RD Sharma 2022 Solutions for Class 9 Maths Chapter 1 - Number System

Answer:

(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, rational numbers are represented in the form of fractions. Integers can be represented in the form of fractions but all fractions are not integers. for example: $\frac{3}{4}$ is a rational number but not an integer.

(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.

Answer:

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,

Answer:

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

Consider,

And

So, five rational numbers between $\frac{6}{6} \mathrm{and} \frac{12}{6}$will be $\frac{7}{6}, \frac{8}{6}, \frac{9}{6}, \frac{10}{6}, \frac{11}{6}$.
Hence 5 rational numbers between 1 and 2 are: OR .

Answer:

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

Consider,

And

So, six rational numbers between $\frac{21}{7} \mathrm{and} \frac{28}{7}$ will be .
Hence 6 rational numbers between 3 and 4 are .

Answer:

We need to find 5 rational numbers betweenand .

Since, LCM of denominators

So, consider

And,

Hence 5 rational numbers between  and are: OR

Answer:

(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, $\frac{15}{4}=3.75$

Answer:

(i) Given rational number is $\frac{2}{3}$.

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

Hence,

(ii) Given rational number is $-\frac{4}{9}$.

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

Hence,

(iii) Given rational number is $\frac{-2}{15}$.

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

Hence, .

(iv) Given rational number is $-\frac{22}{13}$.

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

Hence,

(v) Given rational number is $\frac{437}{999}$.

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

Hence,

Answer:

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.

Answer:

(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, =$\frac{99}{10}$

(vi) Given decimal is

Now we have to find given decimal number into form

Hence,

Answer:

(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,

Answer:

Let $x=0.\overline{7}$
$\Rightarrow x=0.77777...$         (1)
Multiplying both sides by 10, we get
10x = 7.777...                 (2)
Subtract (1) from (2)
10x – x = 7.777... – 0.777...
⇒ 9x = 7
⇒ $x=\frac{7}{9}$

Now, $y=0.4\overline{7}=0.4777...$   (3)
Multiply both sides by 10, we get
10y = 4.777...                    (4)
Again multiply both sides by 100, we get
100y = 47.777...               (5)
Subtract (4) from (5)
100y – 10y = 47.777... – 4.777...
⇒  90y = 43 
$\Rightarrow y=\frac{43}{90}$

$\begin{array}{rcl}\therefore 0.6+0.\overline{7}+0.4\overline{7}& =& \frac{6}{10}+\frac{7}{9}+\frac{43}{90}\\ & =& \frac{54+70+43}{90}\\ & =& \frac{167}{90}\end{array}$

Hence, $0.6+0.\overline{7}+0.4\overline{7}=\frac{167}{90}$.

Answer:

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

Answer:

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.

Answer:

(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  be a rational number.

Since, x 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 $x = \sqrt{5}-2$ be the rational number

Squaring on both sides, we get

$$\begin{gathered} x = \sqrt{5}-2 \\ {x}^2={\left(\sqrt{5}-2\right)}^2 \\ {x}^2=25+4-4\sqrt{5} \\ {x}^2-29=-4\sqrt{5} \\ \frac{x^2-29}{-4}=\sqrt{5} \end{gathered}$$

Now, x is rational

$$\begin{gathered} x^2 \mathrm{is} \mathrm{rational}. \\ \mathrm{So}, {\mathrm{x}}^2-29 \mathrm{is} \mathrm{rational} \\ \frac{{\mathrm{x}}^2-29}{-4} = \sqrt{5} \mathrm{is} \mathrm{rational}. \end{gathered}$$

But, $\sqrt{5}$ is irrational. So we arrive at contradiction

Hence $x = \sqrt{5}-2$ 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 $0.3796$.

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

Answer:

(i) Given number is x =

x = 2, which is a rational number

(ii) Given number is

$3\sqrt{18}=3\sqrt{3\times 3\times 2}=3\times 3\sqrt{2}=9\sqrt{2}$

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

Answer:

(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

Answer:

Let 

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 and

Answer:

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

Answer:

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  and 

Answer:

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

Answer:

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

$$\begin{gathered} a = 0.72072007200072.. \\ b = 0.73073007300073.. \\ c = 0.74074007400074.... \end{gathered}$$

We find that

Hence  are required irrational numbers.

Answer:

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.

Answer:

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.

Answer:

(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

Answer:

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

Answer:

(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.

Answer:

(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.

Answer:

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

Answer:

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

Answer:

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.

Answer:

We know that $5.3\overline{7}$ will lie between 5 and 6. So, we locate $5.3\overline{7}$ 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 $5.3\overline{7}$ . 
$5.3\overline{7}$ lies between 5.37 and 5.38. To visualize $5.3\overline{7}$ 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 
$5.3\overline{7}$ as given in the figure. This is located between 5.3778 and 5.3777. 

Answer:

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.

Answer:

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.

Answer:

$\sqrt{2}$ is an irrational number.

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

Hence, the decimal expansion of $\sqrt{2}$ is non-terminating and non-repeating.

Answer:

$$\begin{gathered} \mathrm{Let} x=1.999..... ...\left(1\right) \\ \mathrm{Multiply} \left(1\right) \mathrm{by} 10 \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ 10x=19.999..... ...\left(2\right) \\ \mathrm{Subtracting} \left(1\right) \mathrm{from} \left(2\right), \mathrm{we} \mathrm{get} \\ 10x-x=19.999....-1.999.... \\ \Rightarrow 9x=18 \\ \Rightarrow x=\frac{18}{9} \\ \Rightarrow x=\frac{2}{1} \end{gathered}$$

Hence, the value of 1.999... in the form of $\frac{m}{n}$, where m and n are integers and n ≠ 0, is $\underline{\frac{2}{1}}$.

Answer:

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.

Answer:

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.

Answer:

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.

Answer:

$$\begin{gathered} \mathrm{Let} x=1.6666..... ...\left(1\right) \\ \mathrm{Multiply} \left(1\right) \mathrm{by} 10 \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ 10x=16.6666..... ...\left(2\right) \\ \mathrm{Subtracting} \left(1\right) \mathrm{from} \left(2\right), \mathrm{we} \mathrm{get} \\ 10x-x=16.6666....-1.6666.... \\ \Rightarrow 9x=15 \\ \Rightarrow x=\frac{15}{9} \\ \Rightarrow x=\frac{5}{3} \end{gathered}$$

Hence, the simplest form of $1.\overline{6}$ is $\underline{\frac{5}{3}}$.

Answer:

$$\begin{gathered} \mathrm{Let} x=0.3333..... ...\left(1\right) \\ \mathrm{Multiply} \left(1\right) \mathrm{by} 10 \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ 10x=3.3333..... ...\left(2\right) \\ \mathrm{Subtracting} \left(1\right) \mathrm{from} \left(2\right), \mathrm{we} \mathrm{get} \\ 10x-x=3.333....-0.333.... \\ \Rightarrow 9x=3 \\ \Rightarrow x=\frac{3}{9} ...\left(3\right) \\ \mathrm{Let} y=0.4444..... ...\left(4\right) \\ \mathrm{Multiply} \left(1\right) \mathrm{by} 10 \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{we} \mathrm{get} \\ 10y=4.4444..... ...\left(5\right) \\ \mathrm{Subtracting} \left(4\right) \mathrm{from} \left(5\right), \mathrm{we} \mathrm{get} \\ 10y-y=4.4444....-0.4444.... \\ \Rightarrow 9y=4 \\ \Rightarrow y=\frac{4}{9} ...\left(6\right) \\ \mathrm{Now}, \\ 0.\overline{3}+0.\overline{4}=x+y \\ =\frac{3}{9}+\frac{4}{9} \left(\mathrm{From} \left(3\right) \mathrm{and} \left(6\right)\right) \\ =\frac{7}{9} \end{gathered}$$

Hence, $0.\overline{3}+0.\overline{4}$ is equal to $\underline{\frac{7}{9}}$.

Answer:

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.

Answer:

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

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