Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol, typically a letter that is used for naming a mathematical object, often a number.^[1][2][3] One say colloqually that the variable represents or denotes the object and that the object is the value of the variable.

Originally, the term "variable" was used primarily for the argument of a function, in which case its value can vary in the 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 y in ${\displaystyle y=f(x).}$

A variable may represent a unspecified number that remain fix during the resolution of a problem; in which case, it is often called a parameter. A variable may denote an unknown number that has to be determined; in which case, it is called a unknown; for example, in the quadratic equation ${\displaystyle ax^{2}+bx+c=0,}$ the variables ${\displaystyle a,b,c}$ are parameters, and ${\displaystyle x}$ is the unknown.

A variable may denote a constant, that is a well defined mathematical object. For example, the Greek letter π is a variable that denotes generally the number π. However, this use of the term "variable" is counterintuitive, and therefore generally avoided at elementary level.

Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc.^[4]

In mathematical logic, a variable is either a symbol representing an unspecified term of the theory, or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation.