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.