In

^{2} + ''bx'' + ''c'', where ''a'', ''b'' and ''c'' are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while ''x'' is called a variable. When studying this polynomial for its polynomial function this ''x'' stands for the function argument. When studying the polynomial as an object in itself, ''x'' is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.

_{0}, ''a''_{1}, ''a''_{2}, ... for situations where distinct letters are inconvenient
* ''a_{i}'' or ''u_{i}'' for the ''i''-th term of a

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a variable (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

'' variabilis'', "changeable") is a symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...

that represents a mathematical object. A variable may represent a number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...

, a vector, a matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

, a function, the argument of a function, a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

, or an element of a set.
Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, ...

solves any quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...

by substituting the numeric values of the coefficients of that equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...

for the variables that represent them in the quadratic formula. In mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, a ''variable'' is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.
History

In ancient works such asEuclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...

's ''Elements'', single letters refer to geometric points and shapes. In the 7th century, Brahmagupta
Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...

used different colours to represent the unknowns in algebraic equations in the '' Brāhmasphuṭasiddhānta''. One section of this book is called "Equations of Several Colours".
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.
In 1637, René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...

"invented the convention of representing unknowns in equations by ''x'', ''y'', and ''z'', and knowns by ''a'', ''b'', and ''c''". Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in a 1887 Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...

article.
Starting in the 1660s, Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...

and Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...

independently developed the infinitesimal calculus, which essentially consists of studying how an infinitesimal variation of a ''variable quantity'' induces a corresponding variation of another quantity which is a '' function'' of the first variable. Almost a century later, Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

fixed the terminology of infinitesimal calculus, and introduced the notation for a function , its variable and its value . Until the end of the 19th century, the word ''variable'' referred almost exclusively to the arguments and the 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 continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

. To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...

by a formal definition. The older notion of limit was "when the ''variable'' varies and tends toward , then tends toward ", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
:$(\backslash forall\; \backslash epsilon\; >0)\; (\backslash exists\; \backslash eta\; >0)\; (\backslash forall\; x)\; \backslash ;,\; x-a,\; <\backslash eta\; \backslash Rightarrow\; ,\; L-f(x),\; <\backslash epsilon,$
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

(e.g., the set of real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s).
Notation

Variables are generally denoted by a single letter, most often from theLatin alphabet
The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and the ...

and less often from the Greek, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in ), another variable (), a word or abbreviation of a word () or a mathematical expression (). Under the influence of computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, some variable names in pure mathematics consist of several letters and digits. Following René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...

(1596–1650), letters at the beginning of the alphabet such as ''a'', ''b'', ''c'' are commonly used for known values and parameters, and letters at the end of the alphabet such as (''x'', ''y'', ''z'') are commonly used for unknowns and variables of functions.Edwards Art. 4 In printed mathematics, the norm is to set variables and constants in an italic typeface.
For example, a general quadratic function is conventionally written as $a\; x^2\; +\; b\; x\; +\; c\backslash ,$, where ''a'', ''b'' and ''c'' are parameters (also called constants
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...

, because they are constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic propertie ...

s), while ''x'' is the variable of the function. A more explicit way to denote this function is $x\backslash mapsto\; a\; x^2\; +\; b\; x\; +\; c\; \backslash ,$, which clarifies the function-argument status of ''x'' and the constant status of ''a'', ''b'' and ''c''. Since ''c'' occurs in a term that is a constant function of ''x'', it is called the constant term.
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 ''x'', ''y'', and ''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
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

and statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

is to use ''X'', ''Y'', ''Z'' for the names of random variables, keeping ''x'', ''y'', ''z'' for variables representing corresponding better-defined values.
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 :$ax^3+bx^2+cx+d=0,$ is interpreted as having five variables: four, , which are taken to be given numbers and the fifth variable, is understood to be an ''unknown'' number. To distinguish them, the variable is called ''an unknown'', and the other variables are called ''parameters'' or '' coefficients'', or sometimes ''constants'', although this last terminology is incorrect for an equation, and should be reserved for the 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", " is the variable of the function ", " is a function of the variable " (meaning that the argument of the function is referred to by the variable ). In the same context, variables that are independent of defineconstant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic propertie ...

s and are therefore called ''constant''. For example, a ''constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...

'' is an arbitrary constant function that is added to a particular antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...

to obtain the other antiderivatives. Because the strong relationship between polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

s and polynomial function, the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.
This use of "constant" as an abbreviation of "constant function" must be distinguished from the normal meaning of the word in mathematics. A constant, or mathematical constant is a well and unambiguously defined number or other mathematical object, as, for example, the numbers 0, 1, and the identity element of a group. Since a variable may represent any mathematical object, a letter that represents a constant is often called a variable. This is, in particular, the case of and , even when they represents Euler's number and
Other specific names for variables are:
* An unknown is a variable in an equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...

which has to be solved for.
* An indeterminate is a symbol, commonly called variable, that appears in a polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

or a formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...

. Formally speaking, an indeterminate is not a variable, but a constant in the polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...

or the ring of formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...

. However, because of the strong relationship between polynomials or power series and the functions that they define, many authors consider indeterminates as a special kind of variables.
* A parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

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
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

the mass and the size of a solid body are ''parameters'' for the study of its movement. In computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...

, ''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
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

and its applications.
All these denominations of variables are of semantic
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...

nature, and the way of computing with them (syntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituenc ...

) is the same for all.
Dependent and independent variables

Incalculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

and its application to physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

and other sciences, it is rather common to consider a variable, say , whose possible values depend on the value of another variable, say . In mathematical terms, the ''dependent'' variable represents the value of a function of . To simplify formulas, it is often useful to use the same symbol for the dependent variable and the function mapping onto . For example, the state of a physical system depends on measurable quantities such as the pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

, the temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...

, 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.
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 , the three variables may be all independent and the notation represents a function of three variables. On the other hand, if and depend on (are ''dependent variables'') then the notation represents a function of the single ''independent variable'' .
Examples

If one defines a function ''f'' from thereal number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s to the real numbers by
:$f(x)\; =\; x^2+\backslash sin(x+4)$
then ''x'' is a variable standing for the argument of the function being defined, which can be any real number.
In the identity
:$\backslash sum\_^n\; i\; =\; \backslash frac2$
the variable ''i'' is a summation variable which designates in turn each of the integers 1, 2, ..., ''n'' (it is also called index because its variation is over a discrete set of values) while ''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 ''ax''Example: the ideal gas law

Consider the equation describing the ideal gas law, $$PV\; =\; Nk\_BT.$$ This equation would generally be interpreted to have four variables, and one constant. The constant is $k\_B$, theBoltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...

. One of the variables, $N$, the number of particles, is a positive integer (and therefore a discrete variable), while the other three, $P,\; V$ and $T$, for pressure, volume and temperature, are continuous variables.
One could rearrange this equation to obtain $P$ as a function of the other variables,
$$P(V,\; N,\; T)\; =\; \backslash frac.$$
Then $P$, as a function of the other variables, is the dependent variable, while its arguments, $V,\; N$ and $T$, are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here $P$ is a function $P:\; \backslash mathbb\_\; \backslash times\; \backslash mathbb\; \backslash times\; \backslash mathbb\_\; \backslash rightarrow\; \backslash mathbb$.
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 $T$. This gives a function
$$P(T)\; =\; \backslash frac,$$
where now $N$ and $V$ are also regarded as constants. Mathematically, this constitutes a partial application of the earlier function $P$.
This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard $k\_B$ as a variable to obtain a function
$$P(V,\; N,\; T,\; k\_B)\; =\; \backslash frac.$$
Moduli spaces

Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for aparabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...

,
$$y\; =\; ax^2\; +\; bx\; +\; c,$$
where $a,\; b,\; c,\; x$ and $y$ are all considered to be real. The set of points $(x,y)$ in the 2D plane satisfying this equation trace out the graph of a parabola. Here, $a,b$ and $c$ are regarded as constants, which specify the parabola, while $x$ and $y$ are variables.
Then instead regarding $a,b$ and $c$ as variables, we observe that each set of 3-tuples $(a,b,c)$ corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas.
Conventional variable names

* ''a'', ''b'', ''c'', ''d'' (sometimes extended to ''e'', ''f'') for parameters or coefficients * ''a''sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

or the ''i''-th coefficient of a series
* ''e'' for Euler's number
* ''f'', ''g'', ''h'' for functions (as in $f(x)$)
* ''i'' for the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...

* ''i'', ''j'', ''k'' (sometimes ''l'' or ''h'') for varying integers or indices in an indexed family, or unit vectors
* ''l'' and ''w'' for the length and width of a figure
* ''l'' also for a line, or in number theory for a prime number not equal to ''p''
* ''n'' (with ''m'' as a second choice) for a fixed integer, such as a count of objects or the degree of an equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in F ...

* ''p'' for a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

or a probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

* ''q'' for a prime power or a quotient
* ''r'' for a radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

, a remainder or a correlation coefficient
* ''t'' for time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

* ''x'', ''y'', ''z'' for the three Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

of a point in Euclidean geometry or the corresponding axes
* ''z'' for a complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

, or in statistics a normal random variable
* ''α'', ''β'', ''γ'', ''θ'', ''φ'' for angle measures
* ''ε'' (with ''δ'' as a second choice) for an arbitrarily small positive number
* ''λ'' for an eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

* Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the standard deviation
* ''μ'' for a mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...

See also

*Lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation th ...

* Observable variable
* Physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...

* Propositional variable
References

Bibliography

* * * * * * * * * * {{DEFAULTSORT:Variable (mathematics) Algebra Calculus Elementary mathematics Syntax (logic) Mathematical logic