# Expression (mathematics)

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An expression combines numbers, operators, and/or free variables and bound variables: bound variables are defined in the expression (they are for internal use), free variables are taken from the context.

For a given combination of values for the free variables, an expression may be evaluated to a value, and is said to have that value, although for some combinations of values of the free variables, the expression may be undefined. Thus an expression represents a function of the values for the free variables.

The evaluation of an expression is dependent on the definition of the mathematical operators and system of values that forms the context of an expression.

Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same value, i.e., they represent the same function.

Example:

The expression

[itex]\sum_{y=1}^{3} 2xy[itex]

has free variable x, bound variable y, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent with the simpler expression 12x. The value for y=3 is 36.

Expressions and their evaluation were formalised by Alonzo Church and Stephen Kleene in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages.

One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable. This is also true of any expression in any system that has power equivalent to the lambda calculus.

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