commutative

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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 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 ...
is commutative if changing the order of the
operand 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 mod ...
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 logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science o ...
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 The plus and minus signs, and , are mathematical symbols used to represent the notion ...
, 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 computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
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 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 m ...
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 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 ...
$*$ 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 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 cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
are commutative in most number systems, and, in particular, between
natural number In mathematics, the natural numbers are those number 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 lang ...
s,
integer An integer is the number zero (), a positive natural number In mathematics, the natural numbers are those number A number is a mathematical object used to count, measure, and label. The original examples are the natural number ...
s,
rational number 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 i ...
s,
real number 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 ...
s and
complex number 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 i ...
s. This is also true in every field. * Addition is commutative in every
vector space In mathematics and 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 th ...
and in every
algebra Algebra () is one of the broad areas of 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 c ...
. *
Union Union commonly refers to: * Trade union A trade union (labor union in American English), often simply referred to as a union, is an organization of workers intent on "maintaining or improving the conditions of their employment Employ ...
and
intersection 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 ...
are commutative operations on sets. * " And" 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 The plus and minus signs, and , are mathematical symbols used to represent the notion ...
is noncommutative, since $0 - 1 \neq 1 - 0$. However it is classified more precisely as anti-commutative, since $0 - 1 = - \left(1 - 0\right)$.
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 functions are noncommutative, since the truth tables 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 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 i ...
of
linear function 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 mo ...
s from the
real numbers 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 ...
to the real numbers is almost always noncommutative. For example, let $f\left(x\right)=2x+1$ and $g\left(x\right)=3x+7$. Then :$\left(f \circ g\right)\left(x\right) = f\left(g\left(x\right)\right) = 2\left(3x+7\right)+1 = 6x+15$ and :$\left(g \circ f\right)\left(x\right) = g\left(f\left(x\right)\right) = 3\left(2x+1\right)+7 = 6x+10$ This also applies more generally for linear and affine transformations from a
vector space In mathematics and 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 th ...
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 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 mo ...
) 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 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 cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
to simplify computing products.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics is an area of knowledge that includes the topics of numbers, f ...
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.

# 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: :$\left(P \lor Q\right) \Leftrightarrow \left(Q \lor P\right)$ and :$\left(P \land Q\right) \Leftrightarrow \left(Q \land P\right)$ where "$\Leftrightarrow$" is a metalogical symbol representing "can be replaced in a proof with".

## Truth functional connectives

''Commutativity'' is a property of some logical connectives of truth functional
propositional logic Propositional calculus is a branch of logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science ...
. The following
logical equivalence In logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow ...
s demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies. ;Commutativity of conjunction:$\left(P \land Q\right) \leftrightarrow \left(Q \land P\right)$ ;Commutativity of disjunction:$\left(P \lor Q\right) \leftrightarrow \left(Q \lor P\right)$ ;Commutativity of implication (also called the law of permutation):$\left(P \to \left(Q \to R\right)\right) \leftrightarrow \left(Q \to \left(P \to R\right)\right)$ ;Commutativity of equivalence (also called the complete commutative law of equivalence):$\left(P \leftrightarrow Q\right) \leftrightarrow \left(Q \leftrightarrow P\right)$

# Set theory

In group and
set theory Set theory is the branch of mathematical logic Mathematical logic is the study of formal logic within mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and ...
, 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 Mathematics is an area of kno ...
and
linear algebra Linear algebra is the branch of 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. Thes ...
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 cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
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, or ''commutative group'' is a group whose group operation is commutative. * A commutative ring is a ring whose
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ( c ...
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 :$f\left(x, y\right) = \frac,$ which is clearly commutative (interchanging ''x'' and ''y'' does not affect the result), but it is not associative (since, for example, $f\left(-4, f\left(0, +4\right)\right) = -1$ but $f\left(f\left(-4, 0\right), +4\right) = +1$). More such examples may be found in commutative non-associative magmas. Furthermore, associativity does not imply commutativity either - for example multiplication of quaternions or of matrices is always associative but not always commutative.

## 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 Mathematics is an area of knowle ...
can be directly linked to commutativity. When a commutative operation is written as a binary function $z=f\left(x,y\right),$ then this function is called a symmetric function, and its graph in three-dimensional space is symmetric across the plane $y=x$. For example, if the function is defined as $f\left(x,y\right)=x+y$ 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 ...
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 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 f ...
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 $x \frac$ 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 In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a ...
$\psi\left(x\right)$: :$x\cdot \psi = x\cdot \psi\text{'} \ \neq \ \psi + x\cdot \psi\text{'} = \left\left( x\cdot \psi \right\right)$ 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.

* Anticommutative property * Centralizer and normalizer (also called a commutant) * Commutative diagram * Commutative (neurophysiology) * Commutator *
Parallelogram law 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 m ...
* 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 *
Trace monoid In computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practica ...
* Commuting probability

# 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