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Encyclopedia > Commutativity
Example showing the commutativity of addition (3 + 2 = 2 + 3)
Example showing the commutativity of addition (3 + 2 = 2 + 3)

Commutativity is a widely used mathematical term that refers to the ability to change the order of something without changing the end result. It is a fundamental property in most branches of mathematics and many proofs depend on it. The commutativity of simple operations 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. Image File history File links Commutative_Addition. ... Image File history File links Commutative_Addition. ... Look up commute in Wiktionary, the free dictionary. ...

Contents

Common uses

The commutative property (or commutative law) is a property associated with binary operations and functions. 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 mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...


In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of math, such as analysis and linear algebra the commutativity of well known operations (such as addition and multiplication on real and complex numbers) is often used (or implicitly assumed) in proofs.[1][2][3] Analysis has its beginnings in the rigorous formulation of calculus. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, multiplication is an elementary arithmetic operation. ...


Mathematical definitions

1. A binary operation ∗ on a set S is said to be commutative if:[4] In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

xy = yx for every x,yS
  • An operation that does not satisfy the above property is called noncommutative.

2. One says that x commutes with y under ∗ if:[5]

xy = yx

3. A binary function f:A×BC is said to be commutative if: In mathematics, a binary function, or function of two variables, is like a function, except that it has two inputs instead of one. ...

f(x,y) = f(y,x) for every xA, yB [citation needed]

History

The first known use of the term was in a French Journal published in 1814
The first known use of the term was in a French Journal published in 1814

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products.[6][7] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[8] 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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...


The first use of the actual term commutative was in a memoir by Francois Servois in 1814,[9][10] 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 in 1844.[11]


Related Properties

Graph showing the symmetry of the addition function
Graph showing the symmetry of the addition function

Image File history File links Symmetry_Of_Addition. ... Image File history File links Symmetry_Of_Addition. ...

Associativity

Main article: 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. In mathematics, associativity is a property that a binary operation can have. ...


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. Symmetry in mathematics occurs not only in geometry, but also in other branches of mathematics. ...


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

Example showing the commutativity of multiplication (3 * 5 = 5 * 3)
Example showing the commutativity of multiplication (3 * 5 = 5 * 3)

Two well-known examples of commutative binary operations are:[12] Image File history File links Commutative_Multiplication. ... Image File history File links Commutative_Multiplication. ...

 y + z = z + y quad forall y,zin mathbb{R}
For example 4 + 5 = 5 + 4, since both expressions equal 9.
 y z = z y quad forall y,zin mathbb{R}
For example, 3 × 5 = 5 × 3, since both expressions equal 15.

3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

Noncommutative operations in everyday life

Concatenation, the act of joining character strings together, is a noncommutative operation.
Concatenation, the act of joining character strings together, is a noncommutative operation.
  • 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 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 don't commute. This is studied in group theory.

Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Concatenation is a standard operation in computer programming languages (a subset of formal language theory). ... Variations of Rubiks Cubes (from left to right: Rubiks Revenge, Rubiks Cube, Professors Cube, & Pocket Cube). ... Group theory is that branch of mathematics concerned with the study of groups. ...

Noncommutative operations in math

Some noncommutative binary operations are:[13]

 begin{bmatrix} 0 & 2  0 & 1 end{bmatrix} = begin{bmatrix} 1 & 1  0 & 1 end{bmatrix} cdot begin{bmatrix} 0 & 1  0 & 1 end{bmatrix} neq begin{bmatrix} 0 & 1  0 & 1 end{bmatrix} cdot begin{bmatrix} 1 & 1  0 & 1 end{bmatrix} = begin{bmatrix} 0 & 1  0 & 1 end{bmatrix}

5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...

Mathematical structures and commutativity

In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, multiplication is an elementary arithmetic operation. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ...

Notes

  1. ^ Axler, p.2
  2. ^ Gallian, p.34
  3. ^ p. 26,87
  4. ^ Krowne, p.1
  5. ^ Weisstein, Commute, p.1
  6. ^ Lumpkin, p.11
  7. ^ Gay and Shute, p.?
  8. ^ O'Conner and Robertson, Real Numbers
  9. ^ Cabillón and Miller, Commutative and Distributive
  10. ^ O'Conner and Robertson, Servois
  11. ^ Cabillón and Miller, Commutative and Distributive
  12. ^ Krowne, p.1
  13. ^ Yark, p.1
  14. ^ Gallian, p.34
  15. ^ Gallian p.236
  16. ^ Gallian p.250
  17. ^ Gallian p.65

References

Books

  • Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2. 
Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
  • Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry, 2e. Prentice Hall. ISBN 0-13-067342-0. 
Abstract algebra theory. Uses commutativity property throughout book.
  • Gallian, Joseph (2006). Contemporary Abstract Algebra, 6e. ISBN 0-618-51471-6. 
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

Articles

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.

The Rhind Mathematical Papyrus ( papyrus British Museum 10057 and pBM 10058), is named after Alexander Henry Rhind, a Scottish 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, Commutative at PlanetMath., Accessed 8 August 2007.
Definition of commutativity and examples of commutative operations
Explanation of the term commute
  • Yark. Examples of non-commutative operations at PlanetMath., Accessed 8 August 2007
Examples proving some noncommutative operations
Article giving the history of the real numbers
  • Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 8 August 2007
Page covering the earliest uses of mathematical terms
Biography of Francois Servois, who first used the term

PlanetMath is a free, collaborative, online mathematics encyclopedia. ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

See also

Look up Commutativity in
Wiktionary, the free dictionary.

  Results from FactBites:
 
Addition - ExampleProblems.com (2022 words)
Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same.
The fact that addition is commutative is known as the "commutative law of addition".
However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".
  More results at FactBites »

 
 

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