# Understanding the properties explaining the effect of mathematical operators

For detailed knowledge of the properties and the respective effect on numbers, it is mandatory to study the basic number properties. Though the idea behind the basic application and understanding is ‘common sense’ of maths, appropriate analysis is needed to impart a perfect understanding of basic maths properties, namely, closure property, commutative property, associative, distributive, and identity. No matter what the set of numbers is, may it be whole numbers or rational numbers or integers or real numbers or any other set, the application of properties remains the same.

It should be noted that these properties hold good only under the basic addition and multiplication and do not apply in the case of subtraction and division. Now let’s take a closer look for a better understanding of the same.

Closure property

The closure property holds good for all sets of numbers and this means that the set is closed under the operation of addition and multiplication. In simple words, whenever the elements of any particular set are added or multiplied, the final result is an element from the same set. For instance, if you add three real numbers then the resultant number also will be a real number.

The knowledge about the operations being closed for some set of numbers helps to understand the nature of the same. As it is easy to understand why the set of natural numbers is more versatile than that of integers.

Commutative property

The commutative property of addition and multiplication can be described and related to the set of numbers as under these two operations the order doesn’t matter, i.e., 5+7 or 7+5 doesn’t make any difference in the result and similarly 5*7 or 7*5 means the same. Commutativity has a more precise and formal explanation in algebra and arithmetic. We have already seen that operations addition and multiplication are commutative. It is important to kn6ow that division, subtraction, and exponents are non-commutative operations.

The commutativity can be assigned to other mathematical elements as well like the respective property is used to simplify various algebraic expressions including like and unlike terms.

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Associative property

The associative property defines addition and multiplication operations and explains that the elements in an operation can be assigned to any of the other elements breaking the sequence or regardless of the group and the result will not vary. For example, 3+(4+5) means the same as (3+4) + 5 likewise for multiplication, 3*(4*5) is as same as (3*4) *5.

This property is more or less like commutative property which includes only two numbers. Here also likewise the order doesn’t matter. The property doesn’t hold well for the operation of subtraction and division.

Distributive property

The distributive property applies to multiple operations and the expressions using parentheses. The BODMAS and other such rules utilize this well. The property is simple to understand and easy to apply. The credit of most frequent usability goes to the distributive property. The property states that in the case of A*(B+C-D) means A*B+A*C-A*C.

This can be applied in the case of algebra, arithmetic, or any other application. The distributive property is one that even includes the operations of subtraction, division, and exponents, etc. This makes it more versatile than all others.

Identity

The property defines that there are some mirrors like numbers which when operated with others, result in the same number as the result. Additive and multiplicative identities are the numbers that don’t affect the numbers irrespective of the operations.

• Additive identity: 0 acts as the additive identity for any number as anything added to zero or zero added to any number results in the same number itself.
• Multiplicative identity: 1 is the multiplicative identity for all numbers as anything multiplied by one doesn’t make any difference.

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