Number Systems ⇒⇒ Exercise 1.1

Question 1

Is zero a rational number ? Can you write it in the form ${ p \over q} $ , where p and q are integers and q ${ \ne } $ 0 ?

Solution :

According to the definition of Rational Number, Any number number that can represent in the form of ${ p \over q} $,
where p and q are both integers and q is not equal to zero.

Now, we can write 0 in the form of ${ p \over q} $

as : ${ 0 \over 1} $ , ${0 \over -3} $

as : ${ 0 \over a} $,

In all of these examples, the numerator (p) is 0, and the denominator (q) is a non-zero integer. This fits the definition perfectly, making zero a rational number.

Question 2

Find six rational numbers between 3 and 4.

Solution :

To find six rational numbers between 3 and 4, we can express both numbers as fractions with a common denominator.

Given numbers = 3 and 4

Hint: To find 'n' rational numbers between any two numbers, multiply & divide both by (n+1)

As we have to find six rational numbers , so multiply the numerator and denominator by (6+1) i.e. 7

as follows:

$$ 3 = { 3 \over 1} × { 7 \over 7} = { 21 \over 7}$$ and

$$ 4 = { 4 \over 1} × { 7 \over 7} = { 28 \over 7}$$

We know that,

21 < 22 < 23 < 24 < 25 < 26 < 27 < 28

So required rational numbers =

$$ { 21 \over 7},{ 22 \over 7},{ 23 \over 7},{ 24 \over 7},{ 25 \over 7},{ 26 \over 7},{ 27 \over 7},{ 28 \over 7} $$

Hence, 6 rational numbers between 3 and 4 are:

$$ 3,{ 22 \over 7},{ 23 \over 7},{ 24 \over 7},{ 25 \over 7},{ 26 \over 7},{ 27 \over 7},4$$

Question 3

Find five rational numbers between ${ 3 \over 5} $ and ${ 4 \over 5} $

Solution :

Given numbers = ${ 3 \over 5} $ and ${ 4 \over 5} $

Hint: To find 'n' rational numbers between any two numbers, multiply & divide both by (n+1)

As we have to find five rational numbers , so multiply the numerator and denominator by (5+1) i.e. 6

as follows:

$$ { 3 \over 5} = { 3 \over 5} × { 6 \over 6} = { 18 \over 30}$$ and

$$ { 4 \over 5} = { 4 \over 5} × { 6 \over 6} = { 24 \over 30}$$

So required rational numbers =

$$ { 18 \over 30},{ 19 \over 30},{ 20 \over 30},{ 21 \over 30},{ 22 \over 30},{ 23 \over 30},{ 24 \over 30}$$

Hence, 5 rational numbers between ${ 3 \over 5} $ and ${ 4 \over 5} $ are:

$$ { 19 \over 30},{ 20 \over 30},{ 21 \over 30},{ 22 \over 30},{ 23 \over 30}$$

Question 4 (i)

State whether the following statements are true or false. Give reasons for your answers
(i) Every natural number is a whole number.

Solution :

(i) True, Every natural number is a whole number.

Because , the set of Natural numbers is represented as N contains numbers 1 to infinity ,
N = {1, 2, 3…….}

and the set of Whole Numbers is represented as W contains numbers 0 to infinity ,
W = {0,1, 2, 3…….}

As you can see, every element in the set of natural numbers is also an element in the set of whole numbers. Therefore, all natural numbers are whole numbers.

Question 4 (ii)

State whether the following statements are true or false. Give reasons for your answers
(ii) Every integer is a whole number.

Solution :

(ii) False, .

Because , the set of Whole Numbers is represented as W contains numbers 0 to infinity ,
W = {0,1, 2, 3…….}

and the set of Integer Numbers is represented as I contains numbers that are both negative and positive and include zero,
I = {-2, -1, 0,1, 2, 3…….}

Here we can see clearly that, negative numbers are missing from whole numbers.

Therefore, not every integer is a whole number.

Thus, it is false that every integer is a whole number.

Question 4 (iii)

State whether the following statements are true or false. Give reasons for your answers
(iii) Every rational number is a whole number.

Solution :

(iii) False, .

Because , the set of Whole Numbers is represented as W contains numbers 0 to infinity ,
W = {0,1, 2, 3…….}

Rational numbers are numbers that can be expressed in ${ p \over q} $ where q${ \ne } $0.
So the rational number contains fraction too.

Therefore, rational numbers may be fractional but whole numbers are not fractional.

Therefore, not every rational number is a whole number.