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Commutative property

Property of some mathematical operations

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated September 12, 2026
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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.

01Definition

A binary operation * on a set S is commutative if x*y=y*x for all 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 * if 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 x*y\neq y*x. This does not exclude the possibility that some pairs of elements commute.

The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.
The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.

02Examples

Commutative operations

Noncommutative operations

A B A ⇒ B B ⇒ A
F F T T
F T T F
T F F T
T T T T
  • The quaternion multiplication is noncommutative, since ⁠\mathbf {ij} =-\mathbf {ji}⁠, where ⁠\mathbf {i}⁠ and ⁠\mathbf {j}⁠ are two of the basis quaternions.
The addition of vectors is commutative, because
The addition of vectors is commutative, because

03Commutative structures

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.

The first known use of the term was in a French Journal published in 1814
The first known use of the term was in a French Journal published in 1814

04History and etymology

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.

The first recorded use of the term commutative was in a memoir by François Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property. Commutative is the feminine form of the French adjective commutatif, which is derived from the French noun commutation and the French verb commuter, meaning "to exchange" or "to switch", a cognate of to commute. The term then appeared in English in 1838 in Duncan Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.

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Sources and credits

This article is adapted from the Wikipedia article “Commutative property”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments.

Canonical commutation relation

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities. For example, [ x ^ , p ^ x ] = i ℏ I {\displaystyle [{\hat {x}},{\hat {p}}_{x}]=i\hbar \mathbb {I} } between the position operator x and momentum operator px in the x direction of a point particle in one dimension, where [x , px] = x px − px x is the commutator of x and px , i is the imaginary unit, and ℏ is the reduced Planck constant (h/2π), and I {\displaystyle \mathbb {I} } is the unit operator.

Centralizer and normalizer

In mathematics, especially group theory, the centralizer of a subset S in a group G is the set C G ⁡ ( S ) {\displaystyle \operatorname {C} _{G}(S)} of elements of G that commute with every element of S, or equivalently, the set of elements g ∈ G {\displaystyle g\in G} such that conjugation by g {\displaystyle g} leaves each element of S fixed. The normalizer of S in G is the set of elements N G ( S ) {\displaystyle \mathrm {N} _{G}(S)} of G that satisfy the weaker condition of leaving the set S ⊆ G {\displaystyle S\subseteq G} fixed under conjugation.