Number System, Class 9 Mathematics R.D Sharma Solutions

Page 1.12 Ex. 1.2

Q1.

(i) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence,

(ii) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence,

(iii) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method as below.

Hence,

Q2.
Answer :

(i) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method 

Hence,

(ii) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(iii) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(iv) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(v) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

(vi) Given rational number is  

Now we have to express this rational number into decimal form. So we will use long division method

Hence,

Page 1.13 Ex. 1.2

Q3.

Prime factorization is the process of finding which prime numbers you need to multiply together to get a certain number. So prime factorization of denominators (q) must have only the power of 2 or 5 or both.

Page 1.22 Ex. 1.3

Q1.

(i) Given decimal is

Now we have to convert given decimal number into the form

Let

Hence,

(ii) Given decimal is

Now we have to convert given decimal number into form

Let

Hence,

(iii) Given decimal is

Now we have to express the given decimal number into form

Let

Hence,

(iv) Given decimal is 7.010

Now we have to express the given decimal number into form

Let

Hence,

(v) Given decimal is

Now we have to find given decimal number into form

Let

Hence,

(vi) Given decimal is

Now we have to find given decimal number into form

Hence,

Q2.

(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,

Page 1.30 Ex. 1.4

Q1.

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

Q2.

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

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.

Q3.

(i) Let

Therefore,

It is non-terminating and non-repeating

Hence is an irrational number

(ii) Let

Therefore,

It is terminating.

Hence is 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 be the rational 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

Since, is rational number,

⇒ x – 6 is a rational nu8mber

⇒is a rational number

⇒is a rational number

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

So is an irrational number

(vii) Let

So is a rational number

(viii) Let be rational number

Using the formula

⇒is a rational number

⇒is a rational number

But we know thatis an irrational number

So, we arrive at a contradiction

So is an irrational number.

(ix) Let be the rational number

Squaring on both sides, we get

Now, x is rational

is rational

is rational

is rational

is rational

But, is irrational. So we arrive at contradiction

Hence 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 that

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.

Q4.

(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.

Page 1.31 Ex. 1.4

Q5.

(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

Q6.

(i) Let  

And, so  

Therefore, andare 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) Let are two irrational numbers and their sum is a rational number

That is

(iv) Let are 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) Let are 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) Let are two irrational numbers and their quotient is an irrational number

That is

Q7.

Let 

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 3 and b has 1. So a > b.

Hence two rational numbers are lying between and

Q8.

Let  and

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 and

Q9.

Let 

Here a and bare 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 and

Q10.

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 and

And irrational number is lying between and

Q11.

Let 

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 are and lying between 0.5 and 0.55.

Q12.

Let 

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 are and lying between 0.1 and 0.12.

Q13.

Given that is an irrational number

Now we have to prove is an irrational number 

Let is a rational

Squaring on both sides

Now   is rational

is rational

is rational

is rational

But, is an irrational

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

Hence is an irrational

Q14.

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

We find that

Hence are required irrational numbers.

Page 1.35 Ex. 1.5

Q1.

(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.

Page 1.36 Ex. 1.5

Q2.

We are asked to represent 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 , 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 that and 

(5) Take A as center and AB₁ as radius and draw an arc which cuts the line AX at A_2.

Here 

So

So A₂ is the representation for

(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

(3) Again draw the perpendicular lineto AX

(4) Take A as center and AB₃ as radius and draw an arc which cuts the horizontal line at A₄ 

Here;

A₄ is basically the representation of

Q3.

We are asked to represent the real numbers on the real number line

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

_(a)

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

(3) Mark a point C on the line such that

(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

(b)

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

(3) Mark a point C on the line such that

(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

(c)

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

(3) Mark a point C on the line such that

(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

Q4.

(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. So is an irrational number.

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

Page 1.39 Formative Assessment_MCQ

Q1.

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 .

Q2.

Since, and are 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 .

Q3.

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 therefore is irrational.

Hence is irrational.

Therefore answer is .

Page 1.40 Formative Assessment_MCQ

Q4.

Since we know that the product of rational and irrational number is always an irrational. For example: Let are rational and irrational numbers respectively and their product is  

Hence the correct option is.

Q5.

Given that

And 7 is not a perfect square.

Hence the correct option is.

Q6.

Given that

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

Hence the correct option is.

Q7.

Given that

Here,, this is the form of . So this is a rational number

Hence the correct option is.

Q8.

Since the given number is repeating, so it is rational number because rational number is always either terminating or repeating

Hence the correct option is.

Q9.

The term “natural number” refers either to a member of the set of positive integer.

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, sometimesis a natural number and sometimes it is an irrational number

Hence the correct option is.

Q10.

Given that

Here is repeating but non-terminating. 

Hence the correct option is.

Q11.

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.

Q12.

Let 

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 OR

Hence the correct answer is.

Q13.

Given decimal numbers are

Here the number is non terminating or non-repeating.

Hence the correct option is.

Q14.

Q15.

Given that

The correct option is.

Page 1.41 Formative Assessment_MCQ

Q16.

Given number is

The correct option is

Q17.

Give number is. Now multiplying by in the given number, we have

Hence the correct option is

Q18.

Given that

Now we have to express this number into form

Let X =

The correct option is

Q19.

Given that

Now we have to express this number into the form of

The correct option is

Q20.

Given that

Now we have to express this number into form

The correct option is

Q21.

Given that

Let

Now we have to find the value of

The correct option is  

Page 1.5 Ex. 1.1

Q1.

Yes, zero is a rational number because it is either terminating or non-terminating so we can write in the form of , where p and q are natural numbers and q is not equal to zero. 

So,

Therefore,

Q2.

We need to find 5 rational numbers between 1 and 2.

Consider,

And

Hence 5 rational numbers between 1 and 2 are: OR .

Q3.

We need to find 6 rational numbers between 3 and 4.

Consider,

And

Hence 6 rational numbers between 3 and 4 are

Q4.

We need to find 5 rational numbers betweenand .

Since, LCM of denominators

So, consider

And,

Hence 5 rational numbers between  and are: OR

Q5.

(i) False, because whole numbers start from zero and natural numbers start from one

(ii) True, because it can be written in the form of a fraction with denominator 1

(iii) False

(iv) True, because natural numbers belong to whole numbers

(v) False, because set of whole numbers contains only zero and set of positive integers, whereas set of integers is the collection of zero and all positive and negative integers.

(vi) False, because rational numbers include fractions but set of whole number does not include fractions.

Page 2.12 Ex. 2.1

Q1.

We have to simplify the following, assuming thatare positive real numbers

(i) Given

As x is positive real number then we have

Hence the simplified value of is

(ii) Given

As x and y are positive real numbers then we can write 

By using law of rational exponents we have 

Hence the simplified value of is

(iii) Given

As x and y are positive real numbers then we have 

By using law of rational exponents we have 

By using law of rational exponents we have

Hence the simplified value of is .

iv)   x-23 y4 ÷ xy-12=x12-23 y412 ÷ x × y-1212=x12×-23 × y4×12×12 × y-12×12    =x-13 × y2x12 × y-14by using the law of rational exponents, am ÷ an = am-n, we have     x-13-12 × y2+14=x-56 × y94 =y94x56v. 243  x10 y5 z105=243 × x10 × y5 × z1015=24315 × x1015 × y5110 × z1015=3515 × x10×15 × y5×15 × z10×15=3 × x2 × y × z2=3x2yz2vi x-4y-1054=x-454y-1054=x-4×54y-10×54=x-5y-252=y252x5

Q2.

(1) Given

By using law of rational exponents we have

Hence the value of is

(ii) Given

Hence the value of is

(iii) Given

The value of is

(iv) Given

Hence the value of is

(v) Given. So,

By using the law of rational exponents

Hence the value of is

(vi) Given . So,

Hence the value of is

Q3.

(i) We have to prove that

By using rational exponent we get,

Hence,

(ii) We have to prove that. So,

Hence,

(iii) We have to prove that

Now,

Hence,

(iv) We have to prove that. So,

Let

Hence,

(v) We have to prove that

Let

Hence,

(vi) We have to prove that . So,

Let

Hence,

(vii) We have to prove that. So let

By taking least common factor we get 

Hence,

(viii) We have to prove that. So,

Let

Hence,

(ix) We have to prove that. So,

Let

Hence,

Q4.

We are given. We have to find the value of

Since

By using the law of exponents we get, 

On equating the exponents we get,

Hence,

Q5.

From the following we have to find the value of x

(i) Given

By using rational exponents

On equating the exponents we get,

The value of x is

(ii) Given

On equating the exponents 

Hence the value of x is

(iii) Given

Comparing exponents we have,

Hence the value of x is

(iv) Given

On equating the exponents of 5 and 3 we get,

And,

The value of x is

(v) Given

On equating the exponent we get

And, 

Hence the value of x is

(vi) Given

On equating the exponents we get 

And, 

Hence the value of x is

Page 2.13 Formative Assessment_VSA

Q1.

We have to writein decimal form. So,

Hence the decimal form of is

Q2.

State the product law of exponents.

If is any real number and , are positive integers, then

By definition, we have 

(factor) ( factor)

to factors

Thus the exponent “product rule” tells us that, when multiplying two powers that have the same base, we can add the exponents. 

Q3.

State the quotient law of exponents.

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If a is a non-zero real number and m, n are positive integers, then

We shall divide the proof into three parts 

(i) when

(ii) when

(iii) when

Case 1 

When

We have 

Case 2 

When

We get

Cancelling common factors in numerator and denominator we get,

By definition we can write 1 as

Case 3 

When

In this case, we have 

Hence, whether, or,

Q4.

State the power law of exponents.

The “power rule” tell us that to raise a power to a power, just multiply the exponents. 

If a is any real number and m, n are positive integers, then

We have,

factors

factors

Hence,

Q5.

We have to find the value of L =

By using rational exponents, we get

By using rational exponents we get 

By definition we can write as 1

Hence the value of expression is .

Q6.

We have to find the value of. So,

By using rational exponents we get 

Hence the simplified value of is

Q7.

We have to simplify. So,

Hence, the value of is

Q8.

We have to simplify. So,

By using rational exponents, we get

Hence the value of is

Page 3.13 Ex. 3.2

Q1.

(i) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor foris. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vii) We know that rationalization factor for is. We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

Page 3.14 Ex. 3.2

Q2.

(i) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(ii) We know that rationalization factor of the denominator is . We will multiply numerator and denominator of the given expression by , to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(iii) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(iv) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(v) Given that

Putting the value of, we get 

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

(vi) We know that rationalization factor of the denominator is. We will multiply numerator and denominator of the given expression by, to get

Putting the value of and, we get

The value of expression can be round off to three decimal places as.

Hence the given expression is simplified to.

Q3.

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(v) We know that rationalization factor for is.We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(vii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(viii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to.

(ix) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified with rational denominator to .

Q4.

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Hence the given expression is simplified to.

Q5.

(i) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(ii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(iii) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(iv) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

(v) We know that rationalization factor forand areand respectively. We will multiply numerator and denominator of the given expression and by and respectively, to get

Hence the given expression is simplified to.

Q6.

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get.

(iii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get. 

(iv) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get.

(v) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get.

(vi) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

On equating rational and irrational terms, we get 

Hence, we get.

Q7.

We know that. We have to find the value of.

As therefore, 

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of and , we get

Hence the value of the given expression.

Page 3.15 Ex. 3.2

Q8.

We know that. We have to find the value of . As therefore,

We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of x and , we get

Hence the given expression is simplified to.

Q9.

We know that rationalization factor for is . We will multiply denominator and numerator of the given expression by , to get

Putting the values of and, we get

Hence value of the given expression is.

Q10.

(i) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the values of, we get 

Hence the given expression is simplified to.

(ii) We know that rationalization factor for is . We will multiply numerator and denominator of the given expression by, to get

Putting the value of, we get 

Hence the given expression is simplified to.

Q11.

We have,

It can be simplified as 

On squaring both sides, we get 

The given equation can be rewritten as.

Therefore, we have 

Hence, the value of given expression is.