# GSEB Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.1

Gujarat Board GSEB Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.1 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 9 Maths Chapter 1 Number Systems Ex 1.1

Question 1.
Is zero a rational number? Can you write it in the form of $$\frac { p }{ q }$$, where p and q are integers and q≠0?
Solution:
Yes, zero is a rational number, because 0 can be written in the form of $$\frac { p }{ q }$$, where p and q are integers and q≠0.
We can write
$$\frac { 0 }{ 1 }$$ = $$\frac { 0 }{ 2 }$$ = $$\frac { 0 }{ 3 }$$, 3 etc

Question 2.
Find six rational numbers between 3 and 4.
Solution:
Infinitely many rational numbers can exist between 3 and 4.
Rational number between 3 and 4
= $$\frac { 1 }{ 2 }$$(a + b) (Where a = 3 and b = 4)
= $$\frac { 1 }{ 2 }$$(3 + 4) = $$\frac { 7 }{ 2 }$$
Rational number between 3 and $$\frac { 7 }{ 2 }$$
= $$\frac { 1 }{ 2 }$$(3 + $$\frac { 7 }{ 2 }$$) (Where a = 3 and b = $$\frac { 7 }{ 2 }$$)
= $$\frac { 1 }{ 2 }$$($$\frac { 6+7 }{ 2 }$$) = $$\frac { 13 }{ 4 }$$
Rational number between 3 and $$\frac { 13 }{ 4 }$$
= $$\frac { 1 }{ 2 }$$(3 + $$\frac { 13 }{ 4 }$$) (Where a = 3 and b = $$\frac { 13 }{ 4 }$$)
= $$\frac { 1 }{ 2 }$$($$\frac { 12+13 }{ 4 }$$) = $$\frac { 25 }{ 8 }$$
Rational number between 3 and $$\frac { 25 }{ 8 }$$
= $$\frac { 1 }{ 2 }$$(3 + $$\frac { 25 }{ 8 }$$) (Where a = 3 and b = $$\frac { 25 }{ 8 }$$)
= $$\frac { 1 }{ 2 }$$($$\frac { 24+25 }{ 8 }$$) = $$\frac { 1 }{ 2 }$$ × $$\frac { 49 }{ 8 }$$ = $$\frac { 49 }{ 16 }$$
Rational number between 3 and $$\frac { 49 }{ 16 }$$
= $$\frac { 1 }{ 2 }$$(3 + $$\frac { 49 }{ 16 }$$) (Where a = 3 and b = $$\frac { 49 }{ 16 }$$)
= $$\frac { 1 }{ 2 }$$($$\frac { 48+49 }{ 16 }$$) = $$\frac { 97 }{ 32 }$$
Rational number between 3 and $$\frac { 97 }{ 32 }$$
= $$\frac { 1 }{ 2 }$$(3 + $$\frac { 97+32 }{ 8 }$$) (Where a = 3 and b = $$\frac { 97 }{ 32 }$$)
= $$\frac { 1 }{ 2 }$$($$\frac { 193 }{ 32 }$$) = $$\frac { 193 }{ 64 }$$
∴ Six rational numbers between 3 and 4 are
$$\frac { 193 }{ 64 }$$, $$\frac { 97 }{ 32 }$$ , $$\frac { 49 }{ 16 }$$, $$\frac { 25 }{ 8 }$$, $$\frac { 13 }{ 4 }$$, $$\frac { 7 }{ 2 }$$
Alternative method:
n = 6 (to be find)
∴ n + 1 = 6 + 1 = 7
Hence, 3 = $$\frac { 3 }{ 1 }$$ = $$\frac { 3×7}{ 1×7 }$$ = $$\frac { 21 }{ 7 }$$
and 4 = $$\frac { 4 }{ 1 }$$ = $$\frac { 4×7 }{ 1×7 }$$ = $$\frac { 28 }{ 7 }$$
Six rational numbers between 3 and 4 are
$$\frac { 22 }{ 7 }$$, $$\frac { 23 }{ 7 }$$, $$\frac { 24 }{ 7 }$$, $$\frac { 25 }{ 7 }$$, $$\frac { 26 }{ 7 }$$, $$\frac { 27 }{ 7 }$$

Question 3.
Find five rational numbers between $$\frac { 3 }{ 5 }$$ and $$\frac { 4 }{ 5 }$$
Solution:
We have to find 5 rational numbers between $$\frac { 3 }{ 5 }$$ and $$\frac { 4 }{ 5 }$$
Here, n = 5 ∴ n + 1 = 5 + 1 = 6
∴ Multiplying by 6 to the numerator and denominator.
$$\frac { 3 }{ 5 }$$ = $$\frac { 3 }{ 5 }$$ x $$\frac { 6 }{ 6 }$$ = $$\frac { 18 }{ 30 }$$
and $$\frac { 4 }{ 5 }$$ = $$\frac { 4 }{ 5 }$$ x $$\frac { 6 }{ 6 }$$ = $$\frac { 24 }{ 30 }$$
Hence rational numbers between $$\frac { 18 }{ 30 }$$ and $$\frac { 24 }{ 30 }$$
are
$$\frac { 19 }{ 30 }$$ < $$\frac { 20 }{ 30 }$$ < $$\frac { 21 }{ 30 }$$, $$\frac { 22 }{ 30 }$$, $$\frac { 23 }{ 30 }$$

Question 4.
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Solution:
(i) True, because the collection (set) of whole numbers contains all the natural numbers

(ii) False, – 1 is an integer but it is not a whole number.

(iii) False, $$\frac { 2 }{ 3 }$$ is a rational number but it is not a whole number.