Commutativity
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{{Other uses|Commute (disambiguation)}}(File:Commutative Addition.svg|right|thumb|280px|Example showing the commutativity of addition (3 + 2 = 2 + 3))In
mathematics,
commutativity is the property that changing the order of something 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.
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 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)Mathematical definitions
{{see|Symmetric function}}The term "commutative" is used in several related senses.
(4)(5)1. A
binary operation ∗ on a
set S is said to be
commutative if:
∀ xy ∈ S: x * y = y * x
- 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 f:
A×A →
B is said to be
commutative if:
∀ xy ∈ A: f (x y) = f(y x)
History and etymology
(File:Commutative Word Origin.PNG|left|thumb|250px|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.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.
Related properties
(File:Symmetry Of Addition.svg|right|thumb|200px|Graph showing the symmetry of the addition function)
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 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.
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
binary relations, a
symmetric relation is analogous to a commutative operation, in that if a relation
R is symmetric, then
a R b ⇔ b R a
.
Examples
Commutative operations in everyday life
- Putting on shoes resembles a commutative operation, since which shoe is put on first is unimportant. Either way, the end result (having both shoes 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 are withdrawn, they always give the same total.
- The arbitrary decision of which food group to eat first, second, etc... off of a plate of multiple separated foods, as long as all foods will be consumed in the end, is an example of the commutative property.
Commutative operations in mathematics
Two well-known examples of commutative binary operations are:
∀ (yz) ∈ R: y + z = z + y
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
File:Noncommutative Example Concatenation.svg|thumb|240px|left|
ConcatenationConcatenation
- 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
Infinite addition is not (necessarily) commutative:
1-1+1-1+1-1+1-1+deriv(⋅)s<1
whereas
1+1-1+1+1-1+1+1-1+deriv(⋅)s=&∈f∈;
Some noncommutative binary operations are:
(11)
- Subtraction is noncommutative since
0-1≠q 1-0
- Division is noncommutative since
1/2≠q 2/1
- Matrix multiplication is noncommutative since
begin{bmatrix}end{bmatrix}
begin{bmatrix}1 & 1 end{bmatrix}cdotbegin{bmatrix}end{bmatrix}neqbegin{bmatrix}end{bmatrix}cdotbegin{bmatrix}1 & 1 end{bmatrix}
begin{bmatrix}end{bmatrix}
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):
x{dover dx}psi = xpsi' neq {dover dx}xpsi = psi + xpsi'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 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.
See also
{{Wiktionary}}
Notes
-
[Axler, p.2]
-
[Gallian, p.34]
-
[p. 26,87]
-
[Krowne, p.1]
-
[Weisstein, Commute, p.1]
-
[Lumpkin, p.11]
-
[Gay and Shute, p.?]
-
[O'Conner and Robertson, Real Numbers]
-
[Cabillón and Miller, Commutative and Distributive]
-
[O'Conner and Robertson, Servois]
-
[Yark, p.1]
-
[Gallian p.236]
-
[Gallian p.250]
References
Books
- BOOK, Sheldon, Axler, Linear Algebra Done Right, 2e, Springer, 1997, 0-387-98258-2,
''Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
- BOOK, Frederick, Goodman, Algebra: Abstract and Concrete, Stressing Symmetry, 2e, Prentice Hall, 2003, 0-13-067342-0,
''Abstract algebra theory. Uses commutativity property throughout book.
- BOOK, Joseph, Gallian, Contemporary Abstract Algebra, 6e, 2006, 0-618-51471-6, Houghton Mifflin, Boston, Mass.,
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
Articles
weblink 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
Online resources
- Krowne, Aaron, {{PlanetMath|title=Commutative|urlname=Commutative}}, Accessed 8 August 2007.
Definition of commutativity and examples of commutative operations
- {{MathWorld|title=Commute|urlname=Commute}}, Accessed 8 August 2007.
Explanation of the term commute
- Yark. {{PlanetMath|title=Examples of non-commutative operations|urlname=ExampleOfCommutative}}, 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
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