comutative

Commutative is the standard term for a property of a binary operation in mathematics. The word is often misspelled as comutative, but the correct term is commutative. An operation is commutative if changing the order of the operands does not change the result.

Formally, a binary operation ∘ on a set S is commutative if a ∘ b = b ∘ a for every a and b in S. This property holds for many familiar operations on numbers and sets.

Common examples include:

- Addition of real or complex numbers: a + b = b + a

- Multiplication of real or complex numbers: a × b = b × a

- Union and intersection of sets: A ∪ B = B ∪ A and A ∩ B = B ∩ A

- Dot product of vectors in Euclidean space: u · v = v · u

- Scalar multiplication in vector spaces: c(u) = (u)c yields a commutative behavior with respect to scalar and vector multiplication in many contexts

Some operations are not commutative:

- Subtraction and division: a − b ≠ b − a and a ÷ b ≠ b ÷ a in general

- Matrix multiplication: AB ≠ BA in general

- Function composition: f ∘ g ≠ g ∘ f in general

- Multiplication of quaternions: the product ab ≠ ba in general

In algebraic structures, commutativity is a defining feature of abelian or commutative structures. An abelian group has a commutative binary operation, and a commutative (or symmetric) ring has a commutative multiplication. In contrast, many algebraic systems are non-commutative, which leads to richer and more complex theory.

The concept is foundational across mathematics, informing both theory and practical computation.