Variable (mathematics)

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol, typically a letter, that holds a place for constants, often numbers.^{[1][2][3][4][5][6]} 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.

Sometimes the same symbol can be used to denote both a variable and a constant, that is a well defined mathematical object. For example, the Greek letter π generally represents the number π, but has also been used to denote a projection. Similarly the letter e often denotes Euler's number, but has been used to denote an unassigned coefficient for quartic function and higher degree polynomials. Even the symbol ⁠${\displaystyle 1}$⁠ has been used to denote an identity element of an arbitrary field. These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant.^[7]

Variables are often used for representing matrices, functions, their arguments, sets and their elements, vectors, spaces, etc.^[8]

In mathematical logic, a variable is either a symbol representing an unspecified constant of the theory, or a variable which is being quantified over.^[9][10][11]