In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers (constants), variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic symbols.
Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example:
would not be considered a mathematical expression but only a meaningless jumble.
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; 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.
Being an expression is a syntactic concept; although different mathematical fields have different notions of valid expressions, the values associated to variables does not play a role. See formal language for general considerations on how expressions are constructed, and formal semantics for questions concerning attaching meaning (values) to expressions.
