Commutative Property of Addition
Commutative Property of Addition
In mathematics, a binary operation is commutative if changing the order of the operands does not
change the 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
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.
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. Euclid is known to
have assumed the commutative property of multiplication in his book Elements. 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.
Know More About :- Commutative Property of Multiplication
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The first recorded use of the term commutative was in a memoir by François 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 in
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
Most commutative operations encountered in practice are also associative. However, commutativity
does not imply associativity. A counterexample is the function which is clearly commutative
(interchanging x and y does not affect the result), but it is not associative (since, for example, but ).
Symmetry:-Graph showing the symmetry of the addition function, Some forms of 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 .
Mathematical structures and commutativity
A commutative semigroup is a set endowed with a total, associative and commutative operation.
If the operation additionally has an identity element, we have a commutative monoid
An abelian group, or commutative group is a group whose group operation is commutative.
A commutative ring is a ring whose multiplication is commutative.
In a field both addition and multiplication are commutative.
Read More About :- What is a Rational Number
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