Expression

In mathematics, the word expression is a term for any well-formed combination of mathematical symbols^{[citation needed]}. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equality sign (=).^{[1][Contradicts Formula]} For example,

x² + 3x − 4

is an expression, while

y = x² + 3x − 4

is an equation but not an expression. Neither is

)x) / y

an expression because the parentheses are not balanced.

Being an expression is a syntactic concept – the meaning of the variables is irrelevant, but different fields have different notions of validity.　See formal language for how expressions are constructed, and formal semantics for meaning.

Variables

Many mathematical expressions include letters called variables. Variables are classified as either free or bound.

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 value of the expression may be undefined. Thus an expression represents a function whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.

For example, the expression

x / y

evaluated for x = 10, y = 5, will give 2; but is undefined for y = 0.

The evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values that is its context. See Formal semantics and Interpretation (logic) for the study of this question in logic.

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:

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.

The '+' and '-' (addition and subtraction) symbols have their usual meanings. division can be expressed either with the '/' or with a horizontal dash, i.e.:

x / 2 or

are perfectly valid. Also, for multiplication one can use the symbols or a "." (dot), or else simply omit it (multiplication is implicit); so:

or or or

are all acceptable (please notice in the first example above how the "times" symbol resembles an "x" and also how the "." symbol resembles a decimal point, so to avoid confusion it's best to use one of the later two forms).

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.

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.

See also

References