SChand CLASS 9 Chapter 1 Rational number and Irrational number Exercise 1(B)

 Exercise 1(B)

Question 1

$\frac{-5}{1},\frac{0}{1}$ , $\sqrt{3},\frac{3}{5}$  , $-\sqrt{9},\sqrt{8} $ , 6.37 , π  , 4 $,\frac{-2}{7} , $ 0.03

Sol :

(i) Rational numbers are:

=-5 , 0 , $\frac{3}{5}, -\sqrt{9}$ , 6.37 , 4 , $\frac{-2}{7}$ , 0.03

(ii) Irrational numbers are :-

√3 , √8 , 𝜋

(iii) Positive Integers is :-

=4

(iv) Negative Integers are :-

=-5 , -√9

(v) The number which is neither positive nor negative is :-

=0

Question 2

(1) All rational number are real numbers.

(2) All real numbers are rational numbers.

(3) Some real numbers are rational numbers.

(4) All integers are rational numbers.

(5) No rational number is also an irrational number.

(6) There exists a whole number that is not a natural number.

Sol :

(i) It is true :- as real numbers includes rational and irrational numbers.

(ii) It is False :- Real numbers includes both rational numbers and irrational numbers.

(iii) It is True :- as some real numbers which are only rational number.

(iv) It is true:- as set of integers is a subset of rational numbers.

(v) It is true:- as by definition , an irrational number is a number which is not a rational number.

(vi) It is true:-as 0 is a whole number not a natural number.

Question 3

Tell whether each decimal numeral represents a rational or an irrational number:

(i) 0.578

(ii) 0.573333…..

(iii) 0.6884344454….

(iv) 0.72737475……

(v) 0.63875471

(vi) 0.4717171….

(vii) 283

(viii) 289.387000…..

(ix) $5.\overline{93}$

(x) 2.30987

(xi) 0.585885888…

Sol :

(i) 0.578

It is terminating . Hence , it is a rational

(ii) 0.573333....

$0.57\overline{3}$

∴It is a repeating decimal

∴It is a rational number

(iii) 0.6884344454...

∵It is neither terminating nor repeating decimal .

∵It is an irrational number.

(iv) 0.72737475...

⇒It is neither terminating nor repeating decimal

⇒It is an irrational number

(v) 0.63875471...

⇒It is neither terminating nor repeating decimals

⇒It is a rational number

(vi) 0.4717171...

⇒It is a repeating decimal

⇒It is a rational number

(vii) 283

⇒It is a rational number

(viii) 289.387000.....

⇒It is a terminating decimal

⇒It is a rational number

(ix) $5.\overline{93}$

⇒It is  a repeating decimal

⇒It is a rational number

(x) 2.30987...

⇒It is a terminating decimal

⇒It is a rational number

(xi) 0.585885888...

⇒It is not repeating decimal non-terminating

⇒It is an irrational number

Question 4

List three distinct irrational numbers.

Sol :

(i) We know that irrational number is non repeating decimal

∴There are 𝜋 ,√5 , √6 , √7 etc

Question 5

Show that (i) √3 , (ii) √5  are not rational numbers.

Sol :

(i) Let assume √3 is a rational number

So, √3$=\frac{p}{q}$ where p and q are integers and they have no common factor and also q≠0

Squaring both sides

⇒$3=\frac{p^2}{q^2}$

⇒p²=3q²...(i)

⇒3q² is multiple of 3

⇒p² is also a multiple of 3

⇒p is a multiple of 3..(ii)

So, p=3m

⇒p²=9m²

∴From (i) and (ii)

3q² =9m² 

or q² =3m² ...(iii)

∵m² is multiple of 3

∴q² is also a multiple of 3

which means q is also multiple of 3...(iv)

From (ii) and (iv)

Both p and q are multiple of 3 which contradicts our assumption that p and q have no common factor

So, √3 is not a rational number 

(ii) Let assume √5 is a rational number

So, √5$=\frac{p}{q}$ where p and q are integers and they have no common factor and also q≠0

Squaring both sides

⇒$5=\frac{p^2}{q^2}$

⇒p²=5q²...(i)

⇒5q² is multiple of 5

⇒p² is also a multiple of 5

⇒p is a multiple of 5..(ii)

So, p=5m

⇒p²=25m²

∴From (i) and (ii)

5q² =25m² 

or q² =5m² ...(iii)

∵m² is multiple of 5

∴q² is also a multiple of 5

which means q is also multiple of 5...(iv)

From (ii) and (iv)

Both p and q are multiple of 5 which contradicts our assumption that p and q have no common factor

So, √5 is not a rational number 

Question 6

Is √100 + √36 same as $\sqrt{100+36}$

Sol :

=√100 + √36

= √(10)²+√(6)²

=10+6=16

and √100+36

=√136=11.66

It is clear that √100+36 and √100+√36 are not same