Accessing OP Malhotra Class 9 Solutions Chapter 1 Rational and Irrational Numbers Ex 1(B) can be a valuable tool for students seeking extra practice.
S Chand Class 9 ICSE Maths Solutions Chapter 1 Rational and Irrational Numbers Ex 1(B)
Question 1.
Look at the following real numbers :
– 5, 0, \(\sqrt{3}, \frac{3}{5},-\sqrt{9}, \sqrt{8}, 6.37, \pi, 4, \frac{-2}{7}, 0.03\)
Tell,
(i) Which are rational?
(ii) Which are irrational?
(iii) Which are positive integers?
(iv) Which are negative integers?
(v) Which number is neither positive nor- negative?
Solution:
(i) Rational numbers are -5, 0, \(\frac { 3 }{ 5 }\), – \(\sqrt{9}\), 6.37, 4, \(\frac { -2 }{ 7 }\), 0.03
(ii) Irrational numbers are \(\sqrt{3}\), \(\sqrt{8}\), π
(iii) Positive integers is 4
(iv) Negative integers are – 5, – \(\sqrt{9}\)
(v) The number which is neither positive nor- negative is 0
Question 2.
Write true or false to describe each sentence:
(i) All rational numbers are real numbers.
(ii) All real numbers are rational numbers.
(iii) Some real numbers are rational numbers.
(iv) All integers are rational numbers.
(v) No rational number is also an irrational number.
(vi) There exists a whole number that is not a natural number.
Solution:
(i) True : as real numbers includes rational and irrational numbers.
(ii) False : ∵ Real number includes both rational numbers and irrational numbers.
(iii) True : as some real numbers which are only rational number.
(iv) True : as set of integers is a subset of rational numbers.
(v) True : as by definition, an irrational number is a number which is not a rational number.
(vi) True : as 0 is a whole number but not a natural number.
Question 3.
Tell whether each decimal numeral represents a rational or an irrational number:
(i) 0.578
(ii) 0.573 333…..
(iii) 0.688 434 4454….
(iv) 0.727 374 75……
(v) 0.638 754 71
(vi) 0.471 7171….
(vii) 283
(viii) 289.387 000…..
(ix) 5.\(\overline{93}\)
(x) 2.309\(\overline{8}\)7
(xi) 0.585 885 888…
Solution:
(i) 0.578
It is a terminating, therefore it a rational
(ii) 0.573 333…..
= 0.57\(\overline{3}\)
∵ It is a repeating decimal
∴ It is a rational number
(iii) 0.688 434 4454….
∵ It is neither terminating not repeating decimal
∴ It is an irrational number
(iv) 0.727 374 75…
∵ It is neither terminating norrepeating decimals
∴ It is an irrational number
(v) 0.638 75471….
∵ It is neither terminating nor-repeating decimals
∴ It is an irrational number
(vi) 0.471 7171
= 0.4\(\overline{71}\)
∵ It is repeating decimal
∴ It is a rational number
(vii) 283
It is rational number
(viii) 289.387000
∵ It is terminating decimal
∴ It is a rational number
(ix) 5.\(\overline{93}\)
∵It is repeating decimal
∴It is a rational number
(x) 2.309\(\overline{8}\)7
∵ It is terminating decimal
∴ It is a rational number
(xi) 0.585885888 = 0.5858858
∵ It is non-repeating decimal norterminating
∴ It is an irrational number
Question 4.
List three distinct irrational numbers.
Solution:
We know that an irrational number is non-repeating decimal
∴ There are π, \(\sqrt{5}\), \(\sqrt{6}\), \(\sqrt{7}\) etc.
Question 5.
Show that (i) \(\sqrt{3}\), (ii) \(\sqrt{5}\) are not rational numbers.
Solution:
(i) Let \(\sqrt{3}\) is a rational number
Let \(\sqrt{3}\) = \(\frac { p }{ q }\) where p and q are integers and have no common factors and also q ≠ 0 Squaring both sides
3 = \(\frac{p^2}{q^2}\) ⇒ p² = 3q² … (i)
∵ 3q² is a multiple of 3
∴ p² is also a multiple of 3
⇒ p is multiple of 3 … (ii)
Let p = 3m (Squaring both sides)
⇒ p² = 9m² … (iii)
∴ From (i) and (iii)
3q² = 9 m²
q² = 3m²
∵ m² is multiple of 3
∴ q² is also multiple of 3
⇒ q is also multiple of 3 … (iv)
From (ii) and (iv)
Both p and q are multiple of 3 which contradicts our supposition that p and q have no common factor
Hence \(\sqrt{3}\) is not a rational number
Hence proved.
(ii) Let \(\sqrt{5}\) is a rational number
Let \(\sqrt{5}\) = \(\frac { p }{ q }\) where p and q are integers and have no common factors and q ≠ 0
Squaring both sides,
5 = \(\frac{p^2}{q^2}\) ⇒ p² = 5q² … (i)
∵ 5q² is multiple of 5
∴ p² is multiple of 5
⇒ p is multiple of 5 … (ii)
Let p = 5m
⇒ p² = 25m² … (iii)
(Squaring both sides)
∵ 25m² is a multiple of 5
∴ p² is a multiple of 5
From (i) and (iii)
25m² = 5 q²
⇒ 5m² = q²
∴ q² is multiple of 5 … (iv)
From (ii) and (iv)
p and q both are multiples of 5 which contradicts our supposition that p and q have no common factors
Hence \(\sqrt{5}\) is not a rational number
Hence proved.
Question 6.
Is \(\sqrt{100}\) + \(\sqrt{36}\) the same as \(\sqrt{100+36}\)?
Give reasons.
Solution:
\(\sqrt{100}\) + \(\sqrt{36}\) = \(\sqrt{(10)^2} + \sqrt{(6)^2}\) = 10 + 6 = 16
and \(\sqrt{100+36}=\sqrt{136}\) = 11.66 (approx)
It is clear that \(\sqrt{100}\) + \(\sqrt{36}\) and \(\sqrt{100+36}\) are not same