In
mathematics, a **variable** (from
Latin *
variabilis*, "changeable") is a
symbol that represents a
mathematical object. A variable may represent a
number, a
vector, a
matrix, a
function, the
argument of a function, a
set, or an
element of a set.^{
[1]}

Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the
quadratic formula solves any
quadratic equation by substituting the numeric values of the coefficients of that
equation for the variables that represent them in the quadratic formula. In
mathematical logic, 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.

In ancient works such as
Euclid's
*Elements*, single letters refer to geometric points and shapes. In the 7th century,
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".^{
[2]}

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.^{
[3]}

In 1637,
René Descartes "invented the convention of representing unknowns in equations by *x*, *y*, and *z*, and knowns by *a*, *b*, and *c*".^{
[4]} 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 article.^{
[5]}

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 *variable quantity* induces a corresponding variation of another quantity which is a *
function* of the first variable. Almost a century later,
Leonhard Euler fixed the terminology of infinitesimal calculus, and introduced the notation *y* = *f*(*x*) for a function *f*, its **variable** *x* and its value *y*. 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. To solve this problem,
Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of
limit by a formal definition. The older notion of limit was "when the *variable* *x* varies and tends toward *a*, then *f*(*x*) tends toward *L*", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula

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 (e.g., the set of real numbers).

Variables are generally denoted by a single letter, most often from the
Latin alphabet and less often from the
Greek, which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in *x*_{2}), another variable (*x*_{i}), a word or abbreviation of a word (*x*_{total}) or a
mathematical expression (*x*_{2i + 1}). Under the influence of
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 *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.^{
[6]} In printed mathematics, the norm is to set variables and constants in an italic typeface.^{
[7]}

For example, a general
quadratic function is conventionally written as , where *a*, *b* and *c* are parameters (also called
constants, because they are
constant functions), while *x* is the variable of the function. A more explicit way to denote this function is , 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.^{
[8]}

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 and
statistics is to use *X*, *Y*, *Z* for the names of
random variables, keeping *x*, *y*, *z* for variables representing corresponding better-defined values.

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

is interpreted as having five variables: four, *a*, *b*, *c*, *d*, which are taken to be given numbers and the fifth variable, *x*, is understood to be an *unknown* number. To distinguish them, the variable *x* 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", "*x* is the variable of the function *f*: *x* ↦ *f*(*x*)", "*f* is a function of the variable *x*" (meaning that the argument of the function is referred to by the variable *x*).

In the same context, variables that are independent of *x* define
constant functions 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 the strong relationship between
polynomials 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,
*π* 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 e and π, even when they represents
Euler's number and 3.14159...

Other specific names for variables are:

- An
**unknown**is a variable in an equation which has to be solved for. - An
**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 in the polynomial ring or the ring of formal power series. 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**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 semantic nature, and the way of computing with them ( syntax) is the same for all.

In
calculus and its application to
physics and other sciences, it is rather common to consider a variable, say *y*, whose possible values depend on the value of another variable, say *x*. In mathematical terms, the *dependent* variable *y* represents the value of a
function of *x*. To simplify formulas, it is often useful to use the same symbol for the dependent variable *y* and the function mapping *x* onto *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.^{
[9]}

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 *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 *y* and *z* depend on *x* (are *dependent variables*) then the notation represents a function of the single *independent variable* *x*.^{
[10]}

If one defines a function *f* from the
real numbers to the real numbers by

then *x* is a variable standing for the
argument of the function being defined, which can be any real number.

In the identity

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*^{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.

Consider the equation describing the ideal gas law,

This equation would generally be interpreted to have four variables, and one constant. The constant is , the
Boltzmann constant. One of the variables, , the number of particles, is a positive integer (and therefore a discrete variable), while the other three, and , for pressure, volume and temperature, are continuous variables.

One could rearrange this equation to obtain as a function of the other variables,

Then , as a function of the other variables, is the dependent variable, while its arguments, and , are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here is a function .

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 . This gives a function

where now and are also regarded as constants. Mathematically, this constitutes a
partial application of the earlier function .

This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard as a variable to obtain a function

Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a parabola,

where and are all considered to be real. The set of points in the 2D plane satisfying this equation trace out the graph of a parabola. Here, and are regarded as constants, which specify the parabola, while and are variables.

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

*a*,*b*,*c*,*d*(sometimes extended to*e*,*f*) for parameters or coefficients*a*_{0},*a*_{1},*a*_{2}, ... for situations where distinct letters are inconvenient*a*or_{i}*u*for the_{i}*i*-th term of a sequence or the*i*-th coefficient of a series*e*for Euler's number*f*,*g*,*h*for functions (as in )*i*for the imaginary unit*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*p*for a prime number or a probability*q*for a prime power or a quotient*r*for a radius, a remainder or a correlation coefficient*t*for time*x*,*y*,*z*for the three Cartesian coordinates of a point in Euclidean geometry or the corresponding axes*z*for a complex number, 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- Σ (capital sigma) for a sum, or σ (lowercase sigma) in statistics for the
standard deviation
^{ [11]} *μ*for a mean

**^**Stover & Weisstein.**^**Tabak 2014, p. 40.**^**Fraleigh 1989, p. 276.**^**Sorell 2000, p. 19.**^***Scientific American*. Munn & Company. September 3, 1887. p. 148.**^**Edwards Art. 4**^**Hosch 2010, p. 71.**^**Foerster 2006, p. 18.**^**Edwards Art. 5**^**Edwards Art. 6**^**Weisstein, Eric W. "Sum".*mathworld.wolfram.com*. Retrieved February 14, 2022.

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