What does commutative mean?
Commutative is chiefly a mathematical adjective describing an operation in which the order of the operands makes no difference to the outcome. Addition and multiplication are commutative because 2 + 3 equals 3 + 2 and 2 × 3 equals 3 × 2; subtraction and division are not, since 5 − 2 differs from 2 − 5. This property — often called the commutative law or commutative property — is one of the foundational axioms taught in arithmetic and algebra, and it underpins more advanced structures such as commutative rings and groups. The term derives from the Latin commutare, 'to exchange', and outside mathematics it occasionally appears in ethical contexts such as commutative justice, which concerns fair exchange between individuals. Because the concept contrasts sharply with noncommutative operations like matrix multiplication, knowing the word helps learners grasp why some calculations are forgiving about order while others are not.
(Of a mathematical operation) such that the result is independent of the order of the operands; satisfying x ∘ y = y ∘ x for all values involved.
"Matrix multiplication is not commutative, since AB does not generally equal BA."
Relating to or involving exchange or interchange between two parties, as in commutative justice.
"The court applied principles of commutative justice in ordering restitution."
In a commutative world, 3 + 5 always equals 5 + 3 — but subtraction breaks that rule, which is why mathematicians had to invent a whole word for it.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of commutative
Commutative comes from Latin commutare, meaning 'to change, alter, or exchange', formed from com- ('together') plus mutare ('to change'). The adjective entered English in the early modern period, with its mathematical sense established by the 19th century as algebra formalized the laws of arithmetic. It shares this root with commute, commuter, and the mathematical noun commutator, all built on the same idea of exchange or interchange.
Related word forms
How commutative is actually used
'Commutative' is primarily a technical term used in mathematics (arithmetic, algebra, group theory), where it names one of the fundamental properties of binary operations. Outside mathematics it appears mainly in philosophy and ethics in phrases like 'commutative justice'. It carries no emotional connotation and belongs to formal register.
Easily confused with commutative
'Communicative' means willing or able to share information readily, while 'commutative' refers to mathematical operations whose result is independent of the order of the operands.