In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations.

The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied.

A binary operation ${\displaystyle *}$ on a set S is commutative if $${\displaystyle x*y=y*x}$$ for all ${\displaystyle x,y\in S}$. An operation that is not commutative is said to be noncommutative.

One says that x commutes with y or that x and y commute under ${\displaystyle *}$ if $${\displaystyle x*y=y*x.}$$

So, an operation is commutative if every two elements commute. An operation is noncommutative if there are two elements such that ${\displaystyle x*y\neq y*x.}$ This does not exclude the possibility that some pairs of elements commute.

Some types of algebraic structures involve an operation that does not require commutativity. If this operation is commutative for a specific structure, the structure is often said to be commutative. So,

However, in the case of algebras, the phrase "commutative algebra" refers only to associative algebras that have a commutative multiplication.

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products. Euclid is known to have assumed the commutative property of multiplication in his book Elements. Formal uses of the commutative property arose in the late 18th and early 19th centuries when mathematicians began to work on a theory of functions. Nowadays, the commutative property is a well-known and basic property used in most branches of mathematics.