It is a fundamental property of many binary operations, and many mathematical proofs depend on it.

The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed.

Thus, this property was not named until the 19th century, when mathematics started to become formalized.

[1][2] A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

An operation that does not satisfy the above property is called noncommutative.

That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.

For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative.

This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation).

Matrix multiplication of square matrices is almost always noncommutative, for example:

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.

[7][8] Euclid is known to have assumed the commutative property of multiplication in his book Elements.

[9] 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.

[2] 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.

" is a metalogical symbol representing "can be replaced in a proof with".

The following logical equivalences demonstrate that commutativity is a property of particular connectives.

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property.

In higher branches of mathematics, such as analysis and linear algebra the commutativity of well-known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.

The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change.

In contrast, the commutative property states that the order of the terms does not affect the final result.

Most commutative operations encountered in practice are also associative.

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example,

More such examples may be found in commutative non-associative magmas.

Some forms of symmetry can be directly linked to commutativity.

When a commutative operation is written as a binary function

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as

(also called products of operators) on a one-dimensional wave function

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely.

For example, the position and the linear momentum in the

, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.