Order did not need a name until algebra got strange
For centuries people quietly used the fact that three plus four equals four plus three. Only in the nineteenth century, as weird new algebraic structures appeared, did mathematicians bother to name commutativity—and to notice how many operations refuse it.
A binary operation is commutative when swapping the two inputs leaves the output unchanged. Addition and multiplication behave this way for natural numbers, integers, rationals, reals, and complex numbers, and in every field. Subtraction and division do not: three minus five is not five minus three. Matrix multiplication of square matrices is generally noncommutative, and the three-dimensional vector cross product is anti-commutative—swapping factors flips the sign.
When a structure's main operation happens to commute, language often flags it: a commutative or abelian group; a commutative ring (ring addition always commutes; the label marks multiplication). For algebras, "commutative algebra" usually means associative algebras whose multiplication commutes. Some truth-functional connectives fail commutativity because their truth tables change when arguments swap—material implication is a standard example.
Ancient practice already leaned on the property. Egyptians used commutativity of multiplication to simplify products; Euclid assumed it for multiplication in the Elements. The adjective arrived late. François Servois used commutatives in an 1814 memoir for functions with the exchange property; the French root points to exchanging or switching. English adopted the word commutative in 1838 via Duncan Gregory's essay on symbolical algebra, printed in 1840 by Edinburgh's Royal Society in its Transactions. Once noncommuting operations flooded research, the everyday arithmetic habit finally needed a label. The long anonymity shows how invisible a law can stay when every schoolbook example obeys it.
Source: Commutative property
