In
mathematics, a
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
is commutative if changing the order of the
operand
In mathematics, an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Example
The following arithmetic expression shows an example of operators and operands:
:3 + 6 = 9
In the above exa ...
s does not change the result. It is a fundamental property of many binary operations, and many
mathematical proof
A mathematical proof is an Inference, inferential Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previo ...
s depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as
division and
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
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 ...
of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.
A similar property exists for
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
s; 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 definit ...
if the relation applies regardless of the order of its operands; for example,
equality is symmetric as two equal mathematical objects are equal regardless of their order.
Mathematical definitions
A
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
on a
set ''S'' is called ''commutative'' if
[Krowne, p.1]
An operation that does not satisfy the above property is called ''non-commutative''.
One says that ''commutes'' with or that and ''commute'' under
if
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'' of ...
and
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 mathematical operations of arithmetic, with the other ones being ad ...
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 language ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 for ...
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 ...
and in every
algebra
Algebra () is one of the areas of mathematics, 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 mathem ...
.
*
Union and
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
are commutative operations on
sets.
* "
And" 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 co ...
s.
Noncommutative operations
Some noncommutative binary operations:
Division, subtraction, and exponentiation
Division is noncommutative, since
.
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
. 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 unswapped ...
, since
.
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 re ...
is noncommutative, since
.
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 on ...
s are noncommutative, since the
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...
s for the functions are different when one changes the order of the operands. For example, the truth tables for and are
:
Function composition of linear functions
Function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
of
linear 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 di ...
s from the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
to the real numbers is almost always noncommutative. For example, let
and
. Then
:
and
:
This also applies more generally for
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generall ...
s from a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 ...
of
square matrices is almost always noncommutative, for example:
:
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 i ...
) 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 unswapped ...
; 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 Med ...
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 mathematical operations of arithmetic, with the other ones being ad ...
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
Validity or Valid may refer to:
Science/mathematics/statistics:
* Validity (logic), a property of a logical argument
* Scientific:
** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments
** ...
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 propos ...
s within
logical expressions in
logical proofs. The rules are:
:
and
:
where "
" is a
metalogical
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
representing "can be replaced in a
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a con ...
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 ...
. 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 ...
s demonstrate that commutativity is a property of particular connectives. The following are truth-functional
tautologies.
;Commutativity of conjunction:
;Commutativity of disjunction:
;Commutativity of implication (also called the law of permutation):
;Commutativity of equivalence (also called the complete commutative law of equivalence):
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 concer ...
, 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 matric ...
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'' of ...
and
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 mathematical operations of arithmetic, with the other ones being ad ...
on real and complex numbers) is often used (or implicitly assumed) in proofs.
Mathematical structures and commutativity
* A
commutative semigroup
In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists o ...
is a set endowed with a total,
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
and commutative operation.
* If the operation additionally has an
identity element
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 s ...
, 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 com ...
, 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 computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
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
:
which is clearly commutative (interchanging ''x'' and ''y'' does not affect the result), but it is not associative (since, for example,
but
). 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 quatern ...
or of
matrices 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
In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs.
Precisely stated, a function f is binary if there exists sets X, Y, Z such that
:\,f \colon X \times Y \rightar ...
then this function is called a
symmetric function, 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 informal ...
is symmetric across the plane
. For example, if the function is defined as
then
is a symmetric function.
For relations, a
symmetric relation
A symmetric relation is a type of binary relation. An example is the relation "is equal to", because if ''a'' = ''b'' is true then ''b'' = ''a'' is also true. Formally, a binary relation ''R'' over a set ''X'' is symmetric if:
:\forall a, b \in X ...
is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then
.
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, q ...
as formulated by
Schrödinger, physical variables are represented by
linear operators such as
(meaning multiply by
), and
. 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 ...
and
(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 m ...
:
:
According to the
uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
of
Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually
complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
in the
-direction of a particle are represented by the operators
and
, respectively (where
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
, 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 (also called a commutant)
*
Commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the ...
*
Commutative (neurophysiology)
*
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
*
Parallelogram law
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 ...
*
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 ...
(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 rel ...
)
*
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