# Properties of Whole Numbers

Properties of whole numbers are defined to operate basic arithmetic operations such as addition and multiplication, in an easy way. Applying these properties helps us to solve such Maths problems quickly, without even using calculators. These properties are almost similar to the properties of integers, although the definition for whole numbers and integers are different.

In this article, we are going to discuss the properties of whole numbers along with examples. Students of Class 6 can learn the properties here at BYJU’S and also solve the mathematical questions based on them. Before we begin with the properties, let us discuss whole numbers.

## What are Whole Numbers?

Whole numbers are the set of positive integers or natural numbers along with the zero. But they cannot be a negative integer. Usually, the set of whole numbers is represented by ‘W’.

W = {0,1,2,3,4,5,6,7,8,9,10,….}

## What are the Properties of Whole Numbers?

Properties of whole numbers are:

• Closure Property
• Commutative Property of Addition and Multiplication
• Associative Property of Addition and Multiplication
• Distributive Property of multiplication over addition
• Identity Property

### Closure Property

According to the Closure Property “Whole numbers are closed under addition and multiplication”. It means, when we add or multiply two whole numbers, then the resulting value is also a whole number.

If A and B are two whole numbers, then,

• A + B → W
• A x B → W

Where W represents whole numbers.

See some examples of closure property below:

 Closure property of Addition Closure property of multiplication 2 + 3 = 5 2 x 3 = 6 10 + 9 = 19 10 x 9 = 90

In the above examples, we can see, the resulting values such as 5, 6, 19 and 90 are also whole numbers. Therefore, the closure property is closed under addition and multiplication.

 Note: Closure property is not applicable for subtraction and division of whole numbers Difference of two whole numbers cannot be necessarily a whole number Division of two whole numbers cannot be necessarily a whole number Division of a whole number by zero (0) is undefined

### Commutativity of Addition and Multiplication

According to the commutative property of whole numbers, if two whole numbers are added or multiplied together, then the change in order of the numbers does not change the result. We can add or multiply two whole numbers in any order.

If A and B are two whole numbers, then;

• A + B = B + A
• A x B = B x A

Examples are given below:

 Commutativity of Addition Commutativity of Multiplication 3 + 5 = 5 + 3 ⇒ 8 3 x 5 = 5 x 3 ⇒ 15 4 + 6 = 6 + 4 ⇒ 10 4 x 6 = 6 x 4 ⇒ 24

As we can see from the above examples, if we change the order of the numbers, the result remains the same.

 Note: Commutativity is not applicable for subtraction and division Changing the order of numbers in subtraction, changes the result (Eg. 3 – 2 = 1, but 2 – 3 = -1, therefore 3 – 2 ≠ 2 -3) Changing the order of numbers in division, changes the result (Eg. 10 ÷ 5 = 2, but 5 ÷ 10 = 0.5, therefore 10 ÷ 5 ≠ 5 ÷ 10)

### Associativity of Addition and Multiplication

The associative property of addition and multiplication states that the regrouping of three whole numbers does not change the result of their sum and product.

Let A, B and C are three whole numbers, then as per associativity,

 A + (B + C) = (A + B) + C A x (B x C) = (A x B) x C

We can understand the above two expressions, with the help of examples given below.

 Associativity of Addition Associativity of Multiplication (2 + 3) + 4 = 2 + (3 + 4) ⇒ 9 (2 x 3) x 4 = 2 x (3 x 4) ⇒ 24 (4 + 4) + 5 = 4 + (4 + 5) ⇒ 13 (4 x 4) x 5 = 4 x (4 x 5) ⇒ 80

From the above examples, we can see, the regrouping of the numbers does not change the actual result.

 Note: Associativity of subtraction and division of whole numbers does not exist.

### Distributivity of Multiplication over Addition

In this property, the multiplication is distributive over addition. It means, if A, B and C are three whole numbers, then;

 A x (B + C) = A x B + A x C

Let us see the examples.

1. 6 × (5 + 3) = (6 × 5) + (6 × 3)

LHS = 6 × (5 + 3) = 6 × 8 = 48

RHS = (6 × 5) + (6 × 3) = 30 + 18 = 48

Hence, proved.

2. 7 × (11 + 9) = 7 × 11 + 7 × 9

LHS = 7 × (11 + 9) = 7 × 20 = 140

RHS = 7 × 11 + 7 × 9 = 77 + 63 = 140

Hence, proved.

### Identity Property (for addition and multiplication)

The identity property of whole numbers for addition and multiplication states that:

 W + 0 = W W x 1 = W

Where W is any whole number.

Thus, by the above given expression, we can conclude that, if zero is added to any whole number, then the value of the original number does not change. Similarly, when we multiply any whole number by 1, then the value of the actual number remains unchanged. Hence, identity property holds both for addition and multiplication.

See some examples here:

 Identity for addition Identity for multiplication 2 + 0 = 2 2 x 1 = 2 5 + 0 = 5 5 x 1 = 5 7 + 0 = 7 7 x 1 = 7 10 + 0 = 10 10 x 1 = 10 50 + 0 = 50 50 x 1 = 50 100 + 0 = 100 100 x 1 = 100

Therefore, 0 is the additive identity and 1 is the multiplicative identity of any whole number.

Also, see:

## Solved Examples

Q.1: Add the numbers 134, 97 and 203, using suitable identity.

Solution: 134 + 97 + 203

= 134 + (97 + 203) [By associative property]

= 134 + 300

= 434

Q.2: Simplify using suitable identity: 126 × 55 + 126 × 45

Solution: 126 × 55 + 126 × 45

= 126 × (55 + 45) [Using distributive property]

= 126 × 100

= 12600

### Practice Questions

1. Find the product by suitable rearrangement:

• 600 × 189 × 16
• 4 × 160 × 25
• 8 × 291 × 100

2. Find the product using identities:

• 8 x 125 x 1294
• 854 x 102

## Frequently Asked Questions on Properties of Whole Numbers

### What are the five properties of whole numbers?

The five properties of whole numbers are:
Commutative property for addition and multiplication
Associative property for addition and multiplication
Distributive property of multiplication over addition

### What is the commutative property of whole numbers?

As per the commutative property of whole numbers, changing the order of the two numbers, when added or multiplied, does not change the result.

### Under which property are the whole numbers closed?

Whole numbers are closed under addition and multiplication. It means the addition and multiplication of two or more whole numbers results in a whole number only.
20 + 30 = 50, a whole number.
20 x 30 = 600, a whole number.

### What is the formula of distributive property?

The formula for distributive property is given by:
A x (B + C) = A x B + A x C
Thus, distributive property is finding the product over the sum.

### What is the identity for the addition of whole numbers?

Zero is the identity for the addition of whole numbers.

### What is the identity for multiplication of whole numbers?

One (1) is the identity for multiplication of whole numbers