Expression (mathematics)

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations, and other aspects of logical syntax.

The use of expressions ranges from the simple:

to the complex:

Forms of mathematical expressions include arithmetic expressions, polynomials, algebraic expressions, closed-form expressions, and analytical expressions. The table below illustrates the elements these forms may contain.

An expression is a syntactic construct, it must be well-formed: the allowed operators must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, have a clear order of operations, etc. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions.

For example, in the usual notation of arithmetic, the expression 1 + 2 x 3 is well-formed, but the following expression is not:

Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions.

In algebra, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 x 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See Order of operations:Calculators).

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