What does commutativity mean?
Commutativity is the mathematical property of an operation that allows its operands to be exchanged without changing the result. An operation is commutative when, for any values a and b it accepts, combining a with b yields exactly the same outcome as combining b with a. Addition and multiplication of numbers are classic examples — 3 + 5 equals 5 + 3, and 3 × 5 equals 5 × 3 — while subtraction, division, and matrix multiplication do not have this property, since reversing the operands changes or invalidates the result. The term belongs mainly to the vocabulary of mathematics, logic, and computer science, where recognising which operations commute often simplifies proofs and calculations considerably. Although abstract, commutativity reflects something familiar from childhood arithmetic, and encountering operations that lack it is frequently a memorable moment in a student's mathematical education. The word derives from the Latin commutare, meaning to change or exchange.
nounThe property of a mathematical operation whereby the order of the operands does not affect the result; formally, an operation * is commutative if a * b = b * a for all relevant values of a and b.
- (Mathematics and logic) The property of a binary operation such that exchanging the order of the operands leaves the result unchanged.
"The commutativity of addition means that 3 + 5 gives the same answer as 5 + 3."
"Because of the commutativity of multiplication, 4 × 9 and 9 × 4 yield the same product."
"In matrix algebra, students are often surprised to discover that multiplication lacks commutativity: AB generally does not equal BA."
Rarely used; the plural appears only in specialised discussions comparing the commutativity properties of several distinct operations.
"The paper examines the commutativities of several tensor products."
Subtraction fails commutativity — 7 − 3 and 3 − 7 give different answers — which is why mathematicians spent centuries treating it as second-class next to addition and multiplication.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of commutativity
Commutativity derives from the Latin verb commutare, meaning 'to change thoroughly' or 'to interchange', formed from com- ('together') plus mutare ('to change'). The related adjective 'commutative' was adopted into mathematics during the early nineteenth century as algebraists began systematically studying the structural properties of operations; the noun 'commutativity' followed as the corresponding abstract property. The same Latin root also gave English 'commute' in its everyday sense of regular travel between home and work, via the notion of exchange — originally the substitution of one payment for another.
Related word forms
How commutativity is actually used
A technical term used chiefly in mathematics, computer science, and logic; it rarely appears in everyday conversation. It is typically discussed in relation to specific operations (addition, multiplication, set union, logical conjunction), some of which possess the property and some of which do not. The adjective form 'commutative' is far more frequent in ordinary mathematical prose.
Easily confused with commutativity
Associativity concerns how operands are grouped, i.e. whether (a * b) * c equals a * (b * c), whereas commutativity concerns whether the order of two operands can be swapped.