RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Mathematics (Maths) Class 9

Class 9 : RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The document RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.
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RD Sharma Solutions: Exercise 1.3 - Number System


Q.1. Express each of the following decimals in the form RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev:

(i) 0.39

(ii) 0.750

(iii) 2.15

(iv) 7.010

(v) 9.90

(vi) 1.0001

Proof: (i) Given decimal is 0.39

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(ii) Given decimal is 0.750

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iii) Given decimal is 2.15

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iv) Given decimal is 7.010

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(v) Given decimal is 9.90

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, 9.90 = 99/10

(vi) Given decimal is 1.0001

Now we have to convert given decimal number into the RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev form

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.2. Express each of the following decimals in the form RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev:

(i) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(ii) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iii) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iv) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(v) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vi) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vii) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Proof: (i) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(ii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 


RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iv) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(v) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vi) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore, 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Since, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore, 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


RD Sharma Solutions: Exercise 1.4 - Number System


Q.1. Define an irrational number.

Proof: An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q.

If the decimal representation of an irrational number is non-terminating and non-repeating, then it is called irrational number. For example RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 


Q.2. Explain, how irrational numbers differ from rational numbers?

Proof: 

Every rational number must have either terminating or non-terminating but irrational number must have non- terminating and non-repeating decimal representation.

A rational number is a number that can be written as simple fraction (ratio) and denominator is not equal to zero while an irrational is a number that cannot be written as a ratio.


Q.3. Examine, whether the following numbers are rational or irrational:

(i) √7

(ii) √4

(iii) 2 + √3

(iv)3 + 2

(v) 3 + 5

(vi) ( 2 – 2)2

(vii) (2−2) (2+2) 

(viii) (2 + 3–√)2

(ix) √5−2

(x) √23

(xi) √225

(xii) 0.3796

(xiii) 7.478478

(xiv) 1.101001000100001

Proof: (i) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Therefore,

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

It is non-terminating and non-repeating

HenceRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

(ii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Therefore,

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

It is terminating.

HenceRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis a rational number.

(iii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev be the rational 

Squaring on both sides

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Since, x is rational 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational

But,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis irrational

So, we arrive at a contradiction.

Hence RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev  is an irrational number

(iv) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev be the rational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Since, x is a rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

But RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev  is an irrational number

So, we arrive at contradiction

Hence RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

(v) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now, x is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is rational number

But
RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

So, we arrive at a contradiction

Hence RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev  is an irrational number

(vi) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev be a rational number.

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Since, x is rational number,

⇒ x – 6 is a rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis a rational number

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis a rational number

But we know that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number, which is a contradiction 

So RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

(vii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Using the formula RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis a rational number

⇒is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev a rational number

But we know that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis an irrational number 

So, we arrive at a contradiction

So RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number.

(ix) Let x = √5−2 be the rational number

Squaring on both sides, we get

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now, x is rational

x2 is rational.

So, x2−29 is rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

But, √5 is irrational. So we arrive at contradiction

Hence x = √5−2 is an irrational number

(x) Let

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

It is non-terminating or non-repeating

Hence RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

(xi) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is a rational number

(xii) Given x= 0.3796.

It is terminating

Hence it is a rational number

(xiii) Given number RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

It is repeating 

Hence it is a rational number

(xiv) Given number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

It is non-terminating or non-repeating

Hence it is an irrational number


Q.4. Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

(i) (√4)

(ii) 3√18

(iii) √1.44

(iv) √927

(v) −√64

(vi) √100

Proof: (i) Given number is x =RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

x = 2, which is a rational number

(ii) Given number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number

(iii) Given number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to check whether it is rational or irrational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is a rational

(iv) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now we have to check whether it is rational or irrational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number

(v) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to check whether it is rational or irrational

Since, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

So it is a rational number

(vi) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to check whether it is rational or irrational

Since, RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

So it is rational number


Q.5. In the following equations, find which variables x, y, z etc. represent rational or irrational numbers:

(i) x2 = 5

(ii) y2 = 9

(iii) z2 = 0.04

(iv) u2 = 17/4

(v) v2 = 3

(vi) w2 = 27

(vii) t2 = 0.4

Proof: (i) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of x

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it x is an irrational number

(ii) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of y

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So y is a rational number

(iii) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of z

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is rational number

(iv) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of u

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number

(v) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of v

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number

(vi) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of w

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number

(vii) Given that RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Now we have to find the value of t

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

So it is an irrational number


Q.6. Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

Proof: Let a = 0.232332333233332

b = 0.212112111211112

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place of a and b have the same digit but digit in the second place of their decimal representation are distinct. And the number a has 3 and b has 1. So a > b.

Hence two rational numbers are RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev  lying between 0.232332333233332... and 0.212112111211112...


Q.7. Give two rational numbers lying between 0.515115111511115...0.5353353335...

Proof: Let a = 0.515115111511115... and b = 0.535335333533335...

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place a and b have the same digit  but digit in the second place of their decimal representation are distinct. And the number a has 1 and b has 3. So a < b.

Hence two rational numbers are RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev lying between 0.515115111511115.. and 0.535335333533335...


Q.8. Find one irrational number between 0.2101 and 0.222... = 0.2¯.

Proof: Let 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Here a and b are rational numbers .Since a has terminating and b has repeating decimal. We observe that in second decimal place a has 1 and b has 2. So a < b.

Hence one irrational number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev lying between 0.2101 and 0.2222...


Q.9. Find a rational number and also an irrational number lying between the numbers 0.3030030003 ... and 0.3010010001 ...

Proof: Let 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Here decimal representation of a and b are non-terminating and non-repeating. So a and b are irrational numbers. We observe that in first two decimal place of a and b have the same digit but digit in the third place of their decimal representation is distinct.

Therefore, a > b.

Hence one rational number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev lying between  0.3030030003... and 0.3010010001...

And irrational number is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev lying between RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev and RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 


Q.10. Find three different irrational numbers between the rational numbers RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev and RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev .

Proof: Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev and RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

Here we observe that in the first decimal x has digit 7 and y has 8. So x < y. In the second decimal place x has digit 1. So, if we considering irrational numbers

a = 0.72072007200072..

b = 0.73073007300073..

c = 0.74074007400074....

We find that

x<a<b<c<y

HenceRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev are required irrational numbers.


Q.11. Give an example of each, of two irrational numbers whose:

(i) difference is a rational number.

(ii) difference is an irrational number.

(iii) sum is a rational number.

(iv) sum is an irrational number.

(v) product is an rational number.

(vi) product is an irrational number.

(vii) quotient is a rational number.

(viii) quotient is an irrational number.

Proof: (i) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

And, so RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevand RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their difference is a rational number

(ii) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their difference is an irrational number

BecauseRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is an irrational number

(iii) LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their sum is a rational number

That isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iv) LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their sum is an irrational number 

That isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

(v) Let RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their product is a rational number

That isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vi) LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their product is an irrational number

That isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(vii) LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev are two irrational numbers and their quotient is a rational number

That isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(viii) LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational numbers and their quotient is an irrational number

That is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.12. Find two irrational numbers between 0.5 and 0.55.

Proof: Let 

a = 0.5

b = 0.55

Here a and b are rational number. So we observe that in first decimal place a and b have same digit .So a < b.

Hence two irrational numbers areRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevandRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

lying between 0.5 and 0.55.


Q.13. Find two irrational numbers lying between 0.1 and 0.12.

Proof: Let 

a = 0.1

b = 0.12

Here a and b are rational number. So we observe that in first decimal place a and b have same digit. So a < b.

Hence two irrational numbers areRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevandRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevlying between 0.1 and 0.12.


Q.14. Prove that √3 + √5 is an irrational number.

Proof: Given that √3 + √5 is an irrational number

Now we have to prove √3 + √5  is an irrational number 

Let x = √3 + √5 is a rational

Squaring on both sides

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now x is rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis rational

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis rational

But,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis an irrational

Thus we arrive at contradiction thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis a rational which is wrong.

HenceRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis an irrational


RD Sharma Solutions: Exercise 1.5 - Number System

Q.1. Complete the following sentences:

(i) Every point on the number line corresponds to a .... number which many 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.

Proof: (i) Every point on the number line corresponds to a real number which may be either rational or an irrational number.

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

(iii) The decimal representation of rational number is either terminating, recurring.

(iv) Every real number is either rational number or an irrational number because rational or an irrational number is a family of real number.


Q.2. Find whether the following statement are true or false.

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

(ii) π is an irrational number.

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

Proof: (i) True, because rational or an irrational number is a family of real number. So every real number is either rational or an irrational number.

(ii) True, because the decimal representation of an irrational is always non-terminating or non-repeating. SoRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis an irrational number.

(iii) False, because we can represent irrational numbers by points on the number line.


Q.3. Represent √6, √7, √8 on the number line.

Proof: We are asked to represent √6, √7, √8 on the number line

We will follow certain algorithm to represent these numbers on real line

We will consider point A as reference point to measure the distance

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(1) First of all draw a line AX and YY’ perpendicular to AX

(2) Consider AO = 2 units and OB = 1 unit, so

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(3) Take A as center and AB as radius, draw an arc which cuts line AX at A1

(4) Draw a perpendicular line A1B1 to AX such thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevand

(5) Take A as center and AB1 as radius and draw an arc which cuts the line AX at A2.

Here 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

SoAA= √6 unit

So A2 is the representation for √6

(1) Draw line A2B2 perpendicular to AX

(2) Take A center and AB2 as radius and draw an arc which cuts the horizontal line at A3 such that

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

So point A3 is the representation of √7

(3) Again draw the perpendicular line A3B3 to AX

(4) Take A as center and ABas radius and draw an arc which cuts the horizontal line at A4

Here;

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev 

A4 is basically the representation of √8


Q.4. Represent √3.5, √9.4, √10.5 on the real number line.

Proof:  We are asked to represent the real numbers √3.5, √9.4, √10.5 on the real number line

We will follow a certain algorithm to represent these numbers on real number line

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(a) √3.5

We will take A as reference point to measure the distance

(1) Draw a sufficiently large line and mark a point A on it

(2) Take a point B on the line such that AB = 3.5 cm

(3) Mark a point C on the line such that BC = 1 cm

(4) Find mid point of AB and let it be O

(5) Take O as center and OC as radius and draw a semi circle. Draw a perpendicular BD which cuts the semi circle at D

(6) Take B as the center and BD as radius, draw an arc which cuts the horizontal line at E 

(7) Point E is the representation of √3.5

(b) √9.4

We will take A as reference point to measure the distance. We will follow the same figure in the part (a) 

(1) Draw a sufficiently large line and mark a point A on it

(2) Take a point B on the line such that AB = 9.4 cm

(3) Mark a point C on the line such that BC = 1 cm

(4) Find mid point of AB and let it be O

(5) Take O as center and OC as radius and draw a semi circle. Draw a perpendicular BC which cuts the semi circle at D

(6) Take B as the center and BD as radius, draw an arc which cuts the horizontal line at E 

(7) Point E is the representation of √9.4

(c) √10.5

We will take A as reference point to measure the distance. We will follow the same figure in the part (a) 

(1) Draw a sufficiently large line and mark a point A on 

(2) Take a point B on the line such that AB = 10.5 cm

(3) Mark a point C on the line such that BC = 1 cm

(4) Find mid point of AB and let it be O

(5) Take O as center and OC as radius and draw a semi circle. Draw a perpendicular BC which cuts the semi circle at D

(6) Take B as the center and BD as radius, draw an arc which cuts the horizontal line at E 

(7) Point E is the representation of √10.5


RD Sharma Solutions: Exercise 1.6 - Number System
 

Q.1. Visualize 2.665 on the number line, using successive magnification.

Proof:

 RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

We know that 2.665 lies between 2 and 3. So, we divide the number line into 10 equal parts and mark each point of division. The first mark on the right of 2 will be 2.1 followed by 2.2 and so on. The point left of 3 will be 2.9. Now, the magnified view of this will show that 2.665 lies between 2.6 and 2.7. So, our focus will be now 2.6 and 2.7. We divide this again into 10 equal parts. The first part will be 2.61 followed by 2.62 and so on. 

We now magnify this again and find that 2.665 lies between 2.66 and 2.67. So, we magnify this portion and divide it again into 10 equal parts. The first part will represent 2.661, next will be 2.662 and so on. So, 2.665 will be 5th mark in this subdivision as shown in the figure.


Q.2. Visualize the representation of 5.37¯ on the number line up to 5 decimal places, that is up to 5.37777.

Proof: 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

We know that 5.37¯ will lie between 5 and 6. So, we locate 5.37¯ between 5 and 6. We divide this portion of the number line between 5 and 6 into 10 equal parts and use a magnifying glass to visualize 5.37¯ . 

5.37¯ lies between 5.37 and 5.38. To visualize 5.37¯ more accurately we use a magnifying glass to visualize between 5.377 and 5.378. Again, we divide the portion between 5.377 and 5.378 into 10 equal parts and visualize more closely to represent 5.37¯ as given in the figure. This is located between 5.3778 and 5.3777. 


MULTIPLE CHOICE QUESTIONS(MCQs)


Q.1. Mark the correct alternative in each of the following:

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 an irrational number is terminating

(d) Decimal expansion of an irrational number is non-terminating and non-repeating

Proof: The decimal expansion of an irrational number is non-terminating and non- repeating. Thus, we can say that a number, whose decimal expansion is non-terminating and non- repeating, called irrational number. And the decimal expansion of rational number is either terminating or repeating. Thus, we can say that a number, whose decimal expansion is either terminating or repeating, is called a rational number.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.2. Which one of the following statements 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

Proof: Since,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevandRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare two irrational number andRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore, sum of two irrational numbers may be rational

Now, letRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevandRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevbe two irrational numbers andRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore, sum of two irrational number may be irrational

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.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 an irrational number

(c) Square of an irrational number is always a rational number

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

Proof: The sum of irrational number and rational number is always irrational number.

Let a be a rational number and b be an irrational number.

Then, 

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

As 2ab is irrational thereforeRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis irrational.

Hence (a + b) is irrational.

Therefore answer isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.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

Proof: Since we know that the product of rational and irrational number is always an irrational. For example: LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevare rational and irrational numbers respectively and their product isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.5. Which of the following is irrational?

(a) √4 / 9

(b) √4 / 5

(c) √7

(d) √81

Proof: Given that

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

And 7 is not a perfect square.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.6. Which of the following is irrational?

(i) 0.14

(ii) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iii) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(iv) 0.1014001400014...

Proof:  Given that

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Here 0.1014001400014...is non-terminating or non-repeating. So it is an irrational number.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.7. Which of the following is rational?

(a) √3

(b) π

(c) 4/0

(d) 0/4

Proof: Given thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Here,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev, this is the form ofRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevSo this is a rational number

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.8. The number 0.318564318564318564 ........ is:

(a) a natural number

(b) an integer

(c) a rational number

(d) an irrational number

Proof: Since the given numberRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis repeating, so it is rational number because rational number is always either terminating or repeating 

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.9. If n is a natural number, then √n is

(a) always a natural number

(b) always an irrational number

(c) always an irrational number

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

Proof: The term “natural number” refers either to a member of the set of positive integer 1,2,3.

And natural number starts from one of counting digit .Thus, if n is a natural number then sometimes n is a perfect square and sometimes it is not.

Therefore, sometimes n is a natural number and sometimes it is an irrational number

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.10. Which of the following numbers can be represented as non-terminating, repeating decimals?

(a) 39/24

(b) 3/16

(c) 3/11

(d) 137/25

Proof: Given that

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

HereRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis repeating but non-terminating.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.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

Proof: 

In basic mathematics, number line is a picture of straight line on which every point is assumed to correspond to real number.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.12. Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(d) 0.5015001500015.

Proof: Given decimal numbers are

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Here the number 0.5015001500015... is non terminating or non-repeating.

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev.


Q.13. The number 0.3¯ in the form p/q, where p and q are integers and q ≠ 0, is

(a) 33/100

(b) 3/10

(c) 1/3

(d) 3/100

Proof: Given number is 0.3¯

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.14. 0.32¯  when expressed in the form p/q (p, q are integers q ≠ 0), is

(a) 8/25

(b) 29/90

(c) 32/99

(d) 32/199

Proof: Given that 0.32¯

Now we have to express this number into p/q form

Let X =0.32¯

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.15. 23.43¯ when expressed in the form p/q (p, q are integers q ≠ 0), is

(a) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(b) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(c) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(d) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Proof: Given thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now we have to express this number into the form ofRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Let

x = 23.43

x = 23+0.4343...

x = 23+43 / 99

x=2277+43 / 99

=2320 / 99

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.16. RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevwhen expressed in the form p/q (p, q are integers, q ≠ 0), is

(a) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(b) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(c) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(d) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Proof: Given thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now we have to express this number intoRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevfrom

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.17. The value ofRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevis

(a) RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(b)RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(c)RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

(d) 0.45

Proof:  Given thatRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

LetRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Now we have to find the value of x

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

The correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.18. An irrational number between 2 and 2.5 is

(a) √11

(b) √5

(c) √22.5

(d) √12.5

Proof: Let 

a = 2

b = 2.5

Here a and b are rational numbers. So we observe that in first decimal place a and b have distinct. According to question a < b.so an irrational number between 2 and 2.5 is 2.236067978 OR 5.

Hence the correct answer isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.19. The number of consecutive zeros in 23 ×34 ×54 ×7, is

(a) 3

(b) 2

(c) 4

(d) 5

Proof: We are given the following expression and asked to find out the number of consecutive zeros

23 ×34 ×54 ×7

We basically, will focus on the powers of 2 and 5 because the multiplication of these two number gives one zero. So

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Therefore the consecutive zeros at the last is 3

So the option (a) is correct


Q.20. The smallest rational number by which 1/3 should be multiplied so that its decimal expansion terminates after one place of decimal, is

(a) 1/10

(b) 3/10

(c)  3

(d) 30

Proof: Give number isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevNow multiplying byRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRevin the given number, we have

RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence the correct option isRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Fill in the Blanks Types Questions(FBQs)


Q.1. The decimal expansion of a rational number is either ______ or _______.

Proof: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, the decimal expansion of a rational number is either terminating or recurring.


Q.2. The decimal expansion of an irrational number is non-terminating and _______.

Proof: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

In case of irrational number, the decimal expansion neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of an irrational number is non-terminating and non-repeating.


Q.3. The decimal expansion of √2 is _______ and  _________.

Proof:  √2  is an irrational number.

The decimal expansion of an irrational number neither terminates nor repeats after finitely many digits.

Hence, the decimal expansion of √2 is non-terminating and non-repeating.



Q.4. The value of 1.999. in the form of m/n, where m and n are integers and n ≠ 0, is _______.

Proof: Let x=1.999.....                ...(1)

Multiply (1) by 10 on both sides, we get

10x=19.999.....                ...(2)

Subtracting (1) from (2), we get

10x−x=19.999....−1.999....

⇒9x=18

⇒x=18/9

⇒x=2/1

Hence, the value of 1.999... in the form of m/n, where m and n are integers and n ≠ 0, is 2/1.


Q.5. Every recurring decimal is a _________ number.

Proof: The decimal expansion of a rational number either terminates after finitely many digits or ends with a repeating sequence.

Hence, every recurring decimal is a rational number.


Q.6. π is an _______ number.

Proof: The decimal expansion of π neither terminates nor repeats after finitely many digits.

Therefore, it is an irrational number.

Hence, π is an irrational number.


Q.7. The product of a non-zero rational number with an irrational number is always an ________ number.

Proof: The product of a non-zero rational number with an irrational number always results in an irrational number.

Hence, the product of a non-zero rational number with an irrational number is always an irrational number.


Q.8. The simplest form of RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is _______.

Proof: Let x=1.6666.....                ...(1)
Multiply (1) by 10 on both sides, we get

10x=16.6666.....                ...(2)

Subtracting (1) from (2), we get

10x−x=16.6666....−1.6666....

⇒9x=15

⇒x=15/9

⇒x=5/3

Hence, the simplest form ofRD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.9. RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev+RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev  is equal to _________.

Proof:  Let x=0.3333.....                ...(1)

Multiply (1) by 10 on both sides, we get

10x=3.3333.....                  ...(2)

Subtracting (1) from (2), we get

10x−x=3.333....−0.333....

⇒9x=3

⇒x=3/9                              ...(3)

Let y=0.4444.....                ...(4)

Multiply (1) by 10 on both sides, we get

10y=4.4444.....                  ...(5)

Subtracting (4) from (5), we get

10y−y=4.4444....−0.4444....

⇒9y=4

⇒y=4/9                              ...(6)

Now, 0.3+0.4 = x+y             

=3/9+4/9            (From (3) and (6))             

=RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev

Hence,RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev + RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev is equal to
RD Sharma Solutions: Number System- 3 Class 9 Notes | EduRev


Q.10. The sum of a rational number and an irrational number is ________ number.

Proof: The sum of a rational number and an irrational number always results in an irrational number.

Hence, the sum of a rational number and an irrational number is an irrational number.


Q.11. Every real number is either ________ or _______ number.

Proof: The real number includes all the rational as well as irrational numbers.

Hence, every real number is either rational or irrational number.

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