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 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. 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.[9][10] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[11] 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 Franรงois Servois in 1814,[12][13] 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 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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