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In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable, respectively. The idea is related to a placeholder (a symbol that will later be replaced by some value), or a wildcard character that stands for an unspecified symbol.

In computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context.

A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a bound variable when we write:

For all x, (x + 1)2 = x2 + 2x + 1.

or

There exists x such that x2 = 2.

In either of these propositions, it does not matter logically whether x or some other letter is used. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use can create an ambiguity with the definition of dummy variables in regression analysis.

## Examples

Before stating a precise definition of free variable and bound variable, the following are some examples that perhaps make these two concepts clearer than the definition would:

In the expression

${\displaystyle \sum _{k=1}^{10}f(k,n),}$

n is a free variable and k is a bound variable; consequently the value of this expression depends on the value of n, but there is nothing called k on which it could depend.

In the expression

${\displaystyle \int _{0}^{\infty }x^{y-1}e^{-x}\,dx,}$

y is a free variable and x is a bound variable; consequently the value of this expression depends on the value of y, but there is nothing called x on which it could depend.

In the expression

${\displaystyle \lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}},}$

x is a free variable and h is a bound variable; consequently the value of this expression depends on the value of x, but there is nothing called h on which it could depend.

In the expression

computer programming, the term free variable refers to variables used in a function that are neither local variables nor parameters of that function. The term non-local variable is often a synonym in this context.

A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a bound variable when we write:

For all x, (x + 1)2 = x2 + 2x + 1.

or

<

There exists x such that x2 = 2.

In either of these propositions, it does not matter logically whether x or some other letter is used. However, it could be confusing to use the same letter again elsewhere in some compound proposition. That is, free variables become bound, and then in a sense retire from being available as stand-in values for other values in the creation of formulae.

The term "d

The term "dummy variable" is also sometimes used for a bound variable (more often in general mathematics than in computer science), but that use can create an ambiguity with the definition of dummy variables in regression analysis.

Before stating a precise definition of free variable and bound variable, the following are some examples that perhaps make these two concepts clearer than the definition would:

In the expression

${\displaystyle \sum _{k=1}^{10}f(k,n),}$

n is a free variable and k is a bound variable; consequently the value of this expression depends on the value of n, but there is nothing called k on which it could depend.

In the expression