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

## 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 :

(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  form

Let

Hence,

(ii) Given decimal is 0.750

Now we have to convert given decimal number into the  form

Let

Hence,

(iii) Given decimal is 2.15

Now we have to convert given decimal number into the  form

Let

Hence,

(iv) Given decimal is 7.010

Now we have to convert given decimal number into the  form

Let

Hence,

(v) Given decimal is 9.90

Now we have to convert given decimal number into the  form

Let

Hence, 9.90 = 99/10

(vi) Given decimal is 1.0001

Now we have to convert given decimal number into the  form

Hence,

Q.2. Express each of the following decimals in the form :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Proof: (i) Let

Hence,

(ii) Let

Hence,

(iii) Let

Hence,

(iv) Let

Hence,

(v) Let

Hence,

(vi) Let

Let

Therefore,

Hence,

(vii) Let

Since,

Therefore,

Hence,

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

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

Therefore,

It is non-terminating and non-repeating

Hence is an irrational number

(ii) Let

Therefore,

It is terminating.

Henceis a rational number.

(iii) Let  be the rational

Squaring on both sides

Since, x is rational

is rational

is rational

is rational

is rational

But,is irrational

So, we arrive at a contradiction.

Hence   is an irrational number

(iv) Let  be the rational number

Squaring on both sides, we get

Since, x is a rational number

is rational number

is rational number

is rational number

is rational number

But   is an irrational number

So, we arrive at contradiction

Hence  is an irrational number

(v) Let  is an irrational number

Squaring on both sides, we get

Now, x is rational number

is rational number

is rational number

is rational number

is rational number

But
is an irrational number

So, we arrive at a contradiction

Hence   is an irrational number

(vi) Let  be a rational number.

Since, x is rational number,

⇒ x – 6 is a rational number

is a rational number

is a rational number

But we know that  is an irrational number, which is a contradiction

So  is an irrational number

(vii) Let

Using the formula

is a rational number

⇒is  a rational number

But we know that is an irrational number

So, we arrive at a contradiction

So  is an irrational number.

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

Squaring on both sides, we get

Now, x is rational

x2 is rational.

So, x2−29 is rational

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

Hence x = √5−2 is an irrational number

(x) Let

It is non-terminating or non-repeating

Hence  is an irrational number

(xi) Let

Hence  is a rational number

(xii) Given x= 0.3796.

It is terminating

Hence it is a rational number

(xiii) Given number

It is repeating

Hence it is a rational number

(xiv) Given number is

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 =

x = 2, which is a rational number

(ii) Given number is

So it is an irrational number

(iii) Given number is

Now we have to check whether it is rational or irrational

So it is a rational

(iv) Given that

Now we have to check whether it is rational or irrational

So it is an irrational number

(v) Given that

Now we have to check whether it is rational or irrational

Since,

So it is a rational number

(vi) Given that

Now we have to check whether it is rational or irrational

Since,

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

Now we have to find the value of x

So it x is an irrational number

(ii) Given that

Now we have to find the value of y

So y is a rational number

(iii) Given that

Now we have to find the value of z

So it is rational number

(iv) Given that

Now we have to find the value of u

So it is an irrational number

(v) Given that

Now we have to find the value of v

So it is an irrational number

(vi) Given that

Now we have to find the value of w

So it is an irrational number

(vii) Given that

Now we have to find the value of t

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   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  lying between 0.515115111511115.. and 0.535335333533335...

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

Proof: Let

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  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

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  lying between  0.3030030003... and 0.3010010001...

And irrational number is  lying between  and

Q.10. Find three different irrational numbers between the rational numbers  and  .

Proof: Let  and

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

Hence 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

And, so

Therefore,and are two irrational numbers and their difference is a rational number

(ii) Let are two irrational numbers and their difference is an irrational number

Because is an irrational number

(iii) Letare two irrational numbers and their sum is a rational number

That is

(iv) Letare two irrational numbers and their sum is an irrational number

That is

(v) Let are two irrational numbers and their product is a rational number

That is

(vi) Letare two irrational numbers and their product is an irrational number

That is

(vii) Let are two irrational numbers and their quotient is a rational number

That is

(viii) Letare two irrational numbers and their quotient is an irrational number

That is

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 areand

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 areandlying 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

Now x is rational

is rational

is rational

is rational

But,is an irrational

Thus we arrive at contradiction thatis a rational which is wrong.

Henceis 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. Sois 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

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

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

(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 thatand

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

Here

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

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;

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

(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:

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:

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 is.

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,andare two irrational number and

Therefore, sum of two irrational numbers may be rational

Now, letandbe two irrational numbers and

Therefore, sum of two irrational number may be irrational

Hence the correct option is.

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,

As 2ab is irrational thereforeis irrational.

Hence (a + b) is irrational.

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: Letare rational and irrational numbers respectively and their product is

Hence the correct option is.

Q.5. Which of the following is irrational?

(a) √4 / 9

(b) √4 / 5

(c) √7

(d) √81

Proof: Given that

And 7 is not a perfect square.

Hence the correct option is.

Q.6. Which of the following is irrational?

(i) 0.14

(ii)

(iii)

(iv) 0.1014001400014...

Proof:  Given that

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

Hence the correct option is.

Q.7. Which of the following is rational?

(a) √3

(b) π

(c) 4/0

(d) 0/4

Proof: Given that

Here,, this is the form ofSo this is a rational number

Hence the correct option is.

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 numberis repeating, so it is rational number because rational number is always either terminating or repeating

Hence the correct option is.

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 is.

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

Hereis repeating but non-terminating.

Hence the correct option is.

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 is.

Q.12. Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c)

(d) 0.5015001500015.

Proof: Given decimal numbers are

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

Hence the correct option is.

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¯

The correct option is

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¯

The correct option is

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

(a)

(b)

(c)

(d)

Proof: Given that

Now we have to express this number into the form of

Let

x = 23.43

x = 23+0.4343...

x = 23+43 / 99

x=2277+43 / 99

=2320 / 99

The correct option is

Q.16. when expressed in the form p/q (p, q are integers, q ≠ 0), is

(a)

(b)

(c)

(d)

Proof: Given that

Now we have to express this number intofrom

The correct option is

Q.17. The value ofis

(a)

(b)

(c)

(d) 0.45

Proof:  Given that

Let

Now we have to find the value of x

The correct option is

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 is

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

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 isNow multiplying byin the given number, we have

Hence the correct option is

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  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 of is

Q.9. +  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))

=

Hence, + is equal to

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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## Mathematics (Maths) Class 9

46 videos|323 docs|108 tests

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