In mathematics, an expression is an arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. 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 usually denote mathematical objects, whereas formulas are statements about mathematical objects. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example,
8
x
−
5
{\displaystyle 8x-5}
and
3
{\displaystyle 3}
are both expressions, while the inequality
8
x
−
5
≥
3
{\displaystyle 8x-5\geq 3}
is a formula. However, formulas are often considered as expressions that can be evaluated to the Boolean values true or false.
To evaluate an expression means to find a numerical value equivalent to the expression. Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression
8
×
2
−
5
{\displaystyle 8\times 2-5}
simplifies to
16
−
5
{\displaystyle 16-5}
, and evaluates to
11.
{\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. For example,
x
↦
x
2
+
1
{\displaystyle x\mapsto x^{2}+1}
and
f
(
x
)
=
x
2
+
1
{\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."
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