Expression (mathematics)

In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can denote numbers (constants), variables, operations, functions. Other symbols include punctuation signs and brackets (often used for grouping, that is for considering a part of the expression as a single symbol).

Many authors distinguish an expression from a formula, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects.^[1] This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, ${\displaystyle 8x-5}$ is an expression, while ${\displaystyle 8x-5\geq 3}$ is a formula.

Expressions can be evaluated or partially evaluated by replacing operations that appear in them with their result. For example, the expression ${\displaystyle 8\times 2-5}$ evaluates partially to ${\displaystyle 16-5}$ and totally to ${\displaystyle 11.}$

An expression that contains variables is often used to define a function, by associating to possible values of the variables the result of the evalution of the expression obtained by replacing each variable with the corresponding value. the variables the result of the evaluation.^[2] For example, ${\displaystyle x\mapsto x^{2}+1}$ and ${\displaystyle f(x)=x^{2}+1}$ define the function that associates to each number its square plus one.