In

multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

on real and complex numbers) is often used (or implicitly assumed) in proofs.

multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

is commutative. (Addition in a ring is always commutative.)
* In a

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, a binary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 example, ...

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
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek ...

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 o ...

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 cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

and addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

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 corresponding property exists for binary relation
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

s; a binary relation is said to be symmetric
Symmetry (from Greek συμμετρία ''symmetria'' "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 pre ...

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.
Common uses

The ''commutative property'' (or ''commutative law'') is a property generally associated with binary operations andfunctions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

. If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to ''commute'' under that operation.
Mathematical definitions

Abinary operation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$*$ on a set ''S'' is called ''commutative'' ifKrowne, p.1
$$x\; *\; y\; =\; y\; *\; x\backslash qquad\backslash mboxx,y\backslash in\; S.$$
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 pair of elements commute.
A binary function
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function (mathematics), function that takes two inputs.
Precisely stated, a function f is binary if there exists Set (mathematics), sets X, Y, Z ...

$f\; \backslash colon\; A\; \backslash times\; A\; \backslash to\; B$ is sometimes called ''commutative'' if
$$f(x,\; y)\; =\; f(y,\; x)\backslash qquad\backslash mboxx,y\backslash in\; A.$$ Such a function is more commonly called a symmetric function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

.
Examples

Commutative operations in everyday life

*Putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same. In contrast, putting on underwear and trousers is not commutative. *The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills are handed over in, they always give the same total.Commutative operations in mathematics

Two well-known examples of commutative binary operations: * Theaddition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s is commutative, since $$y\; +\; z\; =\; z\; +\; y\; \backslash qquad\backslash mboxy,z\backslash in\; \backslash mathbb$$ For example 4 + 5 = 5 + 4, since both expressions equal 9.
* The multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

of real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s is commutative, since $$y\; z\; =\; z\; y\; \backslash qquad\backslash mboxy,z\backslash in\; \backslash mathbb$$ For example, 3 × 5 = 5 × 3, since both expressions equal 15.

As a direct consequence of this, it also holds true that expressions on the form y% of z and z% of y are commutative for all real numbers y and z. For example 64% of 50 = 50% of 64, since both expressions equal 32, and 30% of 50% = 50% of 30%, since both of those expressions equal 15%.

*Some binary

truth function
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

s are also commutative, since the truth table
A truth table is a mathematical table
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy
...

s for the functions are the same when one changes the order of the operands. For example, the

logical biconditional
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stat ...

function p ↔ q is equivalent to q ↔ p. This function is also written as p IFF
IFF, Iff or iff may refer to:
Arts and entertainment
* Simon Iff, a fictional character by Aleister Crowley
* Iff of the Unpronounceable Name, a fictional character in the Riddle-Master trilogy by Patricia A. McKillip
* "IFF", an List of The ...

q, or as p ≡ q, or as E''pq''. The last form is an example of the most concise notation in the article on truth functions, which lists the sixteen possible binary truth functions of which eight are commutative: V''pq'' = V''qp''; A''pq'' (OR) = A''qp''; D''pq'' (NAND) = D''qp''; E''pq'' (IFF) = E''qp''; J''pq'' = J''qp''; K''pq'' (AND) = K''qp''; X''pq'' (NOR) = X''qp''; O''pq'' = O''qp''.

* Further examples of commutative binary operations include addition and multiplication ofcomplex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s, addition and scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

of vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

, and intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

and union of sets.
Noncommutative operations in daily life

*Concatenation
In formal language theory and computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves ...

, the act of joining character strings together, is a noncommutative operation. For example,
*:
*Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
*Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
*The moves of any combination puzzle
A combination puzzle, also known as a sequential move puzzle, is a puzzle
A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together in a logica ...

(such as the twists of a Rubik's Cube
The Rubik's Cube is a 3-D combination puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by Ideal Toy Company, Ideal Toy Corp. ...

, for example) are noncommutative. This can be studied using group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

.
*Thought processes are noncommutative: A person asked a question (A) and then a question (B) may give different answers to each question than a person asked first (B) and then (A), because asking a question may change the person's state of mind.
*The act of dressing is either commutative or non-commutative, depending on the items. Putting on underwear and normal clothing is noncommutative. Putting on left and right socks is commutative.
* Shuffling a deck of cards is non-commutative. Given two ways, A and B, of shuffling a deck of cards, doing A first and then B is in general not the same as doing B first and then A.
Noncommutative operations in mathematics

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 o ...

is noncommutative, since $1\; \backslash div\; 2\; \backslash neq\; 2\; \backslash div\; 1$.
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 $0\; -\; 1\; \backslash neq\; 1\; -\; 0$. However it is classified more precisely as anti-commutative
In mathematics, anticommutativity is a specific property of some non-commutative Operation (mathematics), operations. In mathematical physics, where symmetry (physics), symmetry is of central importance, these operations are mostly called antisymmet ...

, since $0\; -\; 1\; =\; -\; (1\; -\; 0)$.
Exponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

is noncommutative, since $2^3\backslash neq3^2$.
Truth functions

Sometruth function
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...

s are noncommutative, since the truth table
A truth table is a mathematical table
Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy
...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of linear function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s from the real numbers
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

to the real numbers is almost always noncommutative. For example, let $f(x)=2x+1$ and $g(x)=3x+7$. Then
:$(f\; \backslash circ\; g)(x)\; =\; f(g(x))\; =\; 2(3x+7)+1\; =\; 6x+15$
and
:$(g\; \backslash circ\; f)(x)\; =\; g(f(x))\; =\; 3(2x+1)+7\; =\; 6x+10$
This also applies more generally for linear
Linearity is the property of a mathematical relationship (''function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out se ...

and affine transformation
In Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

s from a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

to itself (see below for the Matrix representation).
Matrix multiplication

Matrix multiplication
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of square matrices
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

is almost always noncommutative, for example:
:$\backslash begin\; 0\; \&\; 2\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; =\; \backslash begin\; 1\; \&\; 1\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; \backslash neq\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; \backslash begin\; 1\; \&\; 1\; \backslash \backslash \; 0\; \&\; 1\; \backslash end\; =\; \backslash begin\; 0\; \&\; 1\; \backslash \backslash \; 0\; \&\; 1\; \backslash end$
Vector product

The vector product (orcross product
In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

) of two vectors in three dimensions is anti-commutative
In mathematics, anticommutativity is a specific property of some non-commutative Operation (mathematics), operations. In mathematical physics, where symmetry (physics), symmetry is of central importance, these operations are mostly called antisymmet ...

; i.e., ''b'' × ''a'' = −(''a'' × ''b'').
History and etymology

Records of the implicit use of the commutative property go back to ancient times. TheEgypt
Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country
This is a list of countries located on more than one continent
A continent is one of several large landmasses. Generally identi ...

ians used the commutative property of multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

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
* Produc ...

. Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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
Aberdeen (; sco, Aiberdeen, ; gd, Obar Dheathain ; la, Aberdonia) is a city in northeast Scotland. It is the ...

'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 syste ...

.
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 transposepropositional variable
In mathematical logic
Mathematical logic, also called formal logic, is a subfield of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...

s within logical expressions
Logic (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 milli ...

in logical proofs. The rules are:
:$(P\; \backslash lor\; Q)\; \backslash Leftrightarrow\; (Q\; \backslash lor\; P)$
and
:$(P\; \backslash land\; Q)\; \backslash Leftrightarrow\; (Q\; \backslash land\; P)$
where "$\backslash Leftrightarrow$" is a metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how formal system, logical systems can be used to construct Validity (logic), valid and soundness, sound arguments, metalogic studies the properties of logical systems.Har ...

al symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...

representing "can be replaced in a proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

with".
Truth functional connectives

''Commutativity'' is a property of somelogical connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

s of truth functional propositional logic
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...

. The following logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they are provable from each other under a set of axioms, or have the same truth value in every model (logic), model. The logical equivalence of p and q is sometimes ...

s demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies.
;Commutativity of conjunction:$(P\; \backslash land\; Q)\; \backslash leftrightarrow\; (Q\; \backslash land\; P)$
;Commutativity of disjunction:$(P\; \backslash lor\; Q)\; \backslash leftrightarrow\; (Q\; \backslash lor\; P)$
;Commutativity of implication (also called the law of permutation):$(P\; \backslash to\; (Q\; \backslash to\; R))\; \backslash leftrightarrow\; (Q\; \backslash to\; (P\; \backslash to\; R))$
;Commutativity of equivalence (also called the complete commutative law of equivalence):$(P\; \backslash leftrightarrow\; Q)\; \backslash leftrightarrow\; (Q\; \backslash leftrightarrow\; P)$
Set theory

Ingroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

and set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...

, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

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 matrix (mat ...

the commutativity of well-known operations (such as addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

and Mathematical structures and commutativity

* Acommutative semigroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a set endowed with a total, associative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and commutative operation.
* If the operation additionally has an identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

, we have a commutative monoid
In abstract algebra, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

* An abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, or ''commutative group'' is a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

whose group operation is commutative.
* A commutative ring
In ring theory
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...

is a ring whose 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 grassl ...

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 :$f(x,\; y)\; =\; \backslash frac,$ 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.Distributive

Symmetry

Some forms ofsymmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

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 (mathematics), function that takes two inputs.
Precisely stated, a function f is binary if there exists Set (mathematics), sets X, Y, Z ...

$z=f(x,y),$ then this function is called a symmetric function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, 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 discret ...

in three-dimensional space
Three-dimensional space (also: 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 informal meaning of the ...

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 (mathematics), set ''X'' is symmetric if:
: ...

is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then $a\; R\; b\; \backslash Leftrightarrow\; b\; R\; a$.
Non-commuting operators in quantum mechanics

Inquantum mechanics
Quantum mechanics is a fundamental theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...

as formulated by Schrödinger, physical variables are represented by linear operators
In mathematics, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \rightarrow W between two vector spaces that preserves the operat ...

such as $x$ (meaning multiply by $x$), and $\backslash frac$. These two operators do not commute as may be seen by considering the effect of their compositions $x\; \backslash frac$ and $\backslash frac\; x$ (also called products of operators) on a one-dimensional wave function
A wave function in quantum physics
Quantum mechanics is a fundamental theory
A theory is a rational
Rationality is the quality or state of being rational – that is, being based on or agreeable to reason
Reason is the capacity ...

$\backslash psi(x)$:
:$x\backslash cdot\; \backslash psi\; =\; x\backslash cdot\; \backslash psi\text{'}\; \backslash \; \backslash neq\; \backslash \; \backslash psi\; +\; x\backslash cdot\; \backslash psi\text{'}\; =\; \backslash left(\; x\backslash cdot\; \backslash psi\; \backslash right)$
According to the uncertainty principle
In quantum mechanics
Quantum mechanics is a fundamental Scientific theory, 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 quant ...

of Heisenberg
Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the key pioneers of quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a de ...

, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary
A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be:
* Complement (linguistics), a word or phrase having a particular syntactic role
** Subject complement, a word or phrase addi ...

, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass
Mass is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

in the $x$-direction of a particle are represented by the operators $x$ and $-i\; \backslash hbar\; \backslash frac$, respectively (where $\backslash hbar$ is the reduced Planck constant
The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant i ...

). This is the same example except for the constant $-i\; \backslash hbar$, 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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

(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 (category theory), diagram such that all directed paths in the diagram with the same start an ...

* Commutative (neurophysiology)
*Commutator
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

*Parallelogram law
A parallelogram. The sides are shown in blue and the diagonals in red.
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 ...

*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 Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

)
* Proof that Peano's axioms imply the commutativity of the addition of natural numbers
* Quasi-commutative property
*Trace monoidIn computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of Algorith ...

* 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 theRhind Mathematical Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum
The British Museum, in the Bloomsbury
Bloomsbury is a district in the West End of London
The West End of London (commonly referred to as the West End ...

.''
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