trigonometry |publisher=Britannica Educational Publishing |year=2010 |isbn=978-1-61530-219-2 |url=https://books.google.com/books?id=ad0P0elU1_0C}}
* {{cite journal |last=Menger |first=Karl |author-link=Karl Menger |title=On Variables in Mathematics and in Natural Science |journal=The British Journal for the Philosophy of Science |publisher=University of Chicago Press |volume=5 |issue=18 |pages=134–142 |year=1954 |doi=10.1093/bjps/V.18.134 |jstor=685170}}
* {{cite book |last=Peregrin |first=Jaroslav |author-link=Jaroslav Peregrin |chapter-url=http://w.jarda.peregrin.cz/mybibl/PDFTxt/414.pdf |chapter=Variables in Natural Language: Where do they come from? |editor1-last=Böttner |editor1-first=Michael |editor2-last=Thümmel |editor2-first=Wolf |title=Variable-Free Semantics |year=2000 |pages=46–65 |publisher=Osnabrück Secolo |isbn=978-3-929979-53-4}}
* {{cite journal |last=Quine |first=Willard V. |author-link=Willard Van Orman Quine |url=https://voices.uchicago.edu/wittgenstein/files/2008/05/quine-variables-explained-away.pdf |title=Variables Explained Away |journal=Proceedings of the American Philosophical Society |publisher=American Philosophical Society |volume=104 |number=3 |pages=343–347 |year=1960 |jstor=985250}}
* {{cite book |last=Sorell |first=Tom |author-link=Tom Sorell |title=Descartes: A Very Short Introduction |year=2000 |location=New York |publisher=Oxford University Press |isbn=978-0-19-285409-4 |url=https://books.google.com/books?id=EksSDAAAQBAJ}}
* {{anchor|SW}}<!-- Anchor for citation -->{{cite encyclopedia |last1=Stover |first1=Christopher |last2=Weisstein |first2=Eric W. |author2-link=Eric W. Weisstein |editor-last=Weisstein |editor-first=Eric W. |title=Variable |url=https://mathworld.wolfram.com/Variable.html |access-date=2021-11-22 |encyclopedia=Wolfram MathWorld |publisher=Wolfram Research}}
* {{cite book |last=Tabak |first=John |title=Algebra: Sets, Symbols, and the Language of Thought |date=2014 |publisher=Infobase Publishing |isbn=978-0-8160-6875-3 |url=https://books.google.com/books?id=h-zRieb7VbwC}}
{{refend}}

{{Mathematical logic}}
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{{DEFAULTSORT:Variable (mathematics)}}
[[Category:Variables (mathematics)| ]]
[[Category:Algebra]]
[[Category:Calculus]]
[[Category:Elementary mathematics]]
[[Category:Syntax (logic)]]
[[Category:Mathematical logic]]