Expression (mathematics)

In mathematics, an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions.^[1] Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

Expressions are commonly distinguished from formulas: expressions are a kind of mathematical object, whereas formulas are statements about mathematical objects.^[2] 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 the inequality ${\displaystyle 8x-5\geq 3}$ is a formula.

To evaluate an expression means to find a numerical value equivalent to the expression.^[3][4] Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression ${\displaystyle 8\times 2-5}$ simplifies to ${\displaystyle 16-5}$, and evaluates to ${\displaystyle 11.}$

An expression is often used to define a function, by taking the variables to be arguments, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression.^[5] 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. An expression with no variables would define a constant function. Usually, two expressions are considered equal or equivalent if they define the same function. Such an equality is called a "semantic equality", that is, both expressions "mean the same thing."