**What are Whole Numbers?**

Before we learn about the operations of whole numbers, it is important to recall what is meant by whole numbers. Whole numbers are a set of natural numbers with the number 0. This means that the set of numbers 0, 1, 2, 3 ……. are whole numbers. It is important to note that there are now whole numbers between 0 and 1, 1 and 2 and so on. The set of whole numbers is represented by W. Therefore, we have,

W = { 0, 1, 2, 3, ……… }

The smallest whole number is 0 while there is no largest whole number as the set extends till infinity. Now, let us learn about operations on whole numbers.

**What are Fundamental Operations of Whole Numbers?**

There are four fundamental operations of whole numbers, namely –

- Addition
- Subtraction
- Multiplication
- Division

These four operations also form the basis of any other set of numbers as well such as natural numbers and real numbers. Let us now understand the properties of these four operations on whole numbers.

There are three major properties that define different operations on a set of numbers. The properties are –

**Closure Property** – Closure property states that when an operation is performed on two numbers, the resultant would also be of the same type as the numbers on whom the operation has been performed.

**Commutative Property – **Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter.** **

**Associative Property –** Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter.

Other than the above properties, there are some properties such as identity property, inverse, etc. that are defined for some of the operations only.

It is important to note that not every operation on a set of numbers satisfies all the properties. Let us now understand how well the above properties are satisfied by the four operations on whole numbers.

**Properties of Addition on Whole Numbers**

The addition of whole numbers possesses the following properties –

**Additive Identity**

Additive identity for whole numbers states that –

**If zero is added to any whole number, the sum remains the number itself. **

As we can see that

**0 + a = a = a + 0**, where a is a whole number. Therefore, the number zero is called the additive identity, as it does not change the value of the number when the addition is performed on the number.

**Closure Property**

For whole numbers to satisfy the closure property of addition the following statement should be true –

**If a and b are any two whole numbers then a +b is also a whole number. **

Let us verify this statement.

Let there by two whole numbers say 2 and 7. If we add 2 and 7 we will get 9 which is again a whole number. This is true for any other combination of whole numbers as well.

Therefore, we can say that **addition of whole numbers satisfies the closure property.**

**Commutative Property**

For the commutative property to be true for addition of whole numbers it means that –

**a + b = b + a, for any two whole numbers, a and b.**

Let us verify this statement.

Suppose we have two numbers 32 and 15.

We want to add 15 and 32. The sum will be 15 +32 = 47

Now, if we interchange the order of numbers and we get the problem as 32 + 15.

What would be the answer?

The sum would be 32 + 15 = 47 which is equal to our previous result.

This means that, if we have two numbers, a and b, then

**a + b = b + a**

**Hence, addition is commutative for whole numbers.**

**Associative Property**

For the associative property to be true for addition of whole numbers, the following statement must be true –

If we want to add 3 numbers, two of them can be chosen first. The result of this addition would serve as the first number the original third number will serve as the second number to get the final answer. Let us verify this statement.

Let us take 3 numbers, 8, 5 and 2.

Let us take 8 and 5 first.

We get, 8 + 5 = 13

Now, we add 13 and 2, we get 13 + 2 = 15

Now let us reverse the order and choose 5 and 2 first.

We get, 5 + 2 = 7

Now, we add this result to 8, we get, 7 + 8 = 15

Both the processes give us the same answer.

Therefore, we can say that addition satisfies the associative property. We get

**(a + b) + c = a + (b + c)**

**Hence, addition of whole numbers is associative. **

Let us now look at the properties of subtraction on whole numbers.

**Properties of Subtraction on Whole Numbers**

Subtraction of whole numbers possesses the following properties –

**Closure Property**

For whole numbers to satisfy the closure property of subtraction the following statement should be true –

**If a and b are any two whole numbers then a – b is also a whole number. **

Let us verify this statement.

Let there by two whole numbers say 2 and 7. If we subtract 7 from 2 we will get -5 which is not a whole number. But, if we subtract 2 from 7 we will get 5 which is a whole number. Therefore, the above statement may or not be true for any other combination of whole numbers as well.

Therefore, we can say that **subtraction of whole numbers does not satisfy the closure property.**

**Commutative Property**

For the commutative property to be true for the subtraction of whole numbers it means that if one number is subtracted from the other, it does not matter which number is marked as the minuend and which one is the subtrahend. Let us verify this statement.

Suppose we have two numbers 32 and 15.

We want to subtract 15 from 32. The difference would be 32 – 15 = 17

Now, if we interchange the subtrahend and the minuend, we get the problem as 15 – 32.

What would be the answer?

The difference between 15 and 32 would be -17 which is not equal to 17.

This means that, if we have two numbers, a and b, then

**a – b ≠ b – a**

**Hence, subtraction is not commutative for whole numbers.**

**Associative Property**

For the associative property to be true for subtraction of whole numbers, the following statement must be true –

If we want to find the difference between three numbers, we will have to find the difference between any two first and then find the difference between the result of the first operation and the third number. Will we get the same answer for any order of three numbers a, b and c?

Let us find out.

Suppose, we have three numbers, 5, 8, and 9 and we want to find the difference between them.

We first perform the operation 5 – 8. We get the difference as -3. Now we find out the difference between this result (-3) and 9.

We have

-3 – 9 = -12 which is our final answer.

Now, let us move from right to left. Let us find out the difference between 9 and 8.

We get

9 – 8 = 1

Now we find the difference between 1 and the third number, 5

We have

1 – 5 = -4

Therefore, we can say that subtraction does not satisfy the associative property. We get

**(a – b) – c ≠ a – (b – c)**

**Hence, subtraction of whole numbers is not associative. **

Let us now look at the properties of multiplication on whole numbers.

**Properties of Multiplication on Whole Numbers**

Multiplication of whole numbers possesses the following properties –

**Multiplicative Identity**

A number is said to be an identity for multiplication, if, a number when multiplied by this identity number results in the number itself. Here, **1 is the identity element for multiplication**. Let us see why?

4 x 1 = 4

15 x 1 = 15

20 x 1 = 20

Therefore, any number when multiplied by 1 number results in the number itself. This is the identity property of multiplication.

**Closure Property**

For whole numbers to satisfy the closure property of multiplication the following statement should be true –

**If a and b are any two whole numbers then a x b is also a whole number. **

Let us verify this statement.

Let there by two whole numbers say 2 and 7. If we multiply 2 with 7 we will get 14 which is again a whole number. This is true for any other combination of whole numbers as well.

Therefore, we can say that **multiplication of whole numbers satisfies the closure property.**

**Commutative Property**

For the commutative property to be true for the multiplication of whole numbers it means that if one number is multiplied with the other, it does not matter which number is marked as the multiplier and which number is marked as the multiplicand. Let us verify this statement.

Let us take two numbers 8 and 5.

We first mark 8 as the multiplier and 5 as the multiplicand.

We get, 8 x 5 = 40.

Now let us reverse the order, i.e. 5 becomes the multiplier and 8 becomes the multiplicand.

Now, we get, 5 x 8 = 40.

Both the processes give us the same answer.

Therefore, we can say that multiplication satisfies the communicative property. We get

**a x b = b x a**

Hence, multiplication is communicative.

**Associative Property**

For the associative property to be true for multiplication of whole numbers, the following statement must be true –

If we want to multiply 3 numbers, two of them can be chosen first, one as a multiplier and the second as a multiplicand. The result of the multiplication would serve as a multiplier and the third number as multiplicand to get the final answer. Let us verify this statement.

Let us take 3 numbers, 8, 5 and 2.

Let us take 8 and 5 first.

We first mark 8 as the multiplier and 5 as the multiplicand.

We get, 8 x 5 = 40.

Now, we multiply 40 by 2, we get 40 x 2 = 80

Now let us reverse the order and choose 5 and 2 first.

We get, 5 x 2 = 10

Now, we multiply this result by 8, we get, 10 x 8 = 80

Both the processes give us the same answer.

Therefore, we can say that multiplication satisfies the associative property. We get

**(a x b) x c = a x (b x c)**

Hence, multiplication is associative.

**Distributive Property of Multiplication over Addition / Subtraction**

When two whole numbers are added or subtracted and the result is multiplied by another number, they can be multiplied separately.

Therefore, for any three whole numbers, a, b and c, the distributive property of multiplication over addition states that

**a x ( b + c) = (a x b) + (a x c)**

**For example, let us consider 10 x (18 + 12)**

There are two ways to solve this.

**First Method**

First, we add 18 and 12, we get 30. Now we multiply, 30 and 10, we get 10 x 30 = 300

**Second Method**

Now, we use the distributive property of multiplication over addition.

We have 10 x (18 + 12)

= (10 x 18) + (10 x 12)

= 180 + 120

= 300

By both methods, we get the same answer. Hence,

10 x (18 + 12) = (10 x 18) + (10 x 12)

Similarly, for any three numbers, a, b and c, the distributive property of multiplication over subtraction states that

**a x ( b – c) = (a x b) – (a x c)**

Let us now look at the properties of division on whole numbers.

**Properties of Division on Whole Numbers**

Division of whole numbers possesses the following properties –

**Closure Property**

For whole numbers to satisfy the closure property of subtraction the following statement should be true –

**If a and b are any two whole numbers then a ÷ b is also a whole number. **

Let us verify this statement.

Let there by two whole numbers say 4 and 20. If we divide 4 by 20 we will get 0.2 is not a whole number. But, if we divide 20 by 4 we will get 5 which is a whole number. Therefore, the above statement may or not be true for any other combination of whole numbers as well.

Therefore, we can say that **division of whole numbers does not satisfy the closure property.**

**Commutative Property**

For the commutative property to be true for the multiplication of whole numbers it means that if one number is multiplied with the other, it does not matter which number is marked as the multiplier and which number is marked as the multiplicand. Let us verify this statement.

Let us take two numbers 8 and 5.

We first mark 8 as the multiplier and 5 as the multiplicand.

We get, 8 x 5 = 40.

Now let us reverse the order, i.e. 5 becomes the multiplier and 8 becomes the multiplicand.

Now, we get, 5 x 8 = 40.

Both the processes give us the same answer.

Therefore, we can say that multiplication satisfies the communicative property. We get

**a x b = b x a**

Hence, multiplication is communicative.

**Associative Property**

For the associative property to be true for division of whole numbers, the following statement must be true –

If a, b, and c are whole numbers then (a ÷ b) ÷ c ≠ a ÷ (b ÷ c). Hence, the associative property does not hold good for the division of whole numbers. Let us verify this statement

Suppose we have three numbers, 15, 3 and 5. Now

(15 ÷ 3) ÷ 5 = 5 ÷ 5 = 1

Also,

15 ÷ (3 ÷ 5) = 15 ÷ 3/5 = 15 × 5/3

= 25

We can see that (15 ÷ 3) ÷ 5 ≠ 15 ÷ (3 ÷ 5)

So, we have

**(a ÷ b) ÷ c ≠ a ÷ (b ÷ c)**

**Hence, the division of whole numbers is not associative. **

**Key Facts and Summary**

- Whole numbers are a set of natural numbers with the number 0.
- The smallest whole number is 0 while there is no largest whole number as the set extends till infinity.
- There are four fundamental operations of whole numbers, namely, addition, subtraction, multiplication and division.
- Closure property states that when an operation is performed on two numbers, the resultant would also be of the same type as the numbers on whom the operation has been performed.
- Commutative Property states that when an operation is performed on two numbers, the order in which the numbers are placed does not matter.
- Associative Property states that when an operation is performed on more than two numbers, the order in which the numbers are placed does not matter.
- The addition and multiplication of whole numbers are commutative as well as associative.
- Subtraction and Division of Whole numbers are neither commutative nor associative.
- The number zero is called the additive identity for whole numbers.
- The number 1 is the multiplicative identity for whole numbers.
- Whole numbers satisfy the distributive property of multiplication over addition / subtraction.

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