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In 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 proo ...
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 Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
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 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 ...
if the relation applies regardless of the order of its operands; for example,
equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
is symmetric as two equal mathematical objects are equal regardless of their order.


Mathematical definitions

A binary operation * on a set ''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 system A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, 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 rat ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. * 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 Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
and intersection are commutative operations on sets. * "
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 operation 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.


Noncommutative operations

Some noncommutative binary operations:


Division, subtraction, and exponentiation

Division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
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, 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 r ...
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 argumen ...
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 function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
s 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 In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
of
square matrices In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...
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) of two vectors in three dimensions is anti-commutative; 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 transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Medit ...
ians used the commutative property of multiplication to simplify computing
products Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
.
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's article entitled "On the real nature of symbolical algebra" published in 1840 in the
Transactions of the Royal Society of Edinburgh Transaction or transactional may refer to: Commerce *Financial transaction, an agreement, communication, or movement carried out between a buyer and a seller to exchange an asset for payment *Debits and credits in a Double-entry bookkeeping syst ...
.


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 variable In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of proposit ...
s 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
metalogic Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
al 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 Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
. The following
logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending o ...
s 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 conce ...
, 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 matrices ...
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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
, we have a
commutative monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
* 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 A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
whose group operation is commutative. * A commutative ring is a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
whose multiplication is commutative. (Addition in a ring is always commutative.) * In a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 In mathematics, there exist magma (algebra), magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras. A m ...
. 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 Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is always associative but not always commutative.


Distributive


Symmetry

Some forms of symmetry 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 In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...
, and its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
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 chemistr ...
as formulated by Schrödinger, physical variables are represented by
linear operators In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
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 x \frac and \frac x (also called products of operators) on a one-dimensional
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
\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 the x-direction of a particle are represented by the operators x and -i \hbar \frac, respectively (where \hbar is the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
). 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 *
Centralizer and normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
(also called a commutant) * Commutative diagram * Commutative (neurophysiology) * Commutator *
Parallelogram law In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
*
Particle statistics Particle statistics is a particular description of multiple particles in statistical mechanics. A key prerequisite concept is that of a statistical ensemble (an idealization comprising the state space of possible states of a system, each labeled w ...
(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 r ...
) * Proof that Peano's axioms imply the commutativity of the addition of natural numbers * Quasi-commutative property *
Trace monoid In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certa ...
* 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