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{{Other uses|Commute (disambiguation)}}
{{Other uses|Commute (disambiguation)}}


In [[mathematics]] an operation is '''commutative''' if changing the order of the [[operand]]s does not change the end result. It is a fundamental property of many [[binary operations]], and many [[mathematical proof]]s depend on it. The commutativity of simple operations, such as [[multiplication (mathematics)|multiplication]] and [[addition]] of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, [[division (mathematics)|division]] and [[subtraction]] are ''not'' commutative.
In [[mathematics]] an operation is '''commutative''' if changing the order of the [[operand]]s does not change the end result. It is a fundamental property of many [[binary operations]], and many [[mathematical proof]]s depend on it. The commutativity of simple operations, such as [[multiplication (mathematics)|multiplication]] and [[addition]] of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, [[division (mathematics)|division]] and [[subtraction]] are ''not'' commutative.


==Common uses==
==Common uses==
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The term "commutative" is used in several related senses.<ref name="Krowne, p.1">Krowne, p.1</ref><ref>Weisstein, ''Commute'', p.1</ref>
The term "commutative" is used in several related senses.<ref name="Krowne, p.1">Krowne, p.1</ref><ref>Weisstein, ''Commute'', p.1</ref>


1. A [[binary operation]] ∗ on a [[Set (mathematics)|set]] ''S'' is said to be ''commutative'' if:
1. A [[binary operation]] ∗ on a [[Set (mathematics)|set]] ''S'' is called ''commutative'' if:
:<math>\forall (x,y) \in S: x * y = y * x \,</math>
:<math>\forall (x,y) \in S: x * y = y * x \,</math>


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:<math> x * y = y * x \,</math>
:<math> x * y = y * x \,</math>


3. A [[binary function]] <math>f \colon A \times A \to B</math> is said to be ''commutative'' if:
3. A [[binary function]] <math>f \colon A \times A \to B</math> is said called ''commutative'' if:
:<math>\forall (x,y) \in A: f (x, y) = f(y, x) \,</math>
:<math>\forall (x,y) \in A: f (x, y) = f(y, x) \,</math>


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{{Main|Associativity}}
{{Main|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 in which operations are performed does not affect the final result, as long as the order of terms is not changed. In contrast, the commutative property states that the order of the terms does not affect the final result.
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 doesn't change. In contrast, the commutative property states that the order of the terms does not affect the final result.


===Symmetry===
===Symmetry===
{{Main|Symmetry in mathematics}}
{{Main|Symmetry in mathematics}}


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 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.


For relations, a [[symmetric relation]] is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then <math>a R b \Leftrightarrow b R a</math>.
For relations, a [[symmetric relation]] is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then <math>a R b \Leftrightarrow b R a</math>.
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*Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either way, the end result (having both socks on), is the same.
*Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either way, the end result (having both socks on), is the same.
*The commutativity of addition is observed when paying for an item with cash. Regardless of the order in which the bills handed over, they always give the same total.
*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===
===Commutative operations in mathematics===
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</math>
</math>


According to the [[uncertainty principle]] of [[Werner Heisenberg|Heisenberg]], if the two operators representing a pair of variables do not commute, then that pair of variables are mutually [[complementarity (physics)|complementary]] which means that they cannot be simultaneously measured or known precisely. For example, the position and the linear [[momentum]] of a particle are represented respectively (in the x-direction) by the operators x and (h/2πi)d/dx (where h is [[Planck's constant]]). This is the same example except for the constant (h/2πi), so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.
According to the [[uncertainty principle]] of [[Werner Heisenberg|Heisenberg]], if the two operators representing a pair of variables do not commute, then that pair of variables are mutually [[complementarity (physics)|complementary]], which means they cannot be simultaneously measured or known precisely. For example, the position and the linear [[momentum]] of a particle are represented respectively (in the x-direction) by the operators x and (h/2πi)d/dx (where h is [[Planck's constant]]). This is the same example except for the constant (h/2πi), 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==
==See also==

Revision as of 17:24, 12 September 2011

In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

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 group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, 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]

Mathematical definitions

The term "commutative" is used in several related senses.[4][5]

1. A binary operation ∗ on a set S is called 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 is said called commutative if:

History and etymology

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 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,[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.[9]

Graph showing the symmetry of the addition function

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 doesn't change. 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.

For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then .

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 end result (having both socks on), is the same.
  • 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 are:[4]

For example 4 + 5 = 5 + 4, since both expressions equal 9.
For example, 3 × 5 = 5 × 3, since both expressions equal 15.

Noncommutative operations in everyday life

  • Concatenation, 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.
  • The twists of the Rubik's Cube are noncommutative. This is studied in group theory.

Noncommutative operations in mathematics

Some noncommutative binary operations are:[11]

  • Subtraction is noncommutative since
  • Division is noncommutative since
  • Matrix multiplication is noncommutative since

Mathematical structures and commutativity

Non-commuting operators in quantum mechanics

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and d/dx. These two operators do not commute as may be seen by considering the effect of their products x (d/dx) and (d/dx) x on a one-dimensional wave function ψ(x):

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 of a particle are represented respectively (in the x-direction) by the operators x and (h/2πi)d/dx (where h is Planck's constant). This is the same example except for the constant (h/2πi), 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

Notes

  1. ^ Axler, p.2
  2. ^ a b Gallian, p.34
  3. ^ p. 26,87
  4. ^ a b 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. ^ a b Cabillón and Miller, Commutative and Distributive
  10. ^ O'Conner and Robertson, Servois
  11. ^ Yark, p.1
  12. ^ Gallian p.236
  13. ^ Gallian p.250

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. Boston, Mass.: Houghton Mifflin. 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.

Online resources

Definition of commutativity and examples of commutative operations
Explanation of the term commute
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