class 9 Number Systems Ex 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 ≠ 0?

Solution 1:

Consider the definition of a rational number.

A rational number is the one that can be written in the form of  ^p/_q , where p and q are integers and q 

≠0.

• Zero can be written as \(\displaystyle{{0 \over 1},{0 \over 2},{0 \over 3},{0 \over 4},{0 \over 5}...}\)

• Zero can be written as well \(\displaystyle{{0 \over -1},{0 \over -2},{0 \over -3},{0 \over -4},{0 \over -5}...}\)

So, we arrive at the conclusion that 0 can be written in the form of  ^p/_q , where p and q are integers ( q can be positive or negative integers).

Therefore, zero is a rational number.

Question  2.

Find six rational numbers between 3 and 4. 

Solution 2:

We know that there are infinite rational numbers between any two numbers.

As we have to find 6 rational numbers between 3 and 4 So multiply and divide by 7 (or any number greater than 6)

We get, \(3 = {\displaystyle {3 × 7 \over 7}} = {\displaystyle21 \over 7} \)

                \(4 = \displaystyle {4 × 7 \over 7} = \displaystyle {28 \over 7} \)

Thus the 6 rational numbers are \(\displaystyle{ {22 \over 7},{23 \over 7},{24 \over 7},{25 \over 7},{26 \over 7},{27 \over 7} }\).

Question 3.

Find five rational numbers between ³/₅ and ⁴/₅.

Solution 3:

We know that there are infinite rational numbers between any two numbers.

As we have to find 5 rational numbers between \( \displaystyle{ 3 \over 5}\) and \( \displaystyle{4 \over 5}\)

So, multiply and divide by 6 (Or any number greater than 5) 

\(\displaystyle{ {3 \over 5} = {3 \over 5} × {6 \over 6} = {18 \over 30}}\)

\(\displaystyle{ {4 \over 5} = {4 \over 5} × {6 \over 6} = {24\over 30}}\)

Thus the 5 rational numbers are \(\displaystyle{{19 \over 30}, {20 \over 30} ,{21 \over 30},{22 \over 30},{23 \over 30}}\)

Question 4.

State whether the following statements are true or false. Give reasons for your answers.

(i)Every natural number is a whole number.

 

True,Consider the whole numbers and natural numbers separately.

Whole number series is 0,1, 2, 3, 4,5..... .

Natural number series is 1, 2, 3, 4,5..... .

So every natural number lie in the whole number series.

(ii)Every integer is a whole number.

False,

Series of integers, we have – 4,–3,–2,–1, 0, 1, 2,3, 4..... .

Whole numbers are 0, 1, 2, 3, 4, 5..... .

We can conclude that whole number series lie in the series of integers. But every integer does not appear in the whole number series.

Therefore, we conclude that every integer is not a whole number.

But, clearly every whole number is an integer.

(iii)Every rational number is a whole number. 

False,

Consider the rational numbers and whole numbers separately.

We know that rational numbers are the numbers that can be written in the form \( \displaystyle{p \over q}\) ,where q ≠ 0.

We know that whole numbers are 0, 1, 2, 3, 4, 5..... .

We know that every whole number can be written in the form of  \( \displaystyle{p \over q}\)as follows

\(\displaystyle{ ({0 \over 1},{1 \over 1},{2 \over 1},{3 \over 1},{4 \over 1})...}\).

We conclude that every whole number is a rational number.

But, every rational number  \(\displaystyle{ ({1 \over 2},{1 \over 3},{1 \over 4},{1 \over 5},{1 \over 6})...}\)is not a whole number. 

Therefore, we conclude that every rational number is not a whole number.

But, clearly every whole number is a rational number.