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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many operations throughout mathematics, and many proofs
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
 depend on it. The commutativity of simple operations, such as multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
 or addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
 of numbers, was for many years implicitly assumed and the property was not given a name or attributed until the 19th century when mathematicians began to formalize the theory of mathematics.

Common uses
The commutative property (or commutative law) is a property associated with binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
s and functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, commutativity is the process to change the order of something without changing the end result. It is a fundamental property of many operations throughout mathematics, and many proofs
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
 depend on it. The commutativity of simple operations, such as multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
 or addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
 of numbers, was for many years implicitly assumed and the property was not given a name or attributed until the 19th century when mathematicians began to formalize the theory of mathematics.

Common uses


The commutative property (or commutative law) is a property associated with binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
s and functions
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
. Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation.

In group
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 and set theory
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of math, such as analysis
Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of calculus. It is the branch of mathematics most explicitly concerned with the notion of a limit , whether the limit of a sequence or the limit of a function....
 and linear algebra
Linear algebra

Linear algebra is the branch of mathematics concerned with the study of Euclidean vectors, vector spaces , linear maps , and system of linear equations....
 the commutativity of well known operations (such as addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
 and multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
 on real and complex numbers) is often used (or implicitly assumed) in proofs.

Mathematical definitions

The term "commutative" is used in several related senses.

1. A binary operation
Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
 * on a set S is said to be commutative if:
- An operation that does not satisfy the above property is called noncommutative.


2. One says that x commutes with y under * if:

3. A binary function
Binary function

In mathematics, a binary function, or function of two variables, is a function which takes two inputs.Precisely stated, a function is binary if there exists Set s such that...
 f:A×A ? B is said to be commutative if:

History and etymology


Records of the implicit use of the commutative property go back to ancient times. The Egypt
Egypt

Egypt is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Western Asia. Covering an area of about , Egypt borders the Mediterranean Sea to the north, the Gaza Strip and Israel to the northeast, the Red Sea to the east, Sudan to the south and Libya to the west....
ians used the commutative property of multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
 to simplify computing products. Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
 is known to have assumed the commutative property of multiplication in his book Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
. Formal uses of the commutative property arose in the late 18th and early 19th century 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. Simple versions of the commutative property are usually taught in beginning mathematics courses.

The first use of the actual term commutative was in a memoir by Francois 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 Philosophical Transactions of the Royal Society
Philosophical Transactions of the Royal Society

The Philosophical Transactions of the Royal Society, or Phil. Trans., is a scientific journal published by the Royal Society.Begun in 1665, it is the oldest scientific journal printed in the Anglosphere and the second oldest in the world, after the French Journal des s?avans....
 in 1844.

Related properties


Associativity


The associative property is closely related to the commutative property. The associative property states that the order in which operations are performed does not affect the final result. In contrast, the commutative property states that the order of the terms does not affect the final result.

Symmetry


Symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function which can be seen in the image on the right.

Examples


Commutative operations in everyday life


  • Putting your shoes on resembles a commutative operation since it doesn't matter if you put the left or right shoe on first, the end result (having both shoes on), is the same.
  • When making change we take advantage of the commutativity of addition. It doesn't matter what order we put the change in, it always adds to the same total.


Commutative operations in math


Two well-known examples of commutative binary operations are:
  • The addition
    Addition

    Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
     of real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s, which is commutative since
For example 4 + 5 = 5 + 4, since both expression
Expression (mathematics)

In mathematics, the word expression is a term for any well-formed formula combination of mathematical symbols. For example,is an expression, while...
s equal 9.
  • The multiplication
    Multiplication

    Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
     of real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
    s, which is commutative since
For example, 3 × 5 = 5 × 3, since both expressions equal 15.


  • Further examples of commutative binary operations include addition and multiplication of complex number
    Complex number

    In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
    s, addition of vectors
    Vector space

    File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
    , and intersection
    Intersection (set theory)

    In mathematics, the intersection of two Set A and B is the set that contains all elements of A that also belong to B , but no other elements....
     and union
    Union (set theory)

    In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
     of sets.


Noncommutative operations in everyday life


  • Washing and drying your clothes resembles a noncommutative operation, if you dry first and then wash, you get a significantly different result than if you wash first and then dry.
  • The Rubik's Cube
    Rubik's Cube

    File:Rubik's cube.svgThe Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungary sculptor and professor of architecture Erno Rubik....
     is noncommutative. For example, twisting the front face clockwise, the top face clockwise and the front face counterclockwise (FUF') does not yield the same result as twisting the front face clockwise, then counterclockwise and finally twisting the top clockwise (FF'U). The twists do not commute. This is studied in group theory
    Group theory

    In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
    .


Noncommutative operations in mathematics


Some noncommutative binary operations are:
  • subtraction
    Subtraction

    Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
     is noncommutative since
  • division
    Division (mathematics)

    In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
      is noncommutative since
  • matrix
    Matrix (mathematics)

    In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
     multiplication is noncommutative since

Mathematical structures and commutativity


  • An abelian group
    Abelian group

    An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
     is a group
    Group (mathematics)

    In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
     whose group operation is commutative.
  • A commutative ring
    Commutative ring

    In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
     is a ring
    Ring (mathematics)

    In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
     whose multiplication
    Multiplication

    Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
     is commutative. (Addition in a ring is by definition always commutative.)
  • In a field
    Field (mathematics)

    In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
     both addition and multiplication are commutative.


Books

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.:Abstract algebra theory. Uses commutativity property throughout book.:Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.


Articles

  • Lumpkin, B. (1997). The Mathematical Legacy Of Ancient Egypt - A Response To Robert Palter. Unpublished manuscript.
Article describing the mathematical ability of ancient civilizations.
  • Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
Translation and interpretation of the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scotland antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum....
.


Online Resources


  • Krowne, Aaron, , Accessed 8 August 2007.
Definition of commutativity and examples of commutative operations
Explanation of the term commute
  • . , Accessed 8 August 2007
Examples proving some noncommutative operations
  • O'Conner, J J and Robertson, E F. , Accessed 8 August 2007
Article giving the history of the real numbers
  • Cabillón, Julio and Miller, Jeff. , Accessed 22 November 2008
Page covering the earliest uses of mathematical terms
  • O'Conner, J J and Robertson, E F. , Accessed 8 August 2007
Biography of Francois Servois, who first used the term


See also

  • Anticommutativity
    Anticommutativity

    In mathematics, anticommutativity refers to the property of an Operation being anticommutative, i.e. being non-Commutativity in a precise way....
  • Binary operation
    Binary operation

    In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator....
  • Commutant
    Commutant

    In algebra, the commutant of a subset S of a semigroup A is the subset S′ of elements of A commutative operation with every element of S....
  • Commutative diagram
    Commutative diagram

    In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition....
  • Commutative (neurophysiology)
    Commutative (neurophysiology)

    In neurophysiology, commutation is the process of how the brain's Biological neural network exhibit non-commutativity.Physiologist Douglas B. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state:...
  • Commutator
    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....
  • Distributivity
    Distributivity

    In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra....
  • Particle statistics
    Particle statistics

    Particle statistics refers to the particular description of particles in statistical mechanics....
     (for commutativity in physics
    Physics

    Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
    )