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

Page / Exercise

## Chapter 1 - Number Systems Exercise Ex. 1.1

Question 1
Is zero a rational number? Can you write it in the form , where p and q are integers and q  0?
Solution 1
Yes 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 2
Find five rational numbers between 1 and 2.
Solution 2
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 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 4
Find five rational numbers between .
Solution 4
There 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 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
Solution 1

Question 2
Express the follwoing rational numbers as decimals:

Solution 2

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 3
Solution 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 10
Find 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 answer

Irrational 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 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) 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

Solution 3

Question 4

Solution 4

Question 1

Solution 1

Question 2

Solution 2

## 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

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

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

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

Question 5

Which of the following is irrational?

Solution 5

Question 6

Which of the following is irrational?

Solution 6

Question 7

Which of the following is rational?

Solution 7

Question 8

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

(a) a natural number

(b) an integer

(c) a rational number

(d) an irrational number

Solution 8

Question 9

(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

## Chapter 1 - Number Systems Exercise 1.41

Question 10

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

Solution 10

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

Question 12

Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c)

(d) 0.5015001500015

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

An irrational number between 2 and 2.5 is

Solution 19

Question 20

(a) 3

(b) 2

(c) 4

(d) 5

Solution 20

## Chapter 1 - Number Systems Exercise 1.42

Question 21

Solution 21

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