Expression (math) encyclopedia topics

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.

Types of expressions

Common examples of mathematical expressions include

arithmetic expressions

2 + 3

and algebraic expressions such as

polynomials

x^2 + 3x - 4

rational expressions

2/x + x/2

and equations

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

x^2 + 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.

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 values 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 formal interpretation 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

sum_{n=1}^{3} (2nx)

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.

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.