Mathematical expression
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, punctuation, grouping, and other aspects of logical syntax.
The use of expressions ranges from the simple:
to the complex:
Mathematical expressions include arithmetic expressions, polynomials, algebraic expressions, closed-form expressions, and analytical expressions. The table below highlights some similarities and differences between these different types.
Being an expression is a syntactic concept.
An expression must be well-formed: the operators must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, 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 2 + 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. These semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator to designate an internal direct sum.
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