# NCERT Solutions for Class 9 Maths Chapter 1 Number Systems| PDF Download

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**NCERT Solutions for Class 9 Maths Chapter Number Systems**that will comfort you as you will get to know various concepts needed for solving various questions. These NCERT Solutions are updated as per the latest pattern of 2020-21 syllabus.**Chapter 1 Class 10 Maths NCERT Solutions**Â**PDF Download**can be really helpful if anyone want to understand the detailed solutions and minimise the errors wherever possible.For better understanding and application of concepts, one should first need to focus on the

**Number Systems NCERT Solutions**as it will tell you about the difficulty of questions. You can be able to solve more questions related to chapter after practising more questions and thus fetching more marks.**Exercise 1.1**

1. Is zero a rational number? Can you write it in the form

*p*/

*q*, where

*p*and

*q*are integers and

*q*â‰ 0?

**Answer**

2. Find six rational numbers between 3 and 4.

**Answer**

There are infinite rational numbers in between 3 and 4.

3 and 4 can be represented as 24/8 and 32/8Â respectively.

Therefore, six rational numbers between 3 and 4 are

25/8, 26/8, 27/8, 28/8, 29/8, 30/8.

3. Find five rational numbers between 3/5 and 4/5.

**Answer**

There are infinite rational numbers in between 3/5 and 4/5

3/5 = 3Ã—6/5Ã—6 = 18/30

4/5 = 4Ã—6/5Ã—6 = 24/30

Therefore, five rational numbers betweenÂ 3/5 and 4/5Â are

19/30, 20/30, 21/30, 22/30, 23/30.

**4. State whether the following statements are true or false. Give reasons for your answers.**

(i) Every natural number is a whole number.

â–ºÂ True, since the collection of whole numbers contains all natural numbers.

(ii) Every integer is a whole number.

â–º False, as integers may be negative but whole numbers are always positive.

(iii) Every rational number is a whole number.

â–º False, as rational numbers may be fractional but whole numbers may not be.â–º False, as integers may be negative but whole numbers are always positive.

(iii) Every rational number is a whole number.

Page No: 8

**Exercise 1.2**

**1. State whether the following statements are true or false. Justify your answers.**

(i) Every irrational number is a real number.

â–ºÂ True, since the collection of real numbers is made up of rational and irrational numbers.

(ii) Every point on the number line is of the formâˆš

â–º False, since positive number cannot be expressed as square roots.

(iii) Every real number is an irrational number.

â–ºÂ False, as real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

(ii) Every point on the number line is of the formâˆš

*m*, where m is a natural number.â–º False, since positive number cannot be expressed as square roots.

(iii) Every real number is an irrational number.

â–ºÂ False, as real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.

2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

**Answer**

No, the square roots of all positive integers are not irrational. For example âˆš4 = 2.

3. Show howÂ âˆš5Â can be represented on the number line.

**Answer**

Step 1: Let AB be a line of length 2 unit on number line.

Step 2: At B, draw a perpendicular line BC of length 1 unit. Join CA.

Step 3: Now, ABC is a right angled triangle. Applying Pythagoras theorem,

AB

^{2}+ BC

^{2}= CA

^{2}

â‡’Â 2

^{2}+ 1

^{2}= CA

^{2}

â‡’Â CA

^{2}= 5

â‡’Â CAÂ = âˆš5

Thus, CA is a line of lengthÂ âˆš5Â unit.

Step 4: Taking CA as a radius and A as a centre draw an arc touching

the number line. The point at which number line get intersected by

arc is atÂ âˆš5Â distance from 0 because it is a radius of the circle

whose centre was A.

Thus,Â âˆš5Â is represented on the number line as shown in the figure.

**Exercise 1.3**

**1.Â Write the following in decimal form and say what kind of decimal expansion each has:**

(i) 36/100

= 0.36 (Terminating)

(ii) 1/11

0.09090909... = 0.9 (Non terminating repeating)

(iii)

= 33/8 = 4.125 (Terminating)

(iv) 3/13

= 0.230769230769... =Â 0.230769Â (Non terminating repeating)

(v) 2/11

= 0.181818181818... =Â 0.18Â (Non terminating repeating)

(vi) 329/400

= 0.8225 (Terminating)

2. You know that 1/7 = 0.142857.Can you predict what the decimal expansion of 2/7, 3/7, 4/7, 5/7, 6/7 areÂ without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of 1/7 carefully.]

**Answer**

Yes. We can be done this by:

3. Express the following in the form

*p*/

*q*Â where

*p*and

*q*are integers and

*q*â‰ 0.

(i) 0.6

(ii) 0.47

(iii) 0.001

**Answer**

(i) 0.6 = 0.666...

Let

10

10

9

(ii) 0.47 = 0.4777...

= 4/10Â + 0.777/10

Let

10

10

Let

*x*= 0.666...10

*x*= 6.666...10

*x*= 6Â +*x*9

*x*= 6*x*= 2/3(ii) 0.47 = 0.4777...

= 4/10Â + 0.777/10

Let

*x*= 0.777â€¦10

*x*= 7.777â€¦10

*x*= 7 +*x**x*= 7/9

4/10Â + 0.777.../10 = 4/10Â + 7/90

= 36/90 +7/90 = 43/90

(iii) 0.001 = 0.001001...

Let

1000*x*= 0.001001...*x*= 1.001001â€¦

1000

*x*= 1 +

*x*

999

*x*= 1

*x*= 1/999

4. Express 0.99999â€¦in the formÂ Â

*p*/

*q*. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

**Answer**

Let

*x*= 0.9999â€¦

10

*x*= 9.9999â€¦

10

*x*= 9 +

*x*

9

*x*= 9

*x*= 1

The difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999 is too much near 1, Therefore, the 1 as answer can be justified.Â

**Answer**

1/17 = 0.0588235294117647

There are 16 digits in the repeating block of the decimal expansion of 1/17.

Division Check:

= 0.0588235294117647

6. Look at severalÂ examples of rational numbers in the formÂ

*p*/

*qÂ*(

*qÂ*â‰ 0) whereÂ

*p*Â andÂ

*q*Â are integersÂ withÂ no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what propertyÂ

*q*Â must satisfy?

**Answer**

We observe that when q isÂ 2, 4, 5, 8, 10... then the decimal expansion is terminating. For example:

1/2 = 0.5, denominator

*q*= 2

^{1}

7/8 = 0.875, denominator

*q*= 2

^{3}

4/5 = 0.8, denominator

*q*= 5

^{1}

We can observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.

**Answer**

Three numbers whose decimal expansions are non-terminating non-recurring are:

0.303003000300003...

0.505005000500005...

0.7207200720007200007200000â€¦**Answer**

5/7 = 0.714285

9/11 = 0.81

9/11 = 0.81

Three different irrational numbers are:

0.73073007300073000073â€¦
0.75075007300075000075â€¦

0.76076007600076000076â€¦

(i) âˆš23

(ii)Â âˆš225
(iii)Â 0.3796

(iv) 7.478478Â

(iv) 7.478478Â

(v) 1.101001000100001â€¦

**Answer**

(i) âˆš23 = 4.79583152331...

Since the number is non-terminating non-recurring therefore, it is an irrational number.

(ii)Â âˆš225 = 15 = 15/1

Since the number is rational number as it can represented inÂ

Since the number is rational number as it can represented inÂ

*p*/*q*form.
(iii)Â 0.3796

Since the number is terminating therefore, it is an rational number.

(iv) 7.478478 = 7.478

Since the this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.101001000100001â€¦

Since the number is non-terminating non-repeating, therefore, it is an irrational number.
Page No: 18

**Exercises 1.4**

**Answer**

2. Visualise 4.26 on the number line, up to 4 decimal places.

**Answer**

4.26 = 4.2626

Page No: 24

**Exercise 1.5**

(i) 2 - âˆš5

(ii) (3Â +Â âˆš23) -Â âˆš23
(iii) 2âˆš7/7âˆš7

(iv) 1/âˆš2
(v)Â 2Ï€

**Answer**

(i) 2 - âˆš5 = 2 - 2.2360679â€¦ = - 0.2360679â€¦

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

(ii) (3Â +Â âˆš23) -Â âˆš23 = 3Â +Â âˆš23Â -Â âˆš23Â = 3 = 3/1

Since the number is rational number as it can represented inÂ

(iii) 2âˆš7/7âˆš7 = 2/7

Since the number is rational number as it can represented inÂ

(iv) 1/âˆš2 = âˆš2/2 = 0.7071067811...

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

(v) 2Ï€ = 2Â Ã—Â 3.1415â€¦ =Â 6.2830â€¦

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

2. Simplify each of the following expressions:

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

(ii) (3Â +Â âˆš23) -Â âˆš23 = 3Â +Â âˆš23Â -Â âˆš23Â = 3 = 3/1

Since the number is rational number as it can represented inÂ

*p*/*qÂ*form.(iii) 2âˆš7/7âˆš7 = 2/7

Since the number is rational number as it can represented inÂ

*p*/*qÂ*form.(iv) 1/âˆš2 = âˆš2/2 = 0.7071067811...

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

(v) 2Ï€ = 2Â Ã—Â 3.1415â€¦ =Â 6.2830â€¦

Since the number is is non-terminating non-recurring therefore, it is an irrational number.

2. Simplify each of the following expressions:

(i) (3Â +Â âˆš3) (2 +Â âˆš2)

(ii) (3Â +Â âˆš3) (3 - âˆš3)

(iii) (âˆš5Â +Â âˆš2)

(iv) (âˆš5Â - âˆš2) (âˆš5Â +Â âˆš2)

^{2}(iv) (âˆš5Â - âˆš2) (âˆš5Â +Â âˆš2)

**Answer**

(i) (3Â +Â âˆš3) (2 +Â âˆš2)

â‡’ 3Â Ã—Â 2Â +Â 2 +Â âˆš3 + 3âˆš2+ âˆš3 Ã—âˆš2
â‡’ 6Â + 2âˆš3 +3âˆš2 + âˆš6

(ii) (3Â +Â âˆš3) (3 - âˆš3) [âˆµ (

*a*+*b*) (*a*-*b*) =*a*^{2}-*b*^{2}]
â‡’ 3

â‡’ 9 - 3

â‡’ 6^{2}Â - (âˆš3)^{2}â‡’ 9 - 3

(iii) (âˆš5Â +Â âˆš2)

â‡’ (âˆš5)

â‡’ 5Â + 2Â + 2 Ã— âˆš5Ã— 2Â

^{2Â }[âˆµ (*a*Â +Â*b*)^{2}Â =Â*a*^{2}Â +Â*b*^{2}Â + 2*ab*]â‡’ (âˆš5)

^{2}Â + (âˆš2)^{2}Â + 2 Ã—âˆš5Â Ã— âˆš2â‡’ 5Â + 2Â + 2 Ã— âˆš5Ã— 2Â

â‡’ 7Â +2âˆš10

(iv) (âˆš5Â - âˆš2) (âˆš5Â +Â âˆš2) [âˆµ (

(iv) (âˆš5Â - âˆš2) (âˆš5Â +Â âˆš2) [âˆµ (

*a*Â +Â*b*) (*a*Â -Â*b*) =Â*a*^{2}Â -*b*^{2}]
â‡’ (âˆš5)

^{2}Â - (âˆš2)^{2}Â
â‡’ 5 - 2

â‡’ 3

**Answer**

**Answer**

Step 1: Draw a line segment of unit 9.3. Extend it to C so that BC is of 1 unit.

Step 2: Now, AC = 10.3 units. Find the centre of AC and name it as O.

Step 3: Draw a semi circle with radius OC and centre O.

Step 4: Draw a perpendicular line BD to AC at point B which intersect the semicircle at D. Also, Join OD.

Step 5: Now, OBD is a right angled triangle.

Here, OD = 10.3/2 (radius of semi circle), OC = 10.3/2, BC = 1

OB = OC â€“ BC = (10.3/2) â€“ 1 = 8.3/2

Using Pythagoras theorem,

OD

^{2}= BD

^{2}+ OB

^{2}

â‡’Â (10.3/2)

^{2}= BD2 + (8.3/2)

^{2}

â‡’Â BD

^{2}= (10.3/2)

^{2}- (8.3/2)

^{2}

â‡’Â BD

^{2}= (10.3/2 â€“ 8.3/2) (10.3/2 + 8.3/2)

â‡’Â BD

^{2}= 9.3

â‡’Â BD

^{2}= Â âˆš9.3

Thus, the length of BD isÂ âˆš9.3.

Step 6: Taking BD as radius and B as centre draw an arc which touches the line segment. The point where it touches the line segment is at a distance ofÂ âˆš9.3Â from O as shown in the figure.

(i) 1/âˆš7

(ii)Â 1/âˆš7-âˆš6
(iii) 1/âˆš5+âˆš2

(iv)Â 1/âˆš7-2**Answer**

Page No: 26

**Exercise 1.6**

1. Find:

(i) 64

^{1/2}

(ii) 32

^{1/5}

(iii) 125

^{1/3}

**Answer**

2. Find:

(i) 9

^{3/2}
(ii) 32

^{2/5}
(iii) 16

^{3/4}
(iv) 125

^{-1/3}**Answer**

3. Simplify:

(i) 2

^{2/3}.2^{1/5}
(ii) (1/3

^{3})^{7}
(iii) 11

^{1/2}**/**11^{1/4}
(iv) 7

^{1/2}.8^{1/2}

## NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Number Systems is the first chapter of Class 9 Maths textbook in which we study about natural numbers, whole numbers and integers. This chapter details about rational and irrational numbers. A number â€˜râ€™ is called a rational number, if it can be written in the form p/q, where p and q are integers and q â‰ 0.Â

â€¢ Irrational Numbers: A number â€˜sâ€™ is called irrational, if it cannot be written in the form p/q, where p and q are integers and q â‰ 0. The collection of all rational numbers and irrational numbers together make up what we call the collection of real numbers. Every real number is represented by a unique point on the number line.

â€¢Â Real Numbers and their Decimal Expansions: We will look at the decimal expansions of real numbers and see if we can use the expansions to distinguish between rationals and irrationals and also explain how to visualise the representation of real numbers on the number line using their decimal expansions.Â The decimal expansion of a rational number is either terminating or nonterminating recurring.Â The decimal expansion of an irrational number is non-terminating non-recurring

â€¢ Representing Real Numbers on the Number Line: The process of visualisation of representation of numbers on the number line, through a magnifying glass is known as the process of successive magnification. Every real number is represented by a unique point on the number line. Further, every point on the number line represents one and only one real number.

â€¢Â Operations on Real Numbers: We will be dealing with multiple operations like addition, subtraction, multiplication and division of irrational numbers.

â€¢Â Laws of Exponents for Real Numbers: We will use laws of exponents to solve the questions given in the exercise.

There are total 6 exercises in the chapter which will be help you in revising the chapter properly. We have provided

**Class 9 Chapter 1 Maths NCERT Solutions step by step**on Studyrankers. You only need to click on the exercises given below and start clearing your doubts.- Exercise 1.1 Chapter 1 Class 9 Maths NCERT Solutions
- Exercise 1.2 Chapter 1 Class 9 Maths NCERT Solutions
- Exercise 1.3 Chapter 1 Class 9 Maths NCERT Solutions
- Exercise 1.4 Chapter 1 Class 9 Maths NCERT Solutions
- Exercise 1.5 Chapter 1 Class 9 Maths NCERT Solutions
- Exercise 1.6 Chapter 1 Class 9 Maths NCERT Solutions

Studyrankers experts have prepared Chapter 1 Class 9 Maths NCERT Solutions in well manner so a student can easily clear their misunderstandings any time.

### NCERT Solutions for Class 9 Maths Chapters:

**FAQ onÂ**

**Chapter 1 Number Systems**

#### Why we should solve NCERT Solutions for Chapter 1 Number Systems Class 9?

Chapter 1 Number Systems Class 9 Maths is very important for every student as it will help you in understanding what is given in the chapter and understand the concepts. It is very essential to solve every question before moving further to any other supplementary books.

#### Which three integers are equal to their own cube roots?

â€“1, 0 and 1

#### Find the sum ofÂ (3âˆš3 +7âˆš2) and (âˆš3 - 5âˆš2).

We haveÂ (3âˆš3 +7âˆš2) + (âˆš3 - 5âˆš2)

= âˆš3 3+7âˆš2 + âˆš3 - 5âˆš2

= (3âˆš3+âˆš3) + 7âˆš2 - 5âˆš2)

=âˆš3(3+1) + âˆš2(7-2)

=âˆš3(4) + âˆš2(5) = (4âˆš3 + 2âˆš2)

#### How can I download PDF of Chapter 1 Number Systems Class 9 NCERT Solutions?

You can easily download PDF of Class 9 NCERT Solutions Chapter 1 here through which you can easily get what you're looking for.