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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. COMMON USES The ''commutative property'' (or ''commutative law'') is a property associated with Binary Operation s 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.Axler, p.2Gallian, p.34p. 26,87 MATHEMATICAL DEFINITIONS 1. A Binary Operation ∗ on a set ''S'' is said to be ''commutative'' if:Krowne, p.1 x
2. One says that ''x commutes'' with ''y'' under ∗ if:Weisstein, ''Commute'', p.1 x 3. A Binary Function f:''A''×''B'' → ''C'' is said to be ''commutative'' if: :f(''x'',''y'') = f(''y'',''x'') for every ''x'' ∈ ''A'', ''y'' ∈ ''B'' HISTORY Records of the implicit use of the commutative property go back to ancient times. The Egypt ians used the commutative property of multiplication to simplify computing products.Lumpkin, p.11Gay and Shute, p.? Euclid is known to have assumed the commutative property of multiplication in his book ''Elements'' .O'Conner and Robertson, ''Real Numbers'' 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,Cabillón and Miller, ''Commutative and Distributive''O'Conner and Robertson, ''Servois'' 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.Cabillón and Miller, ''Commutative and Distributive'' RELATED PROPERTIES Associativity See Also: 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 See Also: 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. EXAMPLES Commutative operations in everyday life
Commutative operations in math Two well-known examples of commutative binary operations are:Krowne, p.1
:: :For example 4 + 5 = 5 + 4, since both Expression s equal 9.
:: :For example, 3 × 5 = 5 × 3, since both expressions equal 15.
Noncommutative operations in everyday life , the act of joining character strings together, is a noncommutative operation.]]
Noncommutative operations in math Some noncommutative binary operations are:Yark, p.1
eq 1-0
eq 2/1
: \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} eq \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} MATHEMATICAL STRUCTURES AND COMMUTATIVITY
NOTES REFERENCES 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 Article describing the mathematical ability of ancient civilizations. 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 SEE ALSO |
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