# Property of whole numbers

In this post we will try to understand property of whole numbers.
But first let us define a whole number for our better understanding.

Any number starting from zero which is non negative and without any decimal is a whole number.
Technically whole number can also be written as:
W= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,….}

Apart from knowing what whole numbers are, it is important to understand its properties.

Closure Property for Addition of whole numbers

It means that if you add one whole number with another the sum will always be a whole number

Let us understand with example;
1. 9 + 25 = 34
a. We know that 9 & 25 both are whole numbers
b. The result is 34 which is also a whole number

2. 102 + 500 = 602
a. We know that 102 & 500 both are whole numbers
b. The result is 602 which is also a whole number

3. 99 + 170 = 269
a. We know that 99 & 170 both are whole numbers
b. The result is 269 which is also a whole number

So if we add whole number with whole number, the result will always be whole number

Closure Property for Multiplication of whole numbers

It says that if we multiple one whole number with another the result will always be a whole number

Let us understand with examples:
1. Multiple 2 * 3 = 6
a. Both 2 & 3 are whole numbers
b. The result 6 is also a whole number

2. Multiply 20 * 6 = 120
a. Both 20 & 6 are whole numbers
b. The result 120 is also a whole number

3. Multiply 9 * 8= 72
a. Both 9 & 8 are whole numbers
b. The result 72 is also a whole number

Closure Property does not work for Subtraction and Division

Understand that the closure property works only for addition and multiplication.
When we subtract or divide any two whole numbers the closure property does not always work which means we will not always get whole number in the final result.

Let us understand with examples

1. Divide 5 by 2

a. Both 5 and 2 are whole numbers
b. But the result 2.5 is not a whole number

So dividing two whole numbers will not always get you a whole number

Similar is the case for subtraction

2. Subtract 2 – 5 = -3
a. Both 2 and 5 are whole numbers
b. But -3 is negative number and is not a whole number
Hence closure property does not work

Commutative Property of Whole Numbers

Let us first understand what does commutative property means.
If we can move or change the position of number in any math problem without any effect in end result, it means that that the mathematical operation is commutative

In whole numbers commutative property is valid for addition and multiplication, and not valid for division and subtraction.

1. Commutative property of addition of whole numbers
It means that if we interchange whole numbers in addition, the end result will be the same whole number.
For example
a. we know that 4 + 5 =9
But if we interchange the position of 5 & 4, the result will be the same whole number 9
=> 5 + 4 = 9

2. Commutative property of Multiplication of Whole numbers
It means that if we change the position of whole number in multiplication, the end result will be the same whole number.
For example
a. we know that 5 * 4=20
But if we interchange the position of 5 and 4, the result will be the same whole number 20
so if we do 4 * 5 = 20

But the same commutative property of whole number does not work for subtraction and division
For example
If we subtract 5 – 4 = 1
we get one as a result. But if we interchange the position the result will be different
After interchanging the position we get 4 – 5 => -1, which is completely different non whole number

Similar is the case for division.
Division of whole number is non commutative

Associative Property of Whole Numbers

By Associative property we mean that how different we associate or group numbers in mathematical operation the result will be the same.

In whole numbers, the associative property works for addition and multiplication.

a. Associative property of addition of whole numbers
It means that even if we group numbers differently in mathematical operation, the result will be the same whole number.
Let us understand this with the help of example

We know that addition of 2 + 6 + 5 = 13
But let us see what happens if we group the number differently with the help of brackets

(2 + 6) + 5 => 8 + 5 => 13

Again grouping differently
2 + (6 + 5) => 2 + 11 => 13

You can see that in all the different groupings the final result is same whole number 13.
It means that addition of whole number follow the rule of association

b. Associative Property of Multiplication of whole numbers
Similar to addition, associative property of multiplication of whole numbers also works.
So if we form different form of groups in multiplication of whole numbers, the result will be same

We know that if we multiply 2 * 7 * 5 = 70
But let us see what happens if we group the number differently with the help of brackets

(2 * 7) * 5 => 14 * 5 => 70

Again grouping differently
2 * (7 * 5) => 2 * 35 => 70

You can see that in all the different groupings the final result is same whole number 70.
It means that multiplication of whole number follow the rule of association

The same associative property does not work for subtraction and division. That mean if we form different groupings in subtraction and division, the result will be different

Let us understand with example

Distributive Property of whole numbers

Distributive property means that even if we divide the number into small parts and then again perform the mathematical operation, the result will be the same.

for example
we know that 6 * 8 = 48

Now let us apply distributive property and do the calculation
First break the number 8 into two parts

we can see from above that even if we break the number 8 into parts, the final result of the maths operation is still the same. Hence distributive property of whole number is followed.

Let us look at another example
we know that 12 * 5 = 60
Now let us break 5 into two parts (3 + 2)

Again after breaking 5 into two components, the end result is the same.
I hope you have now understood the working of distributive property of whole numbers. Incase, if you have any doubt feel free to ask in the comment section