RD SHARMA SOLUTION CHAPTER – 1 Number System | CLASS 9TH MATHEMATICS-EDUGROWN

Chapter 1 Number Systems Exercise Ex. 1.1

Question 1Is zero a rational number? Can you write it in the form 

, where p and q are integers and q  0?Solution 1Yes zero is a rational number as it can be represented in the   form,  where p and q are integers and q  0 as  etc.

Concept Insight: Key idea to answer this question is “every integer is a rational number and zero is a non negative integer”.  Also 0 can be expressed in  form in various ways as 0 divided by any number is 0. simplest is   .

Question 2Find five rational numbers between 1 and 2.Solution 2

Question 3Find six rational numbers between 3 and 4.Solution 3There are infinite rational numbers in between 3 and 4.  
3 and 4 can be represented as respectively. Now rational numbers between 3 and 4 are    


Concept Insight:  Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent  form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Question 4Find five rational numbers between  .Solution 4There are infinite rational numbers between 

                 Now rational numbers between are 

Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here.  The trick is to convert the number to equivalent   form by multiplying and dividing by the number at least 1 more than the rational numbers required.

Alternatively for any two rational numbers a and b,    is also a rational number which lies between a and b.

Question 5Are the following statements true or false? Give reasons for you answer.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

(iv) Every natural number is a whole number.

(v) Every integer is whole number.

(vi) Every rational number is whole number.Solution 5(i) False

(ii) True

(iii) False

(iv)True

(v) False

(vi) False

Chapter 1 Number Systems Exercise Ex. 1.2

Question 1

Solution 1

Question 2Express the follwoing rational numbers as decimals:

Solution 2

(i)



(ii)



(iii)

(iv)



(v)



(vi)

Question 3

Solution 3

Chapter 1 Number Systems Exercise Ex. 1.3

Question 1

Solution 1

Question 2

Solution 2

(i)



(ii)



(iii)



(iv)



(v)



(vi)



(vii)

Chapter 1 Number Systems Exercise Ex. 1.4

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.
Question 3(xi)

Solution 3(xi)

Rational number as it can be represented in   form.
Question 3(xii)Examine whether 0.3796 is rational or irrational.Solution 3(xii)0.3796

As decimal expansion of this number is terminating, so it is a rational number.Question 3(xiii)Examine whether 7.478478… is rational or irrational.Solution 3(xiii)
As decimal expansion of this number is non terminating recurring so it is a rational number.
Question 3(xiv)Examine whether 1.101001000100001… is rational or irrational.Solution 3(xiv)

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 4(iv)

Solution 4(iv)

Question 4(v)

Solution 4(v)

Question 4(vi)

Solution 4(vi)

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10Find three different irrational numbers between the rational numbers Solution 10

 3 irrational numbers are –
0.73073007300073000073 … … …0.75075007500075000075 … … …
0.79079007900079000079 … … …

Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them.

Alternatively following result can be used to answerIrrational number between two numbers x and y 

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 1 Number Systems Exercise Ex. 1.5

Question 1Complete the following sentences:

(i) Every point on the number line corresponds to a ___ number which may be either ____ or_____.

(ii) The decimal form of an irrational number is neither ______ nor ______.

(iii) The decimal representation of a rational number is either ____ or _____.

(iv) Every real number is either ______ number or ______ number.Solution 1(i) Real, rational, irrartional.

(ii) terminating, repeating.

(iii) terminating, non-terminating and reccuring.

(iv) rational, an irrational.Question 2

Find whether the following sentences are true or false:

(i) Every real number is either rational or irrational.

(ii)  is an irrational number.

(iii) Irrational numbers cannot be represented by points on the number line.Solution 2

(i) True

(ii) True

(iii) FalseQuestion 3

Solution 3

Question 4

Solution 4

Chapter 1 Number Systems Exercise Ex. 1.6

Question 1

Solution 1

Question 2

Solution 2