An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. Bound variables are assigned values within the expression (they are for internal use) while free variables can take on values from outside the expression. A number is an abstract entity that represents a count or measurement. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the expression may be undefined. Thus an expression represents a function whose inputs are the values assigned the free variables and whose output is the resulting value of the expression. Evaluation is the systematic determination of merit, worth, and significance of something or someone. ... Partial plot of a function f. ...

For example, the expression

x / y

evaluated for x = 10, y = 5, outputs the number 2; but is undefined for y = 0. In mathematics, a division is called a division by zero if the divisor is zero. ...

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

Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

Example:

The expression

has free variable x, bound variable n, 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 x=3 is 36.

An expression must be well-formed. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic. The term well-formed, when used by itself, can refer to: A formula in logic: see WFF The way in which an HTML tag has been used in web page design: see well-formed tag This is a disambiguation page — a navigational aid which lists other pages that might otherwise...

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. Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician and logician who was responsible for some of the foundations of theoretical computer science. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ... Programming language is the current computer science collaboration of the week! Please help improve it to featured article standard. ...

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. In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ...

See also

• Expression (programming)
• Algebraic closure
• Combinator
• Functional programming
• Equation
• Inequation

An expression in a programming language is a combination of values and functions or procedures, interpreted according to the particular rules of precedence and of association for a particular programming language, which computes and then returns another value. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ... This article is about a topic in theoretical computer science, and is not to be confused with combinatorial logic, a topic in electronics. ... Functional programming is a programming paradigm that conceives computation as the evaluation of mathematical functions and avoids state and mutable data. ... In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ... In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ...

External links

• Axiomatic Theory of Formulas - theory of expressions on high abstraction level.
• Plot mathematical expressions this system plots math equations, graphs, diagrams, and even animated cartoons of transformation of math expressions and arithmetic operations. Knowledge of TeX not required.