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. Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, is an expression, while is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to true or false, depending on the values that are given to the variables occurring in the expressions. For example takes the value false if x is given a value less than –1, and the value true otherwise. The use of expressions ranges from the simple: (linear polynomial) (quadratic polynomial) (rational fraction) to the complex: Syntax (logic) 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 × 3 is well-formed, but the following expression is not: Semantics and Formal semantics (logic) 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.