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.