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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
is commutative if changing the order of the
operand In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on. Example The following arithmetic expression shows an example of operators and operands: :3 + 6 = 9 In the above exam ...
s does not change the result. It is a fundamental property of many binary operations, and many
mathematical proof A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every pr ...
s depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
and
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
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. A similar property exists for
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
s; 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.


Mathematical definitions

A
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
* on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''S'' is called ''commutative'' ifKrowne, p.1 x * y = y * x\qquad\mboxx,y\in S. An operation that does not satisfy the above property is called ''non-commutative''. One says that ''commutes'' with or that and ''commute'' under * if x * y = y * x. In other words, an operation is commutative if every two elements commute.


Examples


Commutative operations

*
Addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
are commutative in most number systems, and, in particular, between
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s,
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. This is also true in every field. * Addition is commutative in every
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and in every
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
. * Union and
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
are commutative operations on
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s. * "
And or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
" and " or" are commutative logical operations.


Noncommutative operations

Some noncommutative binary operations:


Division, subtraction, and exponentiation

Division is noncommutative, since 1 \div 2 \neq 2 \div 1.
Subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
is noncommutative, since 0 - 1 \neq 1 - 0. However it is classified more precisely as
anti-commutative 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 unswappe ...
, since 0 - 1 = - (1 - 0).
Exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
is noncommutative, since 2^3\neq3^2.


Truth functions

Some
truth function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly o ...
s are noncommutative, since the
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arg ...
s for the functions are different when one changes the order of the operands. For example, the truth tables for and are :


Function composition of linear functions

Function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
of linear functions from the
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
to the real numbers is almost always noncommutative. For example, let f(x)=2x+1 and g(x)=3x+7. Then :(f \circ g)(x) = f(g(x)) = 2(3x+7)+1 = 6x+15 and :(g \circ f)(x) = g(f(x)) = 3(2x+1)+7 = 6x+10 This also applies more generally for
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generall ...
s from a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
to itself (see below for the Matrix representation).


Matrix multiplication

Matrix multiplication of square matrices is almost always noncommutative, for example: : \begin 0 & 2 \\ 0 & 1 \end = \begin 1 & 1 \\ 0 & 1 \end \begin 0 & 1 \\ 0 & 1 \end \neq \begin 0 & 1 \\ 0 & 1 \end \begin 1 & 1 \\ 0 & 1 \end = \begin 0 & 1 \\ 0 & 1 \end


Vector product

The vector product (or
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
) of two vectors in three dimensions is
anti-commutative 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 unswappe ...
; i.e., ''b'' × ''a'' = −(''a'' × ''b'').


History and etymology

Records of the implicit use of the commutative property go back to ancient times. The
Egypt Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a List of transcontinental countries, transcontinental country spanning the North Africa, northeast corner of Africa and Western Asia, southwest corner of Asia via a land bridg ...
ians used the commutative property of
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
to simplify computing products.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
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. Today 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. 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 1838. in
Duncan Farquharson Gregory Duncan Farquharson Gregory (13 April 181323 February 1844) was a Scottish mathematician. Education Gregory was born in Aberdeen on 13 April 1813, the youngest son of Isabella Macleod (1772–1847) and James Gregory (1753–1821). He was taught ...
's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.


Propositional logic


Rule of replacement

In truth-functional propositional logic, ''commutation'', or ''commutativity'' refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are: :(P \lor Q) \Leftrightarrow (Q \lor P) and :(P \land Q) \Leftrightarrow (Q \land P) where "\Leftrightarrow" is a metalogical
symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
representing "can be replaced in a
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a c ...
with".


Truth functional connectives

''Commutativity'' is a property of some
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies. ;Commutativity of conjunction:(P \land Q) \leftrightarrow (Q \land P) ;Commutativity of disjunction:(P \lor Q) \leftrightarrow (Q \lor P) ;Commutativity of implication (also called the law of permutation):(P \to (Q \to R)) \leftrightarrow (Q \to (P \to R)) ;Commutativity of equivalence (also called the complete commutative law of equivalence):(P \leftrightarrow Q) \leftrightarrow (Q \leftrightarrow P)


Set theory

In group and
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
the commutativity of well-known operations (such as
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
and
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
on real and complex numbers) is often used (or implicitly assumed) in proofs.


Mathematical structures and commutativity

* A commutative semigroup is a set endowed with a total,
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
and commutative operation. * If the operation additionally has an identity element, we have a commutative monoid * An
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
, or ''commutative group'' is a group whose group operation is commutative. * A
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is a ring whose
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
is commutative. (Addition in a ring is always commutative.) * In a field both addition and multiplication are commutative.


Related properties


Associativity

The associative property is closely related to the commutative property. 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. However, commutativity does not imply associativity. A counterexample is the function :f(x, y) = \frac, which is clearly commutative (interchanging ''x'' and ''y'' does not affect the result), but it is not associative (since, for example, f(-4, f(0, +4)) = -1 but f(f(-4, 0), +4) = +1). More such examples may be found in commutative non-associative magmas. Furthermore, associativity does not imply commutativity either - for example multiplication of
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
or of matrices is always associative but not always commutative.


Distributive


Symmetry

Some forms of
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
can be directly linked to commutativity. When a commutative operation is written as a
binary function In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function f is binary if there exists sets X, Y, Z such that :\,f \colon X \times Y \righta ...
z=f(x,y), then this function is called a symmetric function, and its graph in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
is symmetric across the plane y=x. For example, if the function is defined as f(x,y)=x+y then f is a symmetric function. For relations, a
symmetric relation A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if: :\forall a, b \in X ...
is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then a R b \Leftrightarrow b R a.


Non-commuting operators in quantum mechanics

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and \frac. These two operators do not commute as may be seen by considering the effect of their
compositions Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
x \frac and \frac x (also called products of operators) on a one-dimensional wave function \psi(x): : x\cdot \psi = x\cdot \psi' \ \neq \ \psi + x\cdot \psi' = \left( x\cdot \psi \right) According to the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
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 Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
in the x-direction of a particle are represented by the operators x and -i \hbar \frac, respectively (where \hbar is the reduced Planck constant). This is the same example except for the constant -i \hbar, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.


See also

*
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 unswappe ...
* Centralizer and normalizer (also called a commutant) *
Commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
*
Commutative (neurophysiology) In neurophysiology, commutation is the process by which the brain's neural circuits exhibit non-commutativity. Physiologist Douglas B. Tweed and coworkers have considered whether certain neural circuits in the brain exhibit noncommutativity and st ...
*
Commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
* Parallelogram law * Particle statistics (for commutativity in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
) * Proof that Peano's axioms imply the commutativity of the addition of natural numbers *
Quasi-commutative property In mathematics, the quasi-commutative property is an extension or generalization of the general commutative property. This property is used in specific applications with various definitions. Applied to matrices Two matrices p and q are said to ha ...
* Trace monoid * Commuting probability


Notes


References


Books

* *:''Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.'' * * *:''Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.'' * *:''Abstract algebra theory. Uses commutativity property throughout book.'' *


Articles

* *:''Article describing the mathematical ability of ancient civilizations.'' * *:''Translation and interpretation of the
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scotland, Scottish antiquarian, who ...
.''


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'' * , Accessed 8 August 2007 *:''Examples proving some noncommutative operations'' * *:''Article giving the history of the real numbers'' * *:''Page covering the earliest uses of mathematical terms'' * *:''Biography of Francois Servois, who first used the term'' {{Good article Properties of binary operations Elementary algebra Rules of inference Symmetry Concepts in physics Functional analysis