What does associativity mean?
Associativity is a fundamental property in mathematics describing how operands can be grouped within an operation without altering the result. An operation is associative if, for any three values, combining the first two before the third gives the same answer as combining the last two first — formally, (a ∘ b) ∘ c equals a ∘ (b ∘ c). Addition and multiplication are classic examples: whether you compute (2 + 3) + 4 or 2 + (3 + 4), the total is 9. Subtraction and division lack this property, which is why parentheses matter so much in those expressions. Beyond arithmetic, associativity is central to abstract algebra, where it defines semigroups and monoids, and to computer science, where operator associativity governs how expressions are parsed. The word derives from Latin roots meaning 'to join together', reflecting the idea of binding terms into groups.
nounThe mathematical property of an operation whereby the grouping of operands does not affect the result, so that (a ∘ b) ∘ c = a ∘ (b ∘ c). More broadly, the quality or state of being associative.
- In mathematics: the property of a binary operation whereby different groupings of operands yield the same result.
- By extension: the general quality of being associative, as in psychology or linguistics where ideas or elements are said to associate freely.
"Addition and multiplication both exhibit associativity, which is why we can rearrange parentheses without changing the outcome."
"Because matrix multiplication lacks associativity with respect to some operations, programmers must be explicit about evaluation order."
"The associativity of the operator determines how an expression such as a − b − c is parsed."
Rarely pluralized; the plural may appear when comparing distinct operations or systems, each with its own associativity property.
"Different logics can impose different associativities on the same connective."
Subtraction fails the test — because (10 − 5) − 2 and 10 − (5 − 2) give different answers, subtraction is not associative, and that single fact shapes how compilers parse your code.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of associativity
Associativity derives from the Latin verb associare, meaning 'to join with' or 'to unite', itself formed from ad- ('toward') and sociare ('to associate'), from socius ('companion'). English adopted 'associate' from Latin via medieval usage, and the noun 'associativity' was coined in the nineteenth century alongside the formalization of algebraic structures, particularly in the work on group theory. It shares its root with related words such as 'association' and 'associative'.
Related word forms
How associativity is actually used
A technical term used chiefly in mathematics, computer science, logic, and abstract algebra; it is formal in register and rarely appears in everyday speech. In computing, 'left associativity' and 'right associativity' describe how operators of equal precedence are grouped.
Easily confused with associativity
Commutativity concerns the order of operands (a + b = b + a), whereas associativity concerns their grouping under parentheses.
Distributivity describes how one operation interacts with another across a sum, while associativity only concerns regrouping within a single operation.