In mathematics, the word expression is a very general term for any well-formed combination of mathematical symbols. For example, In logic, WFF is an abbreviation for well-formed formula. ...

x² + 3x − 4

is an expression,

)x) / 0

is not, because the parentheses are not balanced and division by 0 is undefined.

Types of expressions

Common examples of mathematical expressions include

arithmetic expressions Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...

2 + 3 = 5,

and algebraic expressions such as This article is about the branch of mathematics. ...

polynomials In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

x² + 3x − 4,

rational expressions In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

2 / x + x / 2,

and equations An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

x + 2 = 5

Manipulating expressions

Just as expressions must be formed according to certain rules (rules which may change from one mathematical specialty to another), expressions can often be given a new form, again following rules, some very general, some specific to a particular area of mathematics. For example, the expression

x² + 3x − 4

is considered equal to, and so in a sense the same as, the expression

(x + 4)(x − 1).

Variables

Many mathematical expressions include letters called variables. Variables are classified as either free or bound. In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ... In computer programming, a free variable is a variable referred to in a function that is not a local variable or an argument of that function. ...

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. Look up evaluation in Wiktionary, the free dictionary. ... This article is about functions in mathematics. ...

For example, the expression

x / y

evaluated for x = 10, y = 5, will give 2; but is undefined for y = 0. For the album by Hux Flux, see Division by Zero (album). ...

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. Example: In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

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. In logic, WFF is an abbreviation for well-formed formula. ...

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. In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ... ‹ The template below (Expand) is being considered for deletion. ... 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. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ...

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

• Algebraic closure
• Combinator
• Defined and undefined
• Expression (programming)
• Equation
• Formula
• Functional programming
• Inequation
• Well-defined

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. ... In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. ... 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. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ... In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...

External links

• Axiomatic Theory of Formulas - theory of expressions on high abstraction level.