Commutativity

Commutativity is a property of a binary operation on a set. An operation is commutative if a * b = b * a for all elements a and b in the set. When this holds, the operation is described as commutative, and the structure is often called abelian, after the Norwegian mathematician Niels Henrik Abel.

Common examples include addition and multiplication of ordinary numbers: a + b = b + a and a × b = b × a. The same is true in many related systems, such as real numbers, integers, and matrices under addition. Logical operations such as AND and OR, as well as the bitwise XOR, are also commutative.

Not all familiar operations are commutative. Subtraction and division depend on order: a − b ≠ b − a, a ÷ b ≠ b ÷ a in general. Matrix multiplication AB ≠ BA in general, though it can be equal for some pairs of matrices. Function composition is typically not commutative: f ∘ g ≠ g ∘ f in general.

Abelian groups are algebraic structures in which the group operation is commutative. A ring is called commutative if its multiplication is commutative. In linear algebra, vector addition is commutative, and scalar multiplication with vectors satisfies (cd)v = c(dv). The commutator [A, B] = AB − BA measures the extent of non-commutativity for matrices.

Commutativity is distinct from associativity, which concerns how operations are grouped. It plays a key role in simplifying expressions, enabling parallel computation, and shaping theories in algebra, geometry, physics, and computer science.