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Number System RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

RD Sharma Solutions: Exercise 1.1 - Number System


Q.1. Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

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

So,

p = 0,q = 1,2,3...

Therefore,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.2. Find five rational numbers between 1 and 2.

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

Consider,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

And

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So, five rational numbers between 6/6 and 12/6  will be 7/6, 8/6, 9/6, 10/6, 11/6.

Hence 5 rational numbers between 1 and 2 are: Number System RD Sharma Solutions | Mathematics (Maths) Class 9OR Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.3. Find six rational numbers between 3 and 4.

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

Consider,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

And,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So, six rational numbers between 21/7 and 28/7 will be Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence 6 rational numbers between 3 and 4 are Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.4. Find five rational numbers between Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: We need to find 5 rational numbers between Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Since, LCM of denominators= LCM (5,5) = 5

So, consider

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

And,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence 5 rational numbers between Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9 are: Number System RD Sharma Solutions | Mathematics (Maths) Class 9 OR Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.5. Are the following statements true or false? Give reasons for your answer.

(i) Every whole number is a natural number.

(ii) Every integer is a rational number.

(iii) Every rational number is an integer.

(iv) Every natural number is a whole number.

(v) Every integer is a whole number.

(vi) Every rational number is a whole number.

Proof: (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: Number System RD Sharma Solutions | Mathematics (Maths) Class 9 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



RD Sharma Solutions: Exercise 1.2 - Number System


Q.1. Express the following rational numbers as decimals:

(i) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii)Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: (i) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence,Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = 3.75


Q.2. Express the following rational numbers as decimals:

(i) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(v) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vi) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: (i) Given rational number is  Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = 0.6666

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Given rational number is  Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore, Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = 0.444

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to express this rational number into decimal form. So we will use long division method
Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = 0.1333

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(v) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to express this rational number into decimal form. So we will use long division method
Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vi) Given rational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to express this rational number into decimal form. So we will use long division method
Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefor Number System RD Sharma Solutions | Mathematics (Maths) Class 9 = Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.3. Look at several examples of rational numbers in the form Number System RD Sharma Solutions | Mathematics (Maths) Class 9 (q ≠0), where p and q are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property q must satisfy?

Proof: 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. 



RD Sharma Solutions: Exercise 1.3 - Number System



Q.1. Express each of the following decimals in the form Number System RD Sharma Solutions | Mathematics (Maths) Class 9:

(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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Given decimal is 0.750

Now we have to convert given decimal number into the Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Given decimal is 2.15

Now we have to convert given decimal number into the Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) Given decimal is 7.010

Now we have to convert given decimal number into the Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(v) Given decimal is 9.90

Now we have to convert given decimal number into the Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, 9.90 = 99/10

(vi) Given decimal is 1.0001

Now we have to convert given decimal number into the Number System RD Sharma Solutions | Mathematics (Maths) Class 9 form

Number System RD Sharma Solutions | Mathematics (Maths) Class 9Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.2. Express each of the following decimals in the form Number System RD Sharma Solutions | Mathematics (Maths) Class 9:

(i) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(v) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vi) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: (i) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(ii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 


Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(v) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vi) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore, 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(vii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Since, Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore, 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence, Number System RD Sharma Solutions | Mathematics (Maths) Class 9


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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 


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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Therefore,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

It is non-terminating and non-repeating

HenceNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

(ii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Therefore,

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

It is terminating.

HenceNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is a rational number.

(iii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 be the rational 

Squaring on both sides

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Since, x is rational 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational

But,Number System RD Sharma Solutions | Mathematics (Maths) Class 9is irrational

So, we arrive at a contradiction.

Hence Number System RD Sharma Solutions | Mathematics (Maths) Class 9  is an irrational number

(iv) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 be the rational number

Squaring on both sides, we get

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Since, x is a rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

But Number System RD Sharma Solutions | Mathematics (Maths) Class 9  is an irrational number

So, we arrive at contradiction

Hence Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

(v) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

Squaring on both sides, we get

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now, x is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is rational number

But
Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

So, we arrive at a contradiction

Hence Number System RD Sharma Solutions | Mathematics (Maths) Class 9  is an irrational number

(vi) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 be a rational number.

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Since, x is rational number,

⇒ x – 6 is a rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is a rational number

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is a rational number

But we know that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number, which is a contradiction 

So Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

(vii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Using the formula Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is a rational number

⇒is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 a rational number

But we know that Number System RD Sharma Solutions | Mathematics (Maths) Class 9is an irrational number 

So, we arrive at a contradiction

So Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number.

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

Squaring on both sides, we get

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now, x is rational

x2 is rational.

So, x2−29 is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Hence x = √5−2 is an irrational number

(x) Let

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

It is non-terminating or non-repeating

Hence Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

(xi) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is a rational number

(xii) Given x= 0.3796.

It is terminating

Hence it is a rational number

(xiii) Given number Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

It is repeating 

Hence it is a rational number

(xiv) Given number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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 =Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

x = 2, which is a rational number

(ii) Given number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is an irrational number

(iii) Given number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to check whether it is rational or irrational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is a rational

(iv) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to check whether it is rational or irrational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is an irrational number

(v) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to check whether it is rational or irrational

Since, Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

So it is a rational number

(vi) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to check whether it is rational or irrational

Since, Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of x

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it x is an irrational number

(ii) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of y

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So y is a rational number

(iii) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of z

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is rational number

(iv) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of u

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is an irrational number

(v) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of v

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is an irrational number

(vi) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of w

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

So it is an irrational number

(vii) Given that Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

Now we have to find the value of t

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9  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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 lying between 0.515115111511115.. and 0.535335333533335...


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

Proof: Let 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 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 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 lying between  0.3030030003... and 0.3010010001...

And irrational number is Number System RD Sharma Solutions | Mathematics (Maths) Class 9 lying between Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9 


Q.10. Find three different irrational numbers between the rational numbers Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9 .

Proof: Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9 and Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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

HenceNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9

And, so Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore,Number System RD Sharma Solutions | Mathematics (Maths) Class 9and Number System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their difference is a rational number

(ii) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their difference is an irrational number

BecauseNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 is an irrational number

(iii) LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their sum is a rational number

That isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their sum is an irrational number 

That isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 

(v) Let Number System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their product is a rational number

That isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

(vi) LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their product is an irrational number

That isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

(vii) LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 are two irrational numbers and their quotient is a rational number

That isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

(viii) LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational numbers and their quotient is an irrational number

That is Number System RD Sharma Solutions | Mathematics (Maths) Class 9


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 areNumber System RD Sharma Solutions | Mathematics (Maths) Class 9andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

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 areNumber System RD Sharma Solutions | Mathematics (Maths) Class 9andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9lying 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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Now x is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is rational

Number System RD Sharma Solutions | Mathematics (Maths) Class 9is rational

But,Number System RD Sharma Solutions | Mathematics (Maths) Class 9is an irrational

Thus we arrive at contradiction thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is a rational which is wrong.

HenceNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is 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. SoNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is 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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(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 thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9and

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

Here 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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;

Number System RD Sharma Solutions | Mathematics (Maths) Class 9 

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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(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:

 Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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: 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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 isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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,Number System RD Sharma Solutions | Mathematics (Maths) Class 9andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are two irrational number andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore, sum of two irrational numbers may be rational

Now, letNumber System RD Sharma Solutions | Mathematics (Maths) Class 9andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9be two irrational numbers andNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Therefore, sum of two irrational number may be irrational

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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, 

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

As 2ab is irrational thereforeNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is irrational.

Hence (a + b) is irrational.

Therefore answer isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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: LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9are rational and irrational numbers respectively and their product isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


Q.5. Which of the following is irrational?

(a) √4 / 9

(b) √4 / 5

(c) √7

(d) √81

Proof: Given that

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

And 7 is not a perfect square.

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


Q.6. Which of the following is irrational?

(i) 0.14

(ii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iii) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(iv) 0.1014001400014...

Proof:  Given that

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


Q.7. Which of the following is rational?

(a) √3

(b) π

(c) 4/0

(d) 0/4

Proof: Given thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Here,Number System RD Sharma Solutions | Mathematics (Maths) Class 9, this is the form ofNumber System RD Sharma Solutions | Mathematics (Maths) Class 9So this is a rational number

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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 numberNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is repeating, so it is rational number because rational number is always either terminating or repeating 

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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 isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

HereNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is repeating but non-terminating.

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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 isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


Q.12. Which of the following is irrational?

(a) 0.15

(b) 0.01516

(c) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(d) 0.5015001500015.

Proof: Given decimal numbers are

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9.


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¯

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

The correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


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¯

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

The correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


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

(a) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(b) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(c) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(d) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: Given thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to express this number into the form ofNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Let

x = 23.43

x = 23+0.4343...

x = 23+43 / 99

x=2277+43 / 99

=2320 / 99

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

The correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.16. Number System RD Sharma Solutions | Mathematics (Maths) Class 9when expressed in the form p/q (p, q are integers, q ≠ 0), is

(a) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(b) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(c) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(d) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Proof: Given thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to express this number intoNumber System RD Sharma Solutions | Mathematics (Maths) Class 9from

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

The correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.17. The value ofNumber System RD Sharma Solutions | Mathematics (Maths) Class 9is

(a) Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(b)Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(c)Number System RD Sharma Solutions | Mathematics (Maths) Class 9

(d) 0.45

Proof:  Given thatNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

LetNumber System RD Sharma Solutions | Mathematics (Maths) Class 9

Now we have to find the value of x

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

The correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


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 isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


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

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

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 isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9Now multiplying byNumber System RD Sharma Solutions | Mathematics (Maths) Class 9in the given number, we have

Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence the correct option isNumber System RD Sharma Solutions | Mathematics (Maths) Class 9


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 Number System RD Sharma Solutions | Mathematics (Maths) Class 9 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 ofNumber System RD Sharma Solutions | Mathematics (Maths) Class 9 is Number System RD Sharma Solutions | Mathematics (Maths) Class 9


Q.9. Number System RD Sharma Solutions | Mathematics (Maths) Class 9+Number System RD Sharma Solutions | Mathematics (Maths) Class 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))             

=Number System RD Sharma Solutions | Mathematics (Maths) Class 9

Hence,Number System RD Sharma Solutions | Mathematics (Maths) Class 9 + Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is equal to
Number System RD Sharma Solutions | Mathematics (Maths) Class 9


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.

The document Number System RD Sharma Solutions | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on Number System RD Sharma Solutions - Mathematics (Maths) Class 9

1. What is the RD Sharma Solutions: Exercise 1.1 - Number System?
Ans. The RD Sharma Solutions: Exercise 1.1 - Number System is a set of solutions provided for the first exercise in the Number System chapter of the RD Sharma textbook. It covers various topics related to number systems, such as natural numbers, whole numbers, integers, rational numbers, etc.
2. How can I access the RD Sharma Solutions: Exercise 1.1 - Number System?
Ans. You can access the RD Sharma Solutions: Exercise 1.1 - Number System by either purchasing the RD Sharma textbook and referring to the solutions provided at the end of the book or by searching for online platforms or websites that offer the solutions for free or for a fee.
3. Are the RD Sharma Solutions: Exercise 1.1 - Number System helpful for exam preparation?
Ans. Yes, the RD Sharma Solutions: Exercise 1.1 - Number System can be highly useful for exam preparation. They provide step-by-step solutions to the exercises in the textbook, helping students understand the concepts and practice solving problems. By using these solutions, students can improve their problem-solving skills and become more confident for exams.
4. Can I rely solely on the RD Sharma Solutions: Exercise 1.1 - Number System for my exam preparation?
Ans. While the RD Sharma Solutions: Exercise 1.1 - Number System can be a valuable resource for exam preparation, it is recommended to use them in conjunction with other study materials. It is essential to understand the concepts thoroughly and practice solving a variety of problems from different sources to ensure a comprehensive preparation for the exam.
5. Are there solutions available for all the exercises in the RD Sharma textbook?
Ans. Yes, solutions are available for almost all the exercises in the RD Sharma textbook. However, it is advisable to check the specific edition or version of the textbook you are using to ensure that the solutions provided match the exercises in your book. Additionally, some advanced or optional exercises may not have solutions provided.
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