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In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Contrariwise if this property is not established the operation is said to be noncommutative. If x * y = y * x for a particular choice of elements x and y, then x and y are said to commute.

The most well known examples of commutative binary operations are addition (a+b) and multiplication (a*b) of real numbers; for example:

• 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
• 2 × 3 = 3 × 2 (since both expressions evaluate to 6)

Among the binary operations that are not commutative are subtraction (a − b), division (a/b), exponentiation (a^b), functional composition (f(g(x)), and tetration (a↑↑b).

Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectorsThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for, and intersection and union of sets. Important non-commutative operations are the multiplication of matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number and the composition of functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtus.

An abelian groupAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo is a groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G whose operation is commutative.

A ring is called commutative if its multiplication is commutative, since the addition is commutative in any ring.

In neurophysiology, commutative has much the same meaning as in algebra.

Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state:

In non-commutative algebra, order makes a difference to multiplication, so that . This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuit s that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models.

(Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.