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Expression (mathematics) - Wikipedia

Expression (mathematics)

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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.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 usually denote mathematical objects, whereas formulas are statements about mathematical objects, such as an equality.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, $8x-5$ and $3$ are both expressions, while the inequality $8x-5\ge 3$ 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.³⁴ Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression $8\times 2-5$ simplifies to $16-5$, and evaluates to $11.$

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, $x\mapsto x^2+1$ and $f(x)=x^2+1$ 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.”

Printed 2026-06-28.

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Footnotes

1. Oxford English Dictionary, s.v. “Expression (n.), sense II.7,” “A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function.” ↩

2. Stoll, Robert R. (1963). Set Theory and Logic. San Francisco, CA: Dover Publications. ISBN 978-0-486-63829-4. ↩

3. Oxford English Dictionary, s.v. “Evaluate (v.), sense a”, “Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).” ↩

4. Oxford English Dictionary, s.v. “Simplify (v.), sense 4.a”, “To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.” ↩

5. Codd, Edgar Frank (June 1970). “A Relational Model of Data for Large Shared Data Banks” (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016. Archived (PDF) from the original on 2004-09-08. Retrieved 2020-04-29. ↩

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