Dictionary Definition
commutative adj : of a binary operation;
independent of order; as in e.g. "a x b = b x a"
User Contributed Dictionary
English
Adjective
commutative, notcomparable (of an operator * ) such that, for any operands a, b , a * b = b * a
 Having a commutative operation.
Translations
 Catalan: commutatiu , commutativa (1, 2), abelià , abeliana (2)
 Czech: komutativní
 Finnish: kommutatiivinen, vaihdannainen
 French: commutatif , commutative (1, 2), abélien , abélienne (2)
 Greek: μεταθετικός (metathetikós) (1,2)
 Hungarian: kommutatív
 Spanish: conmutativo , conmutativa (1, 2)
Related terms
French
 English: commutative
Extensive Definition
In mathematics, commutativity
is the ability to change the order of something without changing
the end result. It is a fundamental property in most branches of
mathematics and many proofs
depend on it. The commutativity of simple operations was for many
years implicitly assumed and the property was not given a name or
attributed until the 19th century when mathematicians began to
formalize the theory of mathematics.
Common uses
The commutative property (or commutative law) is
a property associated with binary
operations and functions.
Similarly, if the commutative property holds for a pair of elements
under a certain binary operation then it is said that the two
elements commute under that operation.
In group and
set
theory, many algebraic structures are called commutative when
certain operands satisfy the commutative property. In higher
branches of math, 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.
Mathematical definitions
The term "commutative" is used in several related senses. An operation that does not satisfy the above property is called noncommutative.
2. One says that x commutes with y under ∗ if:
 x ∗ y = y ∗ x
3. A binary
function f:A×A → B is said to be commutative if:
 f(x,y) = f(y,x) for every x, y ∈ A.
History 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 century when mathematicians began to work on a theory of
functions. Today the commutative property is a well known and basic
property used in most branches of mathematics. Simple versions of
the commutative property are usually taught in beginning
mathematics courses.
The first use of the actual term commutative was
in a memoir by Francois Servois in 1814, which used the word
commutatives when describing functions that have what is now called
the commutative property. The word is a combination of the French
word commuter meaning "to substitute or switch" and the suffix
ative meaning "tending to" so the word literally means "tending to
substitute or switch." The term then appeared in English in
Philosophical Transactions of the Royal Society in 1844.
Related properties
Associativity
The associative property is closely related to
the commutative property. The associative property states that the
order in which operations are performed does not affect the final
result. In contrast, the commutative property states that the order
of the terms does not affect the final result.
Symmetry
Symmetry can be directly linked to commutativity.
When a commutative operator is written as a binary function then
the resulting function is symmetric across the line y = x. As an
example, if we let a function f represent addition (a commutative
operation) so that f(x,y) = x + y then f is a symmetric function
which can be seen in the image on the right.
Examples
Commutative operations in everyday life
 Putting your shoes on resembles a commutative operation since it doesn't matter if you put the left or right shoe on first, the end result (having both shoes on), is the same.
 When making change we take advantage of the commutativity of addition. It doesn't matter what order we put the change in, it always adds to the same total.
Commutative operations in math
Two wellknown examples of commutative binary
operations are:
 The addition of real numbers, which is commutative since

 y + z = z + y \quad \forall y,z\in \mathbb
 For example 4 + 5 = 5 + 4, since both expressions equal 9.
 The multiplication of real numbers, which is commutative since

 y z = z y \quad \forall y,z\in \mathbb
 For example, 3 × 5 = 5 × 3, since both expressions equal 15.
 Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets.
Noncommutative operations in everyday life
 Washing and drying your clothes resembles a noncommutative operation, if you dry first and then wash, you get a significantly different result than if you wash first and then dry.
 The Rubik's Cube is noncommutative. For example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF') does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF'U). The twists do not commute. This is studied in group theory.
Noncommutative operations in math
Some noncommutative binary operations are:
 subtraction is noncommutative since 01\neq 10
 division is noncommutative since 1/2\neq 2/1
 matrix multiplication is noncommutative since
Mathematical structures and commutativity
 An abelian group is a group whose group operation is commutative.
 A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is by definition always commutative.)
 In a field both addition and multiplication are commutative.
 The center is the largest commutative subset of a group.
Notes
References
Books
 Linear Algebra Done Right, 2e
 ''Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
 Algebra: Abstract and Concrete, Stressing Symmetry, 2e
 Abstract algebra theory. Uses commutativity property throughout book.
 Contemporary Abstract Algebra, 6e
 Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
Articles
 http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy Of Ancient Egypt  A Response To Robert Palter. Unpublished manuscript.
 Article describing the mathematical ability of ancient civilizations.
 Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0714109444
 Translation and interpretation of the Rhind Mathematical Papyrus.
Online Resources
 Krowne, Aaron, , Accessed 8 August 2007.
 Definition of commutativity and examples of commutative operations
 , Accessed 8 August 2007.
 Explanation of the term commute
 Yark. , Accessed 8 August 2007
 Examples proving some noncommutative operations
 O'Conner, J J and Robertson, E F. MacTutor history of real numbers, Accessed 8 August 2007
 Article giving the history of the real numbers
 Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 8 August 2007
 Page covering the earliest uses of mathematical terms
 O'Conner, J J and Robertson, E F. MacTutor biography of François Servois, Accessed 8 August 2007
 Biography of Francois Servois, who first used the term''
See also
commutative in Afrikaans: Kommutatiewe
bewerking
commutative in Arabic: عملية تبديلية
commutative in Bulgarian: Комутативност
commutative in Catalan: Propietat
commutativa
commutative in Czech: Komutativita
commutative in Danish: Kommutativitet
commutative in German: Kommutativgesetz
commutative in Estonian: Kommutatiivsus
commutative in Spanish: Conmutatividad
commutative in Esperanto: Komuteco
commutative in Persian: خاصیت جابجایی
commutative in French: Commutativité
commutative in Scottish Gaelic:
Coiomlaideachd
commutative in Korean: 교환 법칙
commutative in Croatian: Komutativnost
commutative in Icelandic: Víxlregla
commutative in Italian: Operazione
commutativa
commutative in Hebrew: חילופיות
commutative in Lithuanian: Komutatyvumas
commutative in Hungarian: Kommutativitás
commutative in Dutch: Commutativiteit
commutative in Japanese: 交換法則
commutative in Norwegian Nynorsk:
Kommutativitet
commutative in Polish: Przemienność
commutative in Portuguese: Comutatividade
commutative in Romanian: Comutativitate
commutative in Russian: Коммутативная
операция
commutative in Slovak: Komutatívnosť
commutative in Slovenian: Komutativnost
commutative in Serbian: Комутативност
commutative in SerboCroatian:
Komutativnost
commutative in Finnish: Vaihdannaisuus
commutative in Swedish: Kommutativitet
commutative in Vietnamese: Giao hoán
commutative in Ukrainian: Комутативність
commutative in Urdu: Commutativity
commutative in Chinese: 交換律