variable

A variable is a symbol that represents a mathematical object.

A free variable is a symbol which denotes positions in an expression where a substitution might take place. The idea is related to a placeholder (symbol to be later replaced by a value).

A bound variable is a symbol has been bound to a specific value or range of values in the universe.

An unknown is a variable in an equation which has to be solved for.
An indeterminate is a symbol which is treated as a variable but does not stand for anything except itself.
A parameter is a symbol present in a function's definition which is not an argument, but allows a definition to capture a broader parametric family of functions. It is a quantity which is fixed, but not necessarily known.
A constant is a variable which represents a function with no arguments, a constant function.
See also: random variable, independent and dependent variables, discrete and continuous variables

Example 1

The quadratic formula solves any quadratic equation by substituting the values of the coefficients of that equation with the variables that represent them in the formula.

A general quadratic function is conventionally written as $a x^{2} + b x + c$ , where $a, b, c$ are parameters (also here constants) while $x$ is the variable of the function. We might write it more explicitly as

x \mapsto a x^{2} + b x + c

which shows more clearly that the function "depends on" a dynamic $x$ as its argument, and $a, b, c$ must be specified to distinguish a particular function from its parametric family.

Example 2

In the expression

\sum_{k = 1}^{10} f (k, n)

$n$ is a free variable and $k$ is a bound variable.