RD SHARMA Solutions for Class 9 Maths Chapter 1 - Number Systems

Chapter 1 - Number Systems Exercise Ex. 1.1

Question 1
Is zero a rational number? Can you write it in the form Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems, where p and q are integers and q Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 0?
Solution 1
Yes zero is a rational number as it can be represented in the Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems  form,  where p and q are integers and q Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 0 as Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 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 Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems form in various ways as 0 divided by any number is 0. simplest is Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems  .

Question 2
Find five rational numbers between 1 and 2.
Solution 2
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3
Find six rational numbers between 3 and 4.
Solution 3
There are infinite rational numbers in between 3 and 4.  
3 and 4 can be represented as Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systemsrespectively.
 
Now rational numbers between 3 and 4 are    
    
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

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 Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted.

Question 4
Find five rational numbers between Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems .
Solution 4
There are infinite rational numbers between Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
         Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
        
Now rational numbers between are Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 
          Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
 
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 equivalentRd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems   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,  Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems  is also a rational number which lies between a and b.


Question 5
Are 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
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 1
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 2
Express the follwoing rational numbers as decimals:

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 2

(i)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(ii)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(iii)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(iv)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(v)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(vi)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 3
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise Ex. 1.3

Question 1
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 1
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 2

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 2

(i)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(ii)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(iii)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(iv)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(v)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(vi)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(vii)

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise Ex. 1.4

Question 1
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 1
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 2
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 2
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(i)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(i)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(ii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(ii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(iii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(iii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(iv)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(iv)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(v)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(v)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(vi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(vi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(vii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(vii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(viii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(viii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(ix)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(ix)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 3(x)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(x)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

As decimal expansion of this number is non-terminating non recurring. So it is an irrational number.

Question 3(xi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 3(xi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Rational number as it can be represented in  Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 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)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

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)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(i)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(i)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(ii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(ii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(iii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(iii)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(iv)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(iv)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(v)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(v)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 4(vi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 4(vi)
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 5
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 5
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 6

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 6

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 7

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 7

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 8

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems 

Solution 8

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 9

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 9

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 10
Find three different irrational numbers between the rational numbers Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Solution 10
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
 
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 answer

Irrational number between two numbers x and y
 
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Question 11

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 11

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 12

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 12

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 13

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 13

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 14

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 14

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise Ex. 1.5

Question 1
Complete 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) Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems is an irrational number.

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

Solution 2

(i) True

(ii) True

(iii) False

Question 3

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 3

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 4

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 4

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise Ex. 1.6

Question 1

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 1

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems
Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 2

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 2

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise 1.40

Question 1

Which one of the following is a correct statement?

(a) Decimal Expansion of a Rational number is terminating.

(b) Decimal Expansion of a Rational number is Non-terminating.

(c) Decimal Expansion of a irrational number is terminating.

(d) Decimal Expansion of a irrational number is Non-terminating and Non-Repeating.

Solution 1

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 2

Which one of the following statement is true?

(a) The sum of two irrational numbers is always an irrational number.

(b) The sum of two irrational numbers is always a rational number.

(c) The sum of two irrational numbers may be a rational number or an irrational number

(d) The sum of two irrational numbers is always an Integer.

Solution 2

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 3

Which of the following is a correct statement?

(a) Sum of two irrational numbers is always irrational.

(b) Sum of a rational and irrational number is always irrational.

(c) Square of an irrational number is always a Rational number.

(d) Sum of two Rational numbers can never be an integer.

Solution 3

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 4

Which of the following statements is true?

(a) Product of two irrational numbers is always irrational

(b) Product of a rational and an irrational number is always irrational

(c) Sum of two irrational numbers can never be irrational

(d) Sum of an integer and a rational number can never be an integer

Solution 4

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 5

Which of the following is irrational?

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 5

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 6

Which of the following is irrational?

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 6

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 7

Which of the following is rational?

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 7

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 8

The number 0.318564318564318564......... is:

(a) a natural number

(b) an integer

(c) a rational number

(d) an irrational number

Solution 8

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 9

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(a) always a natural number

(b) always an rational number

(c) always an irrational number

(d) sometimes a natural number and sometimes an irrational number.

Solution 9

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise 1.41

Question 10

Which of the following numbers can be represented as non-terminating repeating decimals?

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 10

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 11

Every point on a number line represents:

(a) a unique real number

(b) a natural number

(c) a rational number

(d) an irrational number

Solution 11

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 12

Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(d) 0.5015001500015

Solution 12

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 13

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 13

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 14

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 14

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 15

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 15

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 16

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 16

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 17

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 17

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 18

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 18

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 19

An irrational number between 2 and 2.5 is

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 19

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Question 20

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

(a) 3

(b) 2

(c) 4

(d) 5

Solution 20

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Chapter 1 - Number Systems Exercise 1.42

Question 21

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems

Solution 21

Rd-sharma Solutions Cbse Class 9 Mathematics Chapter - Number Systems