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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 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, 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 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. 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.


Mathematical definitions

A binary operation * 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 and multiplication 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 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 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 of linear functions from the real numbers 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 transformations 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 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 representing "can be replaced in a proof 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 and multiplication 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 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 is a ring whose multiplication 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 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 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 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 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 *
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 * 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.''


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