RS Aggarwal Solutions Class 6 Chapter-3 Whole Numbers (Ex 3D) Exercise 3.4

‘Whole Number’ is Chapter 3 of Class 6 Mathematics. In this Chapter, students will learn that the Whole Numbers  are the part of the Number System which includes all the positive Integers, starting from 0 to infinity. It can also be said that, all the Whole Numbers are real numbers, but not all the real numbers are Whole Numbers. Thus, according to RS Aggarwal solutions, Whole Numbers can be defined as a set of natural numbers including 0. Whole Numbers are represented by W, whereas, natural numbers and Integers are represented by N and  Z respectively. In simple words, all the natural numbers are Whole Numbers. All counting numbers are Whole Numbers. All the positive Integers including 0 are Whole Numbers. Also, all Whole Numbers are real numbers. Integers are the negative of natural numbers and the set of Whole Numbers. So, Integers include both positive and negative numbers including 0. Real numbers are a group of all these types of numbers, i.e. Natural numbers, Whole Numbers, Integers and fractions.The full set of natural numbers along with ‘0’ are called Whole Numbers. The Examples are: 0, 11, 251, 3700, 99999, 881100, etc.

There are many properties of Whole Numbers which are based on Mathematical operations such as addition, subtraction, division and multiplication that are defined in RS Aggarwal Chapter 3 Exercise 3.4 solutions. If we add or multiply two Whole Numbers the result will be a Whole Number itself. On the other hand, if we subtract two Whole Numbers, the result  may not be a Whole Number, i.e. it can also be an Integer. Also, upon the division of two Whole Numbers the result can be a fraction sometimes. 

Properties of Whole Numbers.

The properties of Whole Numbers along with their proofs and properties are given below:

Closure Property

According to the Closure Property of Whole Numbers they can be closed under addition and multiplication, which means, if x and y are two Whole Numbers then x(y) or x + y will also be a Whole Number.

Example:

3 and 8 are Whole Numbers.

3 + 8 = 11 (a Whole Number)

3 × 8 = 24 ( a Whole Number)

This proves that the Whole Numbers are closed under addition and multiplication.

Commutative Property of Addition and Multiplication

According to the Commutative property of Addition and Multiplication, the sum and product of two Whole Numbers will be the same even if the order in which they are added or multiplied is changed, i.e., if x and y are two Whole Numbers, then x + y = y + x and x . y = y . x

Proof:

Let us take two Whole Numbers 5 and 7.

9 + 3 = 12

3 + 9 = 12

Thus, 3 + 9 = 9 +3 .

Also,

5 × 7 = 35

7 × 5 = 35

Thus, 5 × 7 = 7 × 5

So the Whole Numbers are commutative under addition and multiplication.

Additive identity

According to the Additive Identity when a Whole Number is added to 0, its value remains unchanged, i.e., if x is a Whole Number then x + 0 = 0 + x = x

Proof: 

Consider two Whole Numbers 0 and 12.

0 + 12 = 12

12 + 0 = 12

Here, 0 + 12 = 12 + 0

Therefore, 0 is called the additive identity of Whole Numbers.

Multiplicative identity

According to the Multiplicative Identity, when a Whole Number is multiplied by 1, its value remains unchanged, i.e., if x is a type of Whole Number then x.1 = x = 1.x

Proof:

Consider two Whole Numbers 1 and 7.

1 × 7 = 7

7 × 1 = 7

Here, 1 × 7 = 7 = 7 × 1

Therefore, 1 is the multiplicative identity of Whole Numbers.

Associative Property

According to the Associative Property, when Whole Numbers are added or multiplied as a set, they can be grouped in any order, and the result will be the same, For Example, if x, y and z are Whole Numbers then x + (y + z) = (x + y) + z and x. (y.z)=(x.y).z

Proof:

Let us take three Whole Numbers 2, 3, and 5.

2 + (3 + 5) = 2 + 8 = 10

(2 + 3) + 5 = 5 + 5 = 10

Thus, 2 + (3 + 5) = (2 + 3) + 5 

2 × (3 × 5) = 2 × 15 = 30

(2 × 3) × 5 = 6 × 5 = 30

Here, 2 × (3 × 5) = (2 × 3) × 5

Hence, the Whole Numbers are associative under addition and multiplication.

Distributive Property

According to the Distributive Property, If x, y and z are three Whole Numbers, the distributive property of multiplication over addition is x. (y + z) = (x.y) + (x.z), and like this, the distributive property of multiplication over subtraction is x. (y – z) = (x.y) – (x.z)

Proof: 

Let us consider three Whole Numbers 9, 10 and 6.

9 × (10 + 6) = 9 × 16 = 144

(9 × 10) + (9 × 6) = 90 + 54 = 144

Here, 9 × (10 + 6) = (9 × 10) + (9 × 6)

Also,

9 × (10 – 6) = 9 × 4 = 36

(9 × 10) – (9 × 6) = 90 – 54 = 36

So, 9 × (10 – 6) = (9 × 10) – (9 × 6)

Hence, the distributive property of Whole Numbers is verified.

Multiplication by zero

If an when a Whole Number is multiplied to 0, the result is always 0, i.e., x.0 = 0.x = 0

Example:

0 × 5 = 0

5 × 0 = 0

Here, 0 × 5 = 5 × 0 = 0

Thus, any Whole Number when multiplied by 0, gives 0.

Division by zero

Division of a Whole Number by o is not defined, i.e., if x is a type of Whole Number then x/0 is not defined.

All the solutions for questions of Exercise 3.4 related to above properties are explained and described in detail in RS Aggarwal Class 6. Students can directly refer to RS Aggarwal for the solutions related to properties questions from Exercise 3.4.