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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a variable is a symbol which works as a placeholder for expression or
quantities Quantity is a property that can exist as a Counting, multitude or Magnitude (mathematics), magnitude, which illustrate discontinuity (mathematics), discontinuity and continuum (theory), continuity. Quantities can be compared in terms of "more", " ...
that may ''vary'' or change; is often used to represent the argument of a function or an arbitrary element of a set. In addition to
number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, 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 un ...
s, variables are commonly used to represent vectors, matrices and functions. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the , which allows one to solve every quadratic equation—by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them. In
mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...
, a ''variable'' is either a symbol representing an unspecified term of the theory (i.e., meta-variable), or a basic object of the theory—which is manipulated without referring to its possible intuitive interpretation.

# Etymology

"Variable" comes from a Latin word, ''variābilis'', with "''vari(us)''"' meaning "various" and "''-ābilis''"' meaning "-able", meaning "capable of changing".

# Genesis and evolution of the concept

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". At the end of the 16th century,
François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Gre ...
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-born philosopher, mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Anc ...
"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. 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 (mathematics), function'' of the first variable. Almost a century later, Leonhard Euler 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 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'' varies and tends toward , then tends toward ", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula :$\left(\forall \epsilon >0\right) \left(\exists \eta >0\right) \left(\forall x\right) \;, x-a, <\eta \Rightarrow , L-f\left(x\right), <\epsilon,$ 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 (e.g., the set of real numbers).

# 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 (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", " 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 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, and the identity element of a group (mathematics), group. 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 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

In calculus and its application to physics 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 (mathematics), 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, 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. 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 the real numbers to the real numbers by :$f\left(x\right) = x^2+\sin\left(x+4\right)$ 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 :$\sum_^n i = \frac2$ 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''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.

# Notation

In mathematics, the variables are generally denoted by a single letter. However, this letter is frequently followed by a subscript, as in , and this subscript may be a number, another variable (), a word or the abbreviation of a word ( and ), and even a mathematical expression. Under the influence of computer science, one may encounter in pure mathematics some variable names consisting in several letters and digits. Following the 17th century French philosopher and mathematician,
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French-born philosopher, mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Anc ...
, letters at the beginning of the alphabet, e.g. ''a'', ''b'', ''c'' are commonly used for known values and parameters, and letters at the end of the alphabet, e.g. ''x'', ''y'', ''z'', and ''t'' are commonly used for unknowns and variables of functions.Edwards Art. 4 In printed
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, the norm is to set variables and constants in an italic typeface. For example, a general quadratic function is conventionally written as: :$a x^2 + b x + c\, ,$ 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 :$x\mapsto a x^2 + b x + c \, ,$ which makes the function-argument status of ''x'' clear, and thereby implicitly 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 usually have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters. 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 actual values. There are many other notational usages. Usually, variables that play a similar role are represented by consecutive letters or by the same letter with different subscript. Below are some of the most common usages. * ''a'', ''b'', ''c'', and ''d'' (sometimes extended to ''e'' and ''f'') often represent parameters or coefficients. * ''a''0, ''a''1, ''a''2, ... play a similar role, when otherwise too many different letters would be needed. * ''ai'' or ''ui'' is often used to denote the ''i''-th term of a sequence or the ''i''-th coefficient of a series (mathematics), series. * ''f'' and ''g'' (sometimes ''h'') commonly denote Function (mathematics), functions. * ''i'', ''j'', and ''k'' (sometimes ''l'' or ''h'') are often used to denote varying integers or indices in an indexed family. They may also be used to denote unit vectors. * ''l'' and ''w'' are often used to represent the length and width of a figure. * ''l'' is also used to denote a line. In number theory, ''l'' often denotes a prime number not equal to ''p''. * ''n'' usually denotes a fixed integer, such as a count of objects or the degree of an equation. ** When two integers are needed, for example for the dimensions of a matrix (mathematics), matrix, one uses commonly ''m'' and ''n''. * ''p'' often denotes a prime numbers or a probability. * ''q'' often denotes a prime power or a quotient * ''r'' often denotes a radius, a remainder or a correlation coefficient. * ''t'' often denotes time. * ''x'', ''y'' and ''z'' usually denote the three Cartesian coordinates of a point in Euclidean geometry. By extension, they are used to name the corresponding axis (mathematics), axes. * ''z'' typically denotes a complex number, or, in statistics, a normal distribution, normal random variable. * ''α'', ''β'', ''γ'', ''θ'' and ''φ'' commonly denote angle measures. * ''ε'' usually represents an arbitrarily small positive number. ** ''ε'' and ''δ'' commonly denote two small positives. * ''λ'' is used for eigenvalues. * ''σ'' often denotes a sum, or, in statistics, the standard deviation. * ''μ'' often denotes a mean. * ''π'' is used for Pi.

* Constant of integration * Constant term, Constant term of a polynomial * Free variables and bound variables (Bound variables are also known as dummy variables) * Indeterminate (variable) * Lambda calculus * Expression (mathematics), Mathematical expression * Observable variable * Physical constant * Variable (computer science)

# Bibliography

* * Karl Menger, "On Variables in Mathematics and in Natural Science", ''The British Journal for the Philosophy of Science'' 5:18:134–142 (August 1954) * Jaroslav Peregrin,
Variables in Natural Language: Where do they come from?
, in M. Boettner, W. Thümmel, eds., ''Variable-Free Semantics'', 2000, pp. 46–65. * W.V. Quine,
Variables Explained Away
, ''Proceedings of the American Philosophical Society'' 104:343–347 (1960).

# References

{{DEFAULTSORT:Variable (Mathematics) Variables (mathematics), Algebra Calculus Elementary mathematics Syntax (logic)