Properties of Whole Number Operations

Maths

definition

Closure property with reference to natural/whole numbers

Closure property says that if for any two natural/whole numbers and is also a natural/whole number then the set of natural/whole numbers is closed under
where represents or
Take any two natural/whole numbers and , add them
So, we get /.

example

Closure property with reference to natural/whole numbers

For example:- Take two numbers and
Now, which is a natural/whole number
Hence, /  is closed under addition.
For subtraction :-
which is not a natural/whole number i.e.
Hence, / is not closed under subtraction.
For multiplication :-
which is a natural number
Hence, / is closed under multiplication.
For division:-
which is not a natural number
Hence, / is not closed under division.

definition

Commutative property of numbers

Take any two numbers and  in your mind. Now add and , comes as .
Add and , comes to be .
Aren't they the same?
Yes, they are equal.
This is because of commutative property.
Commutative Property of numbers says that we can swap the numbers and still we get the same answer.

result

Commutative property of numbers

Take and as two numbers and subtract them i.e. .
Now, subtract from i.e. .
Are they same?
No, they are not equal.
For example:-
So, commutative property does not hold for subtraction.

definition

Associative property of numbers

Definition:- The associative property holds for atleast or more than numbers and it says that the numbers can be added or multiplied regardless of how they are grouped(Group means how we put the brackets). 
For any binary operation , associative property states that

Assume any three numbers and in your mind. Add and and then add to the sum you got i.e. .
Now, add and and then add to the sum you got i.e. to get
Now, we can see that  
Also,
Hence, associative property holds for addition and multiplication.

example

Associative property of numbers

Example:-
We know that, and
So, and satisfy associative property of addition and multiplication.

example

Associative Property in reference to natural/whole numbers

What about subtraction and division ?
Let's take three numbers and
Now, and

Thus,
Also, Now, and

Thus, .
Hence, associative property does not hold for subtraction and division.

example

Distributive property of numbers over addition and subtraction

Definition:- Distributive property is an algebra property which is used to multiply a single term with two or more terms inside the set of parentheses.
i.e.
For any three numbers and , distributive property over addition is given by :
 
For example :-
Similarly, for any three numbers and , distributive property of multiplication over subtraction is given by :

definition

Distributive property of numbers over addition and subtraction

Take A and B as two friends, suppose they have and toffees respectively. What is the total no. of toffees they have together ?
Is it ?
Now, if you have 3 times the no. of toffees with A and B, then how many toffees do you have ?
Then, by using distributivity

definition

Define and identify additive identity for natural/whole numbers

Additive identity of any natural/whole is a number which when added to , leaves it unchanged.
When we add to any natural/whole number , we get

Hence, is the additive identity for natural/whole numbers.

definition

Multiplicative inverse for natural/whole numbers

Multiplicative identity of any natural/whole is a number which when multiplied to , leaves it unchanged.
When we multiply to any natural/whole number , we get

Hence, is the multiplicative identity for natural/whole numbers.

definition

Closure Property with reference to Whole Numbers

System of whole numbers under Addition:
Addition of two whole numbers always results in a whole number.
Eg:
Result is a whole number.
Therefore, system is closed under addition.

System of whole numbers under Subtraction:
Subtraction of two whole numbers does not always results in a whole number.
Eg:
Result is a whole number, but 
Result is not a whole number.
Therefore, system is not closed under subtraction.

System of whole numbers under Multiplication:
Multiplication of two whole numbers always results in a whole number.
Eg:
Result is a whole number.
Therefore, system is closed under Multiplication.

System of whole numbers under Division:
Division of two whole numbers does not always results in a whole number.
Eg:
Result is a whole number, but
Result is not a whole number.
Therefore, system is not closed under division.

definition

Additive Identity

The additive identity of a set which is equipped with the operation of addition is an element which, when added to any element in the set, yields .
If is added to a number, it keeps its identity.
Therefore is an additive identity.