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NCERT Solutions for Class 9 Maths – Class 9 Maths NCERT Solutions
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NCERT Class 9 Maths Solutions – NCERT Solutions Class 9 Maths
 Chapter 1 Number systems
 Chapter 2 Polynomials
 Chapter 3 Coordinate Geometry
 Chapter 4 Linear Equations in Two Variables
 Chapter 5 Introduction to Euclid Geometry
 Chapter 6 Lines and Angles
 Chapter 7 Triangles
 Chapter 8 Quadrilaterals
 Chapter 9 Areas of Parallelograms and Triangles
 Chapter 10 Circles
 Chapter 11 Constructions
 Chapter 12 Heron’s Formula
 Chapter 13 Surface Areas and Volumes
 Chapter 14 Statistics
 Chapter 15 Probability
 Class 9 Maths (Download PDF)
There are 15 chapters in class 9 maths. These chapters lay a foundation for the chapters that will come in class 10. This pdf is accessible to everyone and they can use this pdf based on their convenience. Here below we are helping you with the overview of each and every chapter appearing in the textbook.
NCERT Solutions for Class 9 Maths Chapter 1
 Class 9 Maths Number systems Exercise 1.1
 Class 9 Maths Number systems Exercise 1.2
 Class 9 Maths Number systems Exercise 1.3
 Class 9 Maths Number systems Exercise 1.4
 Class 9 Maths Number Systems Exercise 1.5
 Class 9 Maths Number Systems Exercise 1.6
 Number Systems Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 2
 Class 9 Maths Polynomials Exercise 2.1
 Class 9 Maths Polynomials Exercise 2.2
 Class 9 Maths Polynomials Exercise 2.3
 Class 9 Maths Polynomials Exercise 2.4
 Class 9 Maths Polynomials Exercise 2.5
 Polynomials Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 3
 Class 9 Maths Coordinate Geometry Exercise 3.1
 Class 9 Maths Coordinate Geometry Exercise 3.2
 Class 9 Maths Coordinate Geometry Exercise 3.3
 Coordinate Geometry Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 4
 Class 9 Maths Linear Equations in Two Variables Exercise 4.1
 Class 9 Maths Linear Equations in Two Variables Exercise 4.2
 Class 9 Maths Linear Equations in Two Variables Exercise 4.3
 Class 9 Maths Linear Equations in Two Variables Exercise 4.4
 Linear Equations for Two Variables Class 9 Extra Questions
 Linear Equations in Two Variables Class 9 Word Problems and Important Questions
NCERT Solutions for Class 9 Maths Chapter 5
 Class 9 Maths Introduction to Euclid Geometry Exercise 5.1
 Chapter 5 Introduction to Euclid’s Geometry Ex 5.2
 Introduction to Euclid’s Geometry Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 6
 Class 9 Maths Lines and Angles Exercise 6.1
 Class 9 Maths Lines and Angles Exercise 6.2
 Class 9 Maths Lines and Angles Exercise 6.3
 Lines and Angles Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 7
 Class 9 Maths Triangles Exercise 7.1
 Class 9 Maths Triangles Exercise 7.2
 Class 9 Maths Triangles Exercise 7.3
 Class 9 Maths Triangles Exercise 7.4
 Chapter 7 Triangles Ex 7.5
 Triangles Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 8
 Class 9 Maths Quadrilaterals Exercise 8.1
 Class 9 Maths Quadrilaterals Exercise 8.2
 Quadrilaterals Class 9 Extra Questions
 Quadrilaterals Class 9 Maths Important Questions
NCERT Solutions for Class 9 Maths Chapter 9
 Class 9 Maths Areas of Parallelograms and Triangles Exercise 9.1
 Class 9 Maths Areas of Parallelograms and Triangles Exercise 9.2
 Class 9 Maths Areas of Parallelograms and Triangles Exercise 9.3
 Chapter 9 Areas of Parallelograms and Triangles Ex 9.4
 Areas of Parallelograms and Triangles Class 9 Extra Questions
NCERT Solutions for Class 9 Maths Chapter 10
 Class 9 Maths Circles Exercise 10.1
 Class 9 Maths Circles Exercise 10.2
 Class 9 Maths Circles Exercise 10.3
 Class 9 Maths Circles Exercise 10.4
 Class 9 Maths Circles Exercise 10.5
 Chapter 10 Circles Ex 10.6
 Circles Class 9 Extra Questions
 Circles Class 9 Maths Important Questions with Answers
NCERT Solutions for Class 9 Maths Chapter 11
 Class 9 Maths Constructions Exercise 11.1
 Class 9 Maths Constructions Exercise 11.2
 Constructions Class 9 Extra Questions
 Class 9 Maths Constructions Important Questions
NCERT Solutions for Class 9 Maths Chapter 12
 Class 9 Maths Heron’s Formula Exercise 12.1
 Class 9 Maths Heron’s Formula Exercise 12.2
 Heron’s Formula Class 9 Extra Questions
 Class 9 Areas of Parallelograms and Triangles Worksheets with Solutions
NCERT Solutions for Class 9 Maths Chapter 13
 Class 9 Maths Surface Areas and Volumes Exercise 13.1
 Class 9 Maths Surface Areas and Volumes Exercise 13.2
 Class 9 Maths Surface Areas and Volumes Exercise 13.3
 Class 9 Maths Surface Areas and Volumes Exercise 13.4
 Class 9 Maths Surface Areas and Volumes Exercise 13.5
 Class 9 Maths Surface Areas and Volumes Exercise 13.6
 Class 9 Maths Surface Areas and Volumes Exercise 13.7
 Class 9 Maths Surface Areas and Volumes Exercise 13.8
 Chapter 13 Surface Areas and Volumes Ex 13.9
 Surface Areas and Volumes Class 9 Extra Questions
 Surface Areas and Volumes Word Problems and Important Questions
NCERT Solutions for Class 9 Maths Chapter 14
 Class 9 Maths Statistics Exercise 14.1
 Class 9 Maths Statistics Exercise 14.2
 Class 9 Maths Statistics Exercise 14.3
 Class 9 Maths Statistics Exercise 14.4
 Statistics Class 9 Extra Questions
 Maths Class 9 Statistics Important Questions with solutions
NCERT Solutions for Class 9 Maths Chapter 15
 Class 9 Maths Probability Exercise 15.1
 Probability Class 9 Extra Questions
 Class 9 Probability Important Questions
Maths NCERT Solutions
In this article, we will provide you all the necessary information regarding Class 9th Maths NCERT Solutions. NCERT Maths Class 9 Textbook Solutions is solved by expert teachers provide you a strong foundation in the subject Maths. The 9th CBSE Maths Solutions are solved keeping various parameters in mind such as stepwise marks, formulas, mark distribution, etc., This in turn, helps you not to lose even a single mark.
It is important to build a strong base in maths. This is one subject that will be useful for every student irrespective of their branch. And thus we are helping you with NCERT Solutions of Class 9 Maths. This pdf can guide you to all the solutions given in the NCERT textbook along with the exercise.
Maths plays a major role in every student’s life. Working on NCERT Solutions Class 9 Maths Notes will not only help you to score good marks in the grade 9 but also helps you to clear the toughest competitive exams like JEE, NEET, JEE Advanced etc., Further it is CBSE Class 9 Maths Solutions will also be helpful to clear the exams like Olympiad, NTSE, through which you can easily avail scholarship and make your education journey hassle free. Read on to find out everything about NCERT Class 9th Maths Solutions to secure colorful marks in CBSE grade 9.
CBSE Class 9 Maths Unit Wise Weightage
UNIT I  Number Systems  8 
UNIT II  Algebra  17 
UNIT III  Coordinate Geometry  4 
UNIT IV  Geometry  28 
UNIT V  Mensuration  13 
UNIT VI  Statistics & Probability  10 
NCERT Solutions for Class 9 Maths PDF Download
Browse all 9th NCERT Maths Solutions from your mobile or desktop and gain more marks in your exams. You can also go through the Chapterwise Important Questions for Class 9 Maths which will help you in extra practice and exams. This consists of 1 mark Questions, 2 Mark Numericals Questions, 3 Marks Numerical Questions, 4 Marks Questions, Word Problems, and previous year questions (VSAQ, SAQ, LAQ, and ValueBased Questions) from all chapters in class 9 maths designed according to CBSE Class 9 Maths Syllabus are laid in a sequential manner will help in scoring more marks in your Board Examinations.
Class 9 Maths Chapter 1 Number Systems
This chapter is an extension of the number line you have studied in the previous standards. You will also get know how to place various types of numbers on the number line in this chapter. A total of 6 exercises in this chapter guides you through the representation of terminating or non terminating of the recurring decimals on the number line. Along with the rational numbers, you will also learn where to put the square roots of 2 and 3 on the number line. There are also laws of rational exponents and Integral powers taught in this chapter.
Class 9 Maths Chapter 2 Polynomials
This chapter guides you through algebraic expressions called polynomial and various terminologies related to it. There is plenty to learn in this chapter about the definition and examples of polynomials, coefficient, degrees, and terms in a polynomial. Different types of polynomials like quadratic polynomials, linear constant, cubic polynomials, factor theorems, factorization theorem are taught in this chapter.
Class 9 Maths Chapter 3 Coordinate Geometry
A total of 3 exercises in this chapter will help you understand coordinate geometry in detail. Along with there are concepts like concepts of a Cartesian plane, terms, and various terms associated with the coordinate plane are learned in this chapter. You will also learn about plotting a point in the XY plane and naming process of this point.
Class 9 Maths Chapter 4 Linear Equations in Two Variables
This chapter will introduce to a new equation, ax + by + c = 0 in two variables. The questions in this chapter will be related to proving that a linear number has infinite solutions, using ba graph to plot linear equation, and justifying any point on a line. A total of 4 exercises are there for your practice and understanding.
Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry
The chapter begins with the introduction of Indian geometry as it has some base in Euclid’s geometry. The Introduction of Euclid’s geometry in this chapter helps you with a process of defining geometrical terms and shapes. There are a total of 2 exercises where you will dwell into the relationship between theorems, postulates, and axioms.
Class 9 Maths Chapter 6 Lines and Angles
This chapter in the NCERT textbook also has 2 exercises in it. There are various theorems on angles and lines in this chapter that can be asked in for proof. The first theorem which will be asked for proof is “If the two lines are intersecting each other, then the vertically opposite angles formed will be equal”. Also, the second proof that is asked is, “The sum of all the angles formed in a triangle is 180°”. There are other theorems also given, but these are based on only these two theorems.
Class 9 Maths Chapter 7 Triangles
The contents in this chapter will help in understanding the congruence of triangles along with the rules of congruence. This chapter also has two theorems in it and a total of 5 exercises for students to practice. These two theorems are given as proof while the other is used in the problems or applications. Besides this, there are many properties of inequalities and triangles in this chapter for students to learn.
Class 9 Maths Chapter 8 Quadrilaterals
This chapter is very interesting for students to learn and there are only 2 exercises in it. The questions in this chapter are related to the properties related to quadrilateral and their combinations with the triangles.
Class 9 Maths Chapter 9 Areas of Triangles and Parallelogram
This chapter is important to understand the meaning of the area with this, the areas of the triangle, parallelogram, and their combinations are asked in this chapter along with their proofs. There are also examples of the an which are used as a proof of theorems in this chapter.
Class 9 Maths Chapter 10 Circles
In this chapter, you will get to learn some interesting topics like equal chords and their distance from the center, the chord of a point and angle subtended by it, angles which are subtended by an arc of a circle, and cyclic quadrilaterals. There are also theorems in this chapter which are helpful to prove questions based on quadrilaterals, triangles, and circles.
Class 9 Maths Chapter 11 Constructions
This chapter will help you learn two different categories of construction. One of them is the construction of a triangle along with its base, difference or sum of the remaining two sides, and one base angle with base angle and parameters are given. The other is the construction of bisectors for the line segments and measuring angles that include 45/60/90, etc.
Class 9 Maths Chapter 12 Heron’s Formula
This chapter joins the long list NCERT chapters that also has 2 exercises in it. In this chapter, you will be learning the concepts that are an extension of concepts related to the area of a triangle. Furthermore, you will get to learn about finding the area of triangles, quadrilaterals, and various types of polygons. Along with the, is there is also knowledge of formula for the plane figures given in the chapter.
Class 9 Maths Chapter 13 Surface Areas and Volume
Every one of you has already studied mensuration in previous standards. Thus, you must be aware of surface areas and this chapter is on that. Along with this, this chapter also has a volume of cubes, cylinders, cuboids, cones, hemispheres, and spheres. Also, in this chapter, you will get to know about the conversion of one figure into another, and comparing volumes of two figures.
Class 9 Maths Chapter 14 Statistics
In this chapter, you will get the knowledge about the descriptive statistics and the collection of data based on different aspects of life. This is useful for interpretation and stating the inferences from the data. This chapter gives the basic knowledge of the collection of data as the data is available in raw form. As you move forward and study 5 exercises you will learn about presenting data in tabular form by keeping them together in regular intervals, polygon, histogram, or bar graph drawing. You will also get to the topics like mean, median, and mode and finding the central tendency with the raw data.
Class 9 Maths Chapter 15 Probability
Probability in this book is based on the observation approach or finding the frequency. Questions in this chapter are very intuitive as they are based on daily life or day to day situations. For example, incidents like throwing dice, coin tossing, the probability for a deck of cards and simple events. If you are curious this chapter can be very interesting for you to learn and understand.
There may be a few times where you feel you are stuck and not getting the desired solutions. This is where we can you with NCERT solutions for class 9 maths. You can use this article as a reference for all the chapters in the NCERT book.
FAQs on NCERT Solutions for Class 9 Maths
1. How do I study for the CBSE Class 9 Maths Solutions?
Practice the CBSE 9th Maths Solutions and try covering all the topics and questions carefully.
2. How could I learn Class 9 maths in an efficient and fast way?
The best way to learn fast is to solve NCERT. NCERT has few questions but has great importance in papers. If you can solve the whole NCERT with examples, you can easily score well. If you ample time try referring to RD Sharma too as it’s the best book.
3. Can I get solved math questions for the Class 9 CBSE?
Yes, you can get solved math questions for Class 9 CBSE Exams from our page. Access the direct links available on our page and download them for free of cost.
4. Which is the best maths guide for 9th CBSE?
NCERT Solutions for Class 9 Maths will help you aid your preparation. Get a good grip over the subject by practicing more and more NCERT Solutions prevailing on our page.
5. How can I download the NCERT Solution Book for the CBSE Class 9 Maths?
Aspirants can download the CBSE Class 9 Maths NCERT Solutions by tapping on the direct links available. Lay a stronger foundation of the concepts by referring to the NCERT Solutions.
6. How long should a student of Class 9 practice math?
It’s not about the time limit. Try practicing as much as you can and revise the complete syllabus of Class 9 Maths for the exams to score well.
Now that you are provided all the necessary information regarding NCERT Solutions for class 9 Maths and we hope this detailed article on Class 9 Maths NCERT Solutions is helpful. If you have any doubt regarding this article or NCERT Class 9 Maths Solutions, leave your comments in the comment section below and we will get back to you as soon as possible.
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 NCERT Class 9
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 Chapter 1: Number Systems
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
Ncert solutions class 9 maths chapter 1 – cbse free pdf download.
Download Exclusively Curated Chapter Notes for Class 9 Maths Chapter – 1 Number Systems
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In NCERT Solutions for Class 9 Maths Chapter 1 , students are introduced to several important topics that are considered to be very crucial for those who wish to pursue Mathematics as a subject in their higher classes. Based on these NCERT Solutions , students can practise and prepare for their upcoming CBSE exams, as well as equip themselves with the basics of Class 10. These Maths Solutions of NCERT Class 9 are helpful as they are prepared with respect to the latest update on the CBSE syllabus for 202324 and its guidelines.
 Chapter 1 Number Systems
 Chapter 2 Polynomials
 Chapter 3 Coordinate Geometry
 Chapter 4 Linear Equations in Two Variables
 Chapter 5 Introduction to Euclids Geometry
 Chapter 6 Lines and Angles
 Chapter 7 Triangles
 Chapter 8 Quadrilaterals
 Chapter 9 Areas of Parallelograms and Triangles
 Chapter 10 Circles
 Chapter 11 Constructions
 Chapter 12 Heron’s Formula
 Chapter 13 Surface Areas and Volumes
 Chapter 14 Statistics
 Chapter 15 Inroduction to Probability
NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems
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Access Answers to NCERT Class 9 Maths Chapter 1 – Number Systems
Exercise 1.1 page: 5.
1. Is zero a rational number? Can you write it in the form p/q where p and q are integers and q ≠ 0?
We know that a number is said to be rational if it can be written in the form p/q , where p and q are integers and q ≠ 0.
Taking the case of ‘0’,
Zero can be written in the form 0/1, 0/2, 0/3 … as well as , 0/1, 0/2, 0/3 ..
Since it satisfies the necessary condition, we can conclude that 0 can be written in the p/q form, where q can either be positive or negative number.
Hence, 0 is a rational number.
2. Find six rational numbers between 3 and 4.
There are infinite rational numbers between 3 and 4.
As we have to find 6 rational numbers between 3 and 4, we will multiply both the numbers, 3 and 4, with 6+1 = 7 (or any number greater than 6)
i.e., 3 × (7/7) = 21/7
and, 4 × (7/7) = 28/7. The numbers in between 21/7 and 28/7 will be rational and will fall between 3 and 4.
Hence, 22/7, 23/7, 24/7, 25/7, 26/7, 27/7 are the 6 rational numbers between 3 and 4.
3. Find five rational numbers between 3/5 and 4/5.
There are infinite rational numbers between 3/5 and 4/5.
To find out 5 rational numbers between 3/5 and 4/5, we will multiply both the numbers 3/5 and 4/5
with 5+1=6 (or any number greater than 5)
i.e., (3/5) × (6/6) = 18/30
and, (4/5) × (6/6) = 24/30
The numbers in between18/30 and 24/30 will be rational and will fall between 3/5 and 4/5.
Hence,19/30, 20/30, 21/30, 22/30, 23/30 are the 5 rational numbers between 3/5 and 4/5
4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Natural numbers Numbers starting from 1 to infinity (without fractions or decimals)
i.e., Natural numbers = 1,2,3,4…
Whole numbers – Numbers starting from 0 to infinity (without fractions or decimals)
i.e., Whole numbers = 0,1,2,3…
Or, we can say that whole numbers have all the elements of natural numbers and zero.
Every natural number is a whole number; however, every whole number is not a natural number.
(ii) Every integer is a whole number.
Integers Integers are set of numbers that contain positive, negative and 0; excluding fractional and decimal numbers.
i.e., integers= {…4,3,2,1,0,1,2,3,4…}
Whole numbers Numbers starting from 0 to infinity (without fractions or decimals)
i.e., Whole numbers= 0,1,2,3….
Hence, we can say that integers include whole numbers as well as negative numbers.
Every whole number is an integer; however, every integer is not a whole number.
(iii) Every rational number is a whole number.
Rational numbers All numbers in the form p/q, where p and q are integers and q≠0.
i.e., Rational numbers = 0, 19/30 , 2, 9/3, 12/7…
All whole numbers are rational, however, all rational numbers are not whole numbers.
Exercise 1.2 Page: 8
1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.
i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….
Real numbers – The collection of both rational and irrational numbers are known as real numbers.
i.e., Real numbers = √2, √5, , 0.102…
Every irrational number is a real number, however, every real number is not an irrational number.
(ii) Every point on the number line is of the form √m where m is a natural number.
The statement is false since as per the rule, a negative number cannot be expressed as square roots.
E.g., √9 =3 is a natural number.
But √2 = 1.414 is not a natural number.
Similarly, we know that there are negative numbers on the number line, but when we take the root of a negative number it becomes a complex number and not a natural number.
E.g., √7 = 7i, where i = √1
The statement that every point on the number line is of the form √m, where m is a natural number is false.
(iii) Every real number is an irrational number.
The statement is false. Real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.
Every irrational number is a real number, however, every real number is not irrational.
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.
No, the square roots of all positive integers are not irrational.
For example,
√4 = 2 is rational.
√9 = 3 is rational.
Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).
3. Show how √5 can be represented on the number line.
Step 1: Let line AB be of 2 unit on a number line.
Step 2: At B, draw a perpendicular line BC of length 1 unit.
Step 3: Join CA
Step 4: Now, ABC is a right angled triangle. Applying Pythagoras theorem,
AB 2 +BC 2 = CA 2
2 2 +1 2 = CA 2 = 5
⇒ CA = √5 . Thus, CA is a line of length √5 unit.
Step 4: Taking CA as a radius and A as a center draw an arc touching
the number line. The point at which number line get intersected by
arc is at √5 distance from 0 because it is a radius of the circle
whose center was A.
Thus, √5 is represented on the number line as shown in the figure.
4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP 1 of unit length (see Fig. 1.9). Now draw a line segment P 2 P 3 perpendicular to OP 2 . Then draw a line segment P 3 P 4 perpendicular to OP 3 . Continuing in Fig. 1.9 :
Constructing this manner, you can get the line segment P n1 Pn by square root spiral drawing a line segment of unit length perpendicular to OP n1 . In this manner, you will have created the points P 2 , P 3 ,….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …
Step 1: Mark a point O on the paper. Here, O will be the center of the square root spiral.
Step 2: From O, draw a straight line, OA, of 1cm horizontally.
Step 3: From A, draw a perpendicular line, AB, of 1 cm.
Step 4: Join OB. Here, OB will be of √2
Step 5: Now, from B, draw a perpendicular line of 1 cm and mark the end point C.
Step 6: Join OC. Here, OC will be of √3
Step 7: Repeat the steps to draw √4, √5, √6….
Exercise 1.3 Page: 14
1. Write the following in decimal form and say what kind of decimal expansion each has :
= 0.36 (Terminating)
= 4.125 (Terminating)
(vi) 329/400
= 0.8225 (Terminating)
2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?
[Hint: Study the remainders while finding the value of 1/7 carefully.]
3. Express the following in the form p/q, where p and q are integers and q 0.
Assume that x = 0.666…
Then,10 x = 6.666…
10 x = 6 + x
(ii) \(\begin{array}{l}0.4\overline{7}\end{array} \)
= (4/10)+(0.777/10)
Assume that x = 0.777…
Then, 10 x = 7.777…
10 x = 7 + x
(4/10)+(0.777../10) = (4/10)+(7/90) ( x = 7/9 and x = 0.777…0.777…/10 = 7/(9×10) = 7/90 )
= (36/90)+(7/90) = 43/90
Assume that x = 0.001001…
Then, 1000 x = 1.001001…
1000 x = 1 + x
4. Express 0.99999…. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Assume that x = 0.9999…..Eq (a)
Multiplying both sides by 10,
10 x = 9.9999…. Eq. (b)
Eq.(b) – Eq.(a), we get
10 x = 9.9999
– x = 0.9999…
_____________
The difference between 1 and 0.999999 is 0.000001 which is negligible.
Hence, we can conclude that, 0.999 is too much near 1, therefore, 1 as the answer can be justified.
5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17 ? Perform the division to check your answer.
Dividing 1 by 17:
There are 16 digits in the repeating block of the decimal expansion of 1/17.
6. Look at several examples of rational numbers in the form p/q (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
We observe that when q is 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 observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both.
7. Write three numbers whose decimal expansions are nonterminating nonrecurring.
We know that all irrational numbers are nonterminating nonrecurring. three numbers with decimal expansions that are nonterminating nonrecurring are:
 √3 = 1.732050807568
 √26 =5.099019513592
 √101 = 10.04987562112
8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.
Three different irrational numbers are:
 0.73073007300073000073…
 0.75075007300075000075…
 0.76076007600076000076…
9. Classify the following numbers as rational or irrational according to their type:
√23 = 4.79583152331…
Since the number is nonterminating and nonrecurring therefore, it is an irrational number.
√225 = 15 = 15/1
Since the number can be represented in p/q form, it is a rational number.
(iii) 0.3796
Since the number,0.3796, is terminating, it is a rational number.
(iv) 7.478478
The number,7.478478, is nonterminating but recurring, it is a rational number.
(v) 1.101001000100001…
Since the number,1.101001000100001…, is nonterminating nonrepeating (nonrecurring), it is an irrational number.
Exercise 1.4 Page: 18
1. Visualise 3.765 on the number line, using successive magnification.
Exercise 1.5 Page: 24
1. Classify the following numbers as rational or irrational:
We know that, √5 = 2.2360679…
Here, 2.2360679…is nonterminating and nonrecurring.
Now, substituting the value of √5 in 2 –√5, we get,
2√5 = 22.2360679… = 0.2360679
Since the number, – 0.2360679…, is nonterminating nonrecurring, 2 –√5 is an irrational number.
(ii) (3 +√23) √23
(3 + √ 23) –√23 = 3+ √ 23–√23
Since the number 3/1 is in p/q form, ( 3 +√23) √23 is rational.
(iii) 2√7/7√7
2√7/7√7 = ( 2/7)× (√7/√7)
We know that (√7/√7) = 1
Hence, ( 2/7)× (√7/√7) = (2/7)×1 = 2/7
Since the number, 2/7 is in p/q form, 2√7/7√7 is rational.
Multiplying and dividing numerator and denominator by √2 we get,
(1/√2) ×(√2/√2)= √2/2 ( since √2×√2 = 2)
We know that, √2 = 1.4142…
Then, √2/2 = 1.4142/2 = 0.7071..
Since the number , 0.7071..is nonterminating nonrecurring, 1/√2 is an irrational number.
We know that, the value of = 3.1415
Hence, 2 = 2×3.1415.. = 6.2830…
Since the number, 6.2830…, is nonterminating nonrecurring, 2 is an irrational number.
2. Simplify each of the following expressions:
(i) (3+√3)(2+√2)
(3+√3)(2+√2 )
Opening the brackets, we get, (3×2)+(3×√2)+(√3×2)+(√3×√2)
= 6+3√2+2√3+√6
(ii) (3+√3)(3√3 )
(3+√3)(3√3 ) = 3 2 (√3) 2 = 93
(iii) (√5+√2) 2
(√5+√2) 2 = √5 2 +(2×√5×√2)+ √2 2
= 5+2×√10+2 = 7+2√10
(iv) (√5√2)(√5+√2)
(√5√2)(√5+√2) = (√5 2 √2 2 ) = 52 = 3
3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter, (say d). That is, π =c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
There is no contradiction. When we measure a value with a scale, we only obtain an approximate value. We never obtain an exact value. Therefore, we may not realize whether c or d is irrational. The value of π is almost equal to 22/7 or 3.142857…
4. Represent (√9.3) on the number line.
Step 1: Draw a 9.3 units long line segment, AB. Extend AB to C such that BC=1 unit.
Step 2: Now, AC = 10.3 units. Let the centre of AC be O.
Step 3: Draw a semicircle of radius OC with centre O.
Step 4: Draw a BD perpendicular to AC at point B intersecting the semicircle at D. Join OD.
Step 5: OBD, obtained, is a right angled triangle.
Here, OD 10.3/2 (radius of semicircle), OC = 10.3/2 , BC = 1
OB = OC – BC
⟹ (10.3/2)1 = 8.3/2
Using Pythagoras theorem,
OD 2 =BD 2 +OB 2
⟹ (10.3/2) 2 = BD 2 +(8.3/2) 2
⟹ BD 2 = (10.3/2) 2 (8.3/2) 2
⟹ (BD) 2 = (10.3/2)(8.3/2)(10.3/2)+(8.3/2)
⟹ BD 2 = 9.3
⟹ BD = √9.3
Thus, the length of BD is √9.3.
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 of √9.3 from O as shown in the figure.
5. Rationalize the denominators of the following:
Multiply and divide 1/√7 by √7
(1×√7)/(√7×√7) = √7/7
(ii) 1/(√7√6)
Multiply and divide 1/(√7√6) by (√7+√6)
= (√7+√6)/(76)
= (√7+√6)/1
(iii) 1/(√5+√2)
Multiply and divide 1/(√5+√2) by (√5√2)
= (√5√2)/(52)
= (√5√2)/3
(iv) 1/(√72)
Multiply and divide 1/(√72) by (√7+2)
1/(√72)×(√7+2)/(√7+2) = (√7+2)/(√72)(√7+2)
= (√7+2)/(74)
Exercise 1.6 Page: 26
64 1/2 = (8×8) 1/2
32 1/5 = (2 5 ) 1/5
(iii)125 1/3
(125) 1/3 = (5×5×5) 1/3
= 5 1 (3×1/3 = 3/3 = 1)
9 3/2 = (3×3) 3/2
= (3 2 ) 3/2
(ii) 32 2/5
32 2/5 = (2×2×2×2×2) 2/5
= (2 5 ) 2⁄5
(iii)16 3/4
16 3/4 = (2×2×2×2) 3/4
= (2 4 ) 3⁄4
(iv) 125 1/3
125 1/3 = (5×5×5) 1/3
= (5 3 ) 1⁄3
3. Simplify :
(i) 2 2/3 ×2 1/5
(ii) (1/3 3 ) 7
(iii) 11 1/2 /11 1/4
11 1/2 /11 1/4 = 11 (1/2)(1/4)
(iv) 7 1/2 ×8 1/2
NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems
As the Number System is one of the important topics in Maths, it has a weightage of 8 marks in Class 9 Maths CBSE exams. On an average three questions are asked from this unit.
 One out of three questions in part A (1 marks).
 One out of three questions in part B (2 marks).
 One out of three questions in part C (3 marks).
This chapter talks about:
 Introduction of Number Systems
 Irrational Numbers
 Real Numbers and their Decimal Expansions
 Representing Real Numbers on the Number Line.
 Operations on Real Numbers
 Laws of Exponents for Real Numbers
List of Exercises in NCERT Solutions for Class 9 Maths Chapter 1:
Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)
Exercise 1.2 Solutions 4 Questions ( 3 long, 1 short)
Exercise 1.3 Solutions 9 Questions ( 9 long)
Exercise 1.4 Solutions 2 Questions ( 2 long)
Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)
Exercise 1.6 Solutions 3 Questions ( 3 long)
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
NCERT Solutions for Class 9 Maths Chapter 1 Number System is the first chapter of Class 9 Maths. The Number System is discussed in detail in this chapter. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers.
The chapter starts with the introduction of Number Systems in section 1.1, followed by two very important topics in sections 1.2 and 1.3
 Irrational Numbers – The numbers which can’t be written in the form of p/q.
 Real Numbers and their Decimal Expansions – Here, you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrational.
Next, it discusses the following topics.
 Representing Real Numbers on the Number Line – In this, the solutions for 2 problems in Exercise 1.4.
 Operations on Real Numbers – Here, you explore some of the operations like addition, subtraction, multiplication and division on irrational numbers.
 Laws of Exponents for Real Numbers – Use these laws of exponents to solve the questions.
Explore more about Number Systems and learn how to solve various kinds of problems only on NCERT Solutions For Class 9 Maths . It is also one of the best academic resources to revise for your CBSE exams.
Key Advantages of NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems
 These NCERT Solutions for Class 9 Maths help you solve and revise the whole CBSE syllabus of Class 9.
 After going through the stepwise solutions given by our subject expert teachers, you will be able to score more marks in the board exams.
 It follows NCERT guidelines.
 It contains all the important questions from the examination point of view.
The faculty have curated the solutions in a lucid manner to improve the problemsolving abilities of the students. For a more clear idea about Number Systems, students can refer to the study materials available at BYJU’S.
 RD Sharma Solutions for Class 9 Maths Number Systems
Disclaimer:
Dropped Topics – 1.4 Representing real numbers on the number line.
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CBSE Class 9 Maths Worksheet Chapter 1 Number System
 Maths Worksheets
 Number System
CBSE Class 9 Maths Worksheet Chapter 1 Number System  Download Free PDF with Solution
When it comes to Maths as a whole, not many people excel in this subject as it is a subject that solely relies on the logical reasoning functions of the brain. That is why the number system can be intimidating to most students. In Chapter 1, Number System for Class 9, students will learn the number system and their types and how to solve the equations.
So, what is the number system, and what does the number system syllabus contain? A number system can be defined as an arithmetic system or practice of writing numbers to express them. It is the mathematical notation for continuously representing numbers of any given set by using a certain set of digits, symbols, or other characters. It offers a unique representation of every number. It signifies the arithmetic and algebraic structure of the given figures, permitting us to carry out mathematical calculations such as addition, subtraction, and division.
All these figures carry their values, which can be determined by looking at the digit, the position in the number, and the base of the number. A number is a mathematical value used to count, measure, or label objects. Regarding the number system, these numbers are used as digits.
With the help of worksheets such as the Number System Class 9 worksheet, Class 9 Maths Chapter 1 worksheet pdf, and worksheet for Class 9 Maths Chapter 1 with solutions and the operations on Real Numbers Class 9 worksheet, students will have a better understanding of what number systems are and how to solve them accurately.
Access Worksheet for Class 9 Maths Number System
1. It is impossible to represent a rational number in decimal form.
Terminating
Non terminating
Repeating or Non Terminating
Nonrepeating or Non terminating
2. Between two rational numbers
There is no rational number.
There is exactly one rational number.
There are infinitely many rational numbers.
There are only rational numbers and no irrational numbers.
3. The product of any two irrational numbers,
is always an irrational number.
is always a rational number.
is always an integer.
can be rational or irrational.
4. Which of the following is irrational?
$\sqrt{81}$
$\dfrac{\sqrt{12}}{\sqrt{3}}$
$\dfrac{\sqrt{4}}{9}$
5. What is the value is $\sqrt{4} \times \sqrt{81}$?
6. Fill in the blanks;
Any two integers are separated by a finite number of others …..
There are an ….. amount of rational numbers between 15 and 18.
X+Y is a rational number if x and y are both ……
Value of $\sqrt[3]{8}$ …….
7. Match the Column:
Column I  Column II 
Value of 1.9999…..  $\dfrac{3}{7}$ 
The Simplest form of a rational number $\dfrac{177}{413}$  Recurring decimal and Non Terminating 
0.36  2 
0.18181818………  Terminating Decimal 
8. Using two irrational numbers as an example:
Product is an irrational number.
Difference is an irrational number.
Division is an irrational number.
9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}\sqrt{6})$.
10. Simplify; $\sqrt[3]{1331}\sqrt{100}+\sqrt{81}$.
11. Calculate the value of $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$.
12. Calculate the $\dfrac{x}{y}$ form of $0.777 . . . . .$, where $\mathbf{x}$ and $\mathbf{y}$ are integers and $\mathbf{y}$ does not equal to zero.
13. Find three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$.
14. The value of $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$.
15. The value of $a^b+b^a$, if $\mathbf{a}=2$ and $\mathbf{b}=3$
16. Simplify; $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$
17. Find the value of $\dfrac{1}{a^b+b^a}$, where $a=5, \mathbf{b}=2$
18. Arrange in ascending order $\sqrt[3]{2}, \sqrt{3}, \sqrt[6]{5} \text {. }$
19. Simplify $(4 \sqrt{5}+3 \sqrt{7})^2$
20. Find the value of a, If $\left(\dfrac{y}{x}\right)^{2a8}=\left(\dfrac{x}{y}\right)^{a1}$.
21. Rationalize the denominators of $\dfrac{1}{\sqrt{7}}$.
22. Recall, $\pi$ is defined as the ratio of circumference (say c) to its diameter (say d). That is $\pi=\dfrac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?
23. Express $0 . \overline{001}$ in the form of $\dfrac{p}{q}$, where $\mathrm{p}$ and $\mathrm{q}$ are integers and $\mathrm{q} \neq 0$.
24. Find five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$
25. Find six rational numbers between 3 and 4.
Answers to the Worksheet:
A rational number cannot have a nonterminating or nonrepeating decimal form.
2. (c)
Between two rational numbers, there are infinitely many rational numbers.
E.g. $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are two rational numbers, then $\dfrac{31}{50} \dfrac{32}{50} \dfrac{33}{50} \dfrac{34}{50} \dfrac{35}{50} \ldots$ are infinite rational number between them.
3. (d)
The product of two irrational numbers can be rational or irrational depending on the two numbers.
For example, $\sqrt{3} \times \sqrt{3}$ is 3 which is a rational number whereas $\sqrt{2} \times \sqrt{4}$ is $\sqrt{8}$ which is an irrational number. As $\sqrt{3}, \sqrt{2}, \sqrt{4}$ are irrational.
Hence, option D is correct.
4. (a) $\sqrt{7}$ is an irrational number.
5. (b)
$\sqrt{4} \times \sqrt{81}$ $= \sqrt{2^2} \times \sqrt{9^2}$ $= 2 \times 9$ = 18
6. Fill in the blanks.
Any two integers are separated by a finite number of other integers .
There are an endless amount of rational numbers between 15 and 18 .
$\mathrm{X}+\mathrm{Y}$ is a rational number if $\mathrm{x}$ and $\mathrm{y}$ are both rational numbers .
Value of $\sqrt[3]{8}$ is $\underline{2}$
7. Match The Column:
Column I  Column II 
Value of 1.9999…..  2 
The Simplest form of a rational number $\dfrac{177}{413}$  $\dfrac{3}{7}$ 
0.36  Terminating Decimal 
0.18181818………  Recurring decimal and Non Terminating 
Explanation:
Explanation  
Value of 1.9999…..  Let, $x=1.999$ …(1) Since only 1 digit is repeating. So, by multiplying $x$ by 10 , we get $10 x=19.99$ …(2) Subtracting equation (1) from (2), we get $9 x=18$ $\Rightarrow x=\dfrac{18}{9}$ $\Rightarrow x=2$ The value of $1.999 \ldots$ in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers an $q \neq 0$, is 2 . 
The Simplest form of a rational number $\dfrac{177}{413}$  $\dfrac{177}{413}$ = $\dfrac{3 \times 59}{7 \times 59}$ 59 will cancel out, therefore, we get = $\dfrac{3}{7}$ 
0.36  A terminating decimal is a decimal, that has an end digit. It is a decimal, which has a finite number of digits(or terms). Hence, 0.36 is terminating decimal. 
0.18181818………  Nonterminating decimals are the one that does not have an end term. Hence, 0.18181818……… is nonterminating decimal. 
8. Given an example of two irrational numbers whose;
Product is an irrational number $\sqrt{6} \times \sqrt{3}=\sqrt{6 \times 3}=\sqrt{18}=3 \sqrt{2}$
Difference is a irrational number $\sqrt{6}\sqrt{3}$ = $\sqrt{3}$
Division is an irrational number $\dfrac{\sqrt{6}}{\sqrt{3}}=\sqrt{\dfrac{6}{3}}=\sqrt{2}$
9. Simplify; $(\sqrt{5}+\sqrt{6})(\sqrt{5}\sqrt{6})$
We know that, $(a+b)(ab)=a^2b^2$
= $\left((\sqrt{5})^2(\sqrt{6})^2\right)$
10. $\sqrt[3]{1331}\sqrt{100}+\sqrt{81}$
= $\sqrt[3]{11^3}\sqrt{10^2}+\sqrt{9^2}$
= $1110+9$
11. $\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}$
$\dfrac{11^{\dfrac{1}{2}}}{11^{\dfrac{1}{4}}}=11^{\dfrac{1}{2}\dfrac{1}{4}}$
$=11^{\dfrac{21}{4}}$
$=11^{\dfrac{1}{4}}$
12. Let,
$p= 0.777…$ .... (1)
Multiply both side in above equation 10
Then,
$10p= 7.777…$ ….(2)
Subtracting equation (1) from (2), we get;
$10pp= 7.777…  0.777…$
$p= \dfrac{7}{9}$
13. Three rational number between $\dfrac{9}{11}$ and $\dfrac{5}{11}$
Rational number of $\dfrac{9}{11}$ and $\dfrac{5}{11}$ is denominator same
$= \dfrac{9}{11}, \dfrac{8}{11}, \dfrac{7}{11}, \dfrac{6}{11}, \dfrac{5}{11}$
14. $\dfrac{\sqrt{8}+\sqrt{12}}{\sqrt{32}+\sqrt{48}}$
$= \dfrac{\sqrt{2^3}+\sqrt{4 \times 3}}{\sqrt{8 \times 4}+\sqrt{8 \times 6}}$
$= \dfrac{2 \sqrt{2}+2 \sqrt{3}}{4 \sqrt{2}+4 \sqrt{3}}$
$= \dfrac{2(\sqrt{2}+\sqrt{3})}{4(\sqrt{2}+\sqrt{3})}$
$= \dfrac{(\sqrt{2}+\sqrt{3})}{2(\sqrt{2}+\sqrt{3})}$
$= \dfrac{1}{2}$
15. If $a=2$ and $b=3$
The value of $a^b+b^a$
$= 2^3+3^2$
16. $2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}$
$2^{\dfrac{2}{3}} \cdot 2^{\dfrac{1}{5}}=2^{\dfrac{2}{3}+\dfrac{1}{5}} \quad \because a^p \cdot a^q=a^{p+q}$
$=2^{\dfrac{10+3}{15}}$
$=2^{\dfrac{13}{15}}$
17. Value of $\dfrac{1}{a^b+b^a}$, where $a=5, b=2$
$= \dfrac{1}{5^2+2^5}$
$= \dfrac{1}{25+32}$
$= \dfrac{1}{57}$
18. Here we have : $\sqrt[3]{2}, \sqrt{3}, \sqrt[5]{5}$
We can also write the expression in simpler form as follows:
$2^{\dfrac{1}{3}}, 3^{\dfrac{1}{2}}, 5^{\dfrac{1}{6}}$
Now we can see that in the denominators of the exponents we have: $3,2,6$
We will now take the LCM of $3,2,6$, which is 6 .
Now we will make all the denominators equal to 6 , so we have to multiply by the multiples in both numerator and denominator.
We can write the numbers as:
$2^{\dfrac{1}{3}} \times \dfrac{2}{2}=2^{\dfrac{2}{6}}$
For the second number we can write:
$3 \dfrac{1}{2} \times \dfrac{3}{3}=3 \dfrac{3}{6}$
Since in the third number we already have the desired denominator, so the third number is
$5^{\dfrac{1}{6}}$
Now we will again write the numbers in the root under, but we have to keep in mind that the numerator will turn as the exponential powers inside the root.
So we have the numbers as:
$\sqrt[6]{2^2}, \sqrt[5]{3^3}, \sqrt[5]{5}$
We will simplify the values inside the root, so we have:
$\sqrt[5]{4}, \sqrt[6]{27}, \sqrt[5]{5}$
From this we can write the smaller value in the front and then the larger value:
$\sqrt[5]{4}, \sqrt[6]{5}, \sqrt[5]{27}$
Hence the original numbers in ascending form are:
$\sqrt[3]{2}, \sqrt[6]{5}, \sqrt{3}$
19. $(4 \sqrt{5}+3 \sqrt{7})^2$
We know that,
$(a+b)^2=a^2+b^2+2 a b$
$=(4 \sqrt{5})^2+(3 \sqrt{7})^2+2 \times (4 \sqrt{5}) (3 \sqrt{7})$
$=80+63+24 \sqrt{5 \times 7}$
$=143+24 \sqrt{35}$
20. $\left(\dfrac{y}{x}\right)^{2 a8}=\left(\dfrac{x}{y}\right)^{a1}$
$\left(\dfrac{y}{x}\right)^{2 a8}=\left(\dfrac{x}{y}\right)^{82 a}$ $ \because (x)^{a}=\dfrac{1}{x^a}$
$\left(\dfrac{x}{y}\right)^{82 a}=\left(\dfrac{x}{y}\right)^{a1}$
When the bases of both sides of an equation are the same, then their exponents are also equal.
$\Rightarrow 82 a=a1$
$\Rightarrow 2 a+a=8+1$
$\Rightarrow 3 a=9$
$\Rightarrow a=\dfrac{9}{3}$
$\Rightarrow a=3$
21. $\dfrac{1}{\sqrt{7}}=\dfrac{1}{\sqrt{7}} \times \dfrac{\sqrt{7}}{\sqrt{7}}$
(Dividing and multiplying by $\sqrt{7}$ )
$=\dfrac{\sqrt{7}}{7}$
22. Writing $\pi$ as $\dfrac{22}{7}$ is only an approximate value and so we can't conclude that it is in the form of a rational. In fact, the value of $\pi$ is calculating as nonterminating, nonrecurring decimal as $\pi=3.14159$ Whereas
If we calculate the value of $\dfrac{22}{7}$ it gives $3.142857$ and hence $\pi \neq \dfrac{22}{7}$
In conclusion $\pi$ is an irrational number.
23. Let $x=0.001001 \ldots \ldots$ (1)
Since 3 digits are repeated multiply both the sides of (1) by 1000
$1000 x=1.001001 \ldots$
$1000 x=1+0.001001 \ldots$
$1000 x=1+x$
$1000 xx=1$
$x=\dfrac{1}{999}$
$\therefore 0 . \overline{001}=\dfrac{1}{999}$
24. Since we make the denominator the same first, then
$\dfrac{3}{4}=\dfrac{3 \times 5}{4 \times 5}=\dfrac{15}{20}$
$\dfrac{4}{5}=\dfrac{4 \times 4}{5 \times 4}=\dfrac{16}{20}$
Now we need to find 5 rational no.
$\dfrac{15}{20} =\dfrac{15 \times 6}{20 \times 6}=\dfrac{90}{120}$
$\dfrac{16}{20}=\dfrac{16 \times 6}{20 \times 6}=\dfrac{96}{120}$
$\therefore$ Five rational numbers between $\dfrac{3}{4}$ and $\dfrac{4}{5}$ are $\dfrac{91}{120}, \dfrac{92}{120}, \dfrac{93}{120}, \dfrac{94}{120}$ and $\dfrac{95}{120}$
25. We can find any number of rational numbers between two rational numbers. First of all, we make the denominators same by multiplying or dividing the given rational numbers by a suitable number. If denominator is already same then depending on number of rational no. we need to find in question, we add one and multiply the result by numerator and denominator.
$3=\dfrac{3 \times 7}{7} \text { and } \quad 4=\dfrac{4 \times 7}{7}$
$3=\dfrac{21}{7} \quad \text { and } \quad 4=\dfrac{28}{7}$
We can choose 6 rational numbers as: $\dfrac{22}{7}, \dfrac{23}{7}, \dfrac{24}{7}, \dfrac{25}{7}, \dfrac{26}{7}$ and $\dfrac{27}{7}$
Benefits of Learning Number System in Class 9 Chapter 1 Maths Worksheet The Class 9 Maths Chapter 1 worksheet pdf contains more than enough material to help students better understand what number systems are and how to solve them. The worksheets come with extensive questions, attempting to clear any doubts the students might have about the number system and their types.
The Maths assignment for Class 9 Number System list of questions and answers provide thorough insights on the topic’s resources and offers easy tricks to identify quicker ways to solve the questions faster while also being more aware and making sure students don’t go wrong or commit any silly mistakes in their solutions.
All of these worksheets have been developed by the best mathematicians and experienced arithmetic representatives who are very aware of the needs and requirements of the students of Class 9.
Examples of Usage of Number System for Class 9
These are a few examples of Maths assignments for Class 9 Number System exercises’ examples :
Answer the following.
Find two irrational numbers and two rational numbers between 0.7 and 0.77.
Every integer is not a whole number. True or false?
Find at least 7 rational numbers between 2 and 9.
Write down 4567 in the decimal and binary number systems.
Is 0 a rational number? State your reasons based on your answer.
Interesting Facts About Number System for Class 9
There are nine types of number systems in mathematics. They are :
Natural numbers
Whole numbers
Rational numbers
Irrational numbers
Real numbers
Imaginary numbers
Prime and composite numbers
Natural numbers are the root forms of numbers between 0 to infinity. They are also named “positive numbers” or “counting numbers” and are represented by the symbol N. (1, 2, 3, 4, 5 and so on)
Whole numbers are natural numbers, with the only difference being the inclusion of 0. They are represented by the symbol W. (0, 1, 2, 3, 4, 5 and so on)
Integers contain whole numbers and the negative values of natural numbers and don’t include fractions, so their numbers can’t be written in the “a/b” format. It ranges from infinity at the negative end to infinity at the positive end, including 0 and is represented by Z. (...3, 2, 1, 0, 1, 2, 3… and so on)
Fractions are numbers written in the “a/b” format, where “a” (numerator) is a whole number, and “b” (denominator) is a natural number. Hence, the denominator can never be 0. (2/4, 0/10, 5/7, etc.)
Rational numbers can be written in fractions where “a” and “b” are both integers and b ≠ 0. All fractions are rational numbers, but all rational numbers are not fractions.(5/9, 3/9, 8/14, etc)
Irrational numbers are numbers that can’t be written in fractional forms. (√8, √.127, √3.209, etc)
Real numbers can be written in decimals, including whole numbers, integers, fractions, etc. All integers belong to real numbers, but not all real numbers belong to integers. (1.25, 0.467, 8.9, etc.)
Imaginary numbers are not real numbers, resulting in negative numbers when squared or put together. They are also named complex numbers and are represented by the symbol i. (√3, √16, √1, etc.)
Numbers that don’t have other factors except 1 are called prime numbers, and the rest of the numbers  except 0  are called composite numbers, as 0 is neither a prime nor a composite number. ( 2, 3, 5… are prime numbers whereas 4, 6, 8… are composite numbers)
Other definitions and elaborated explanations will be provided in the operations on real numbers class 9 worksheet pdf and more .
Important Topics for Class 9 Number System
The important topics students will have to learn in the number system syllabus for Class 9 are as follows :
What are number systems, and how to solve them?
What are the four types of number systems?
How to convert one number system to another number system.
Solving the problems and choosing the correct answer.
Various other exercises in the Number System Class 9 worksheet
What does the PDF Consist of?
Most schools have syllabuses that don’t include just spoonfeeding the information to the students.
It only means that the students must learn by themselves, with the teachers guiding and aiding them throughout their learning process.
With technology being part of most school curriculums, a huge part of their assignments, tests, and worksheets are online, favoured as pdfs.
Vedantu’s pdf format is highly soughtafter as it is used for creating, editing, highlighting, saving, and sharing content.
The worksheet for Class 9 Maths Chapter 1 with Solutions pdfs is free for download at Vedantu’s website.
Rest assured, all the worksheets adhere to the CBSE guidelines' strict, updated, and revised rules.
Many other Number Systems Class 9 worksheet pdfs are present at Vedantu’s platform, created by their own arithmetic subject matter experts, ensuring that the students receive the best training and exercises needed to test their skills and excel in their examinations.
FAQs on CBSE Class 9 Maths Worksheet Chapter 1 Number System
1. Which number system is frequently used?
The decimal number system is the most widely used.
2. How are the values of various figures calculated?
All these figures carry their values, and these values can be determined by:
looking at the digit
the position in the number
the base of the number.
3. Can rational numbers be whole numbers?
No, since rational numbers may be fractional and whole numbers are not.
NCERT Books
NCERT EXEMPLAR BOOK CLASS 9 Maths
Cbse ncert exemplar book class 9 maths.
Mathematics can be easier if the students are strong with the fundamentals. Thus, NCERT Exemplar Book Class 9 Maths comes to the rescue. Firstly, the practice book will help the students progress in the mathematics subject. Secondly, The conceptual sums can be easily decoded if the students spend enough in time solving the NCERT Exemplar Book Class 9 Maths. The questions are varied: from multiplechoice to twisted. The conceptual sums of NCERT Exemplar Books help the student increase his or her analytical skills.
NCERT Exemplar Maths Book Class 9 will help the students to be thorough with formulas and theories. In fact, the books will help them to solve the questions in more than one way. More importantly, students will also get to know the shortcuts and tips when it comes to solving twisted problems. Each chapter is described in details. Therefore, it provides an opportunity for the students to have more focus on the application of concepts, rather than learning the concepts.
In total, there are 14 chapters in NCERT Exemplar Book Class 9 Maths. It covers the basics and fundamentals of mathematics on all topics. In addition, it has added information on a higher level for various competitive examinations, such as NTSE, NIMO, etc. NCERT Exemplar Book Class 9 Maths prepare the students for competitive exams like JEE Main, JEE Advanced, NEET etc because they are specifically designed for entrance exams. In conclusion, the NCERT Exemplar Book Class 9 Maths will make everyone fall in love with Mathematics.
Download NCERT Exemplar Book Class 9 Maths PDF
We have provided free download links to the NCERT Exemplar Book Class 9 Maths questions. Click on the required link to download the NCERT Exemplar Maths book for class 9 in PDF Format.
1  Number Systems  
2  Polynomials  
3  Coordinate Geometry  
4  Linear Equation in Two Variable  
5  Introduction to Euclid’s Geometry  
6  Lines and Angles  
7  Triangles  
8  Quadrilaterals  
9  Areas of Parallelograms and Triangles  
10  Circles  
11  Constructions  
12  Heron’s Formula  
13  Surface Area and Volumes  
14  Statistics and Probability 
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Chapter 8 – Linear Graphs and Their Applications
Exercise 8.1, exercise 8.2, exercise 8.3, review exercise, multiple choice questions, this post has 6 comments.
Nice 👍 Notes
bhi bohth achay notes hai main zyda is ko follow kerta hon
Good effort 🙂
ap please 7 chapter ke notes post karden please
very good notes
I like these notes
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Class 9 Assignments Download Pdf
Please refer to Assignments for Class 9 for all subjects given below. We have given below free printable Assignments for Class 9 for easy and free download in PDF. The assignments have been developed based on the current year’s NCERT Books for Class 9. These Assignments for Grade 9 cover all critical topics which can come in your standard 9 class tests and exams. Free printable Assignments for CBSE Class 9 , school and class worksheets, and practice test papers have been developed by our experienced class 9 teachers. You can free download CBSE NCERT printable assignments for Class 9 with solutions and answers. All assignments and test sheets have been prepared by expert faculty as per the latest curriculum in Class 9. Students can simply click on the links below and download all Pdf assignments for class 9 for free. Latest Kendriya Vidyalaya Class 9 assignments and test papers are provided below.
Assignments for Class 9 Pdf Download
We have provided the largest collection of free CBSE NCERT KVS Assignments for Class 9 . You can download all free assignments in Pdf for standard 9th. Our expert faculty members have covered Class 9 questions and answers for all subjects as per the latest curriculum for the current academic year. All practice test papers and question banks for Class 9 subjects and CBSE Assignments for Class 9 will be really useful for students to properly prepare for the upcoming exams. Class 9th students are advised to free download in Pdf all printable question banks given below.
Subjectwise Assignments for Class 9 for Download in Pdf
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 By clicking on the links above you can access the largest collection of assignments for Grade 9, you will be able to find critical and scoring questions which can come in your exams and school tests.
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We provide here Standard 9 subjectwise assignments which can be easily downloaded in Pdf format for free. Our teachers have provided these Grade 9 test sheets for all subjects given in your books.
You can click on the links above and get assignments for all subjects in Grade 9, all subject and topicwise question banks with solutions have been provided here. You can click on the links to download in Pdf.
We have provided here subjectwise Grade 9 question banks, revision notes and questions for all difficult topics, and other study material. You can download it all without any charge by clicking on the links provided above.
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All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Class 9. Students can click on the links below and download all Pdf Assignments for Mathematics class 9 for free. All latest Kendriya Vidyalaya Class 9 Mathematics Assignments with Answers and test papers are given below.
Free PDF download of Extra Questions for CBSE Class 9 Maths prepared by expert Mathematics teachers from the latest edition of CBSE (NCERT) books.
NCERT Solutions for class 9 Maths solved by expert LearnCBSE.in Maths teachers from latest edition books and as per NCERT (CBSE) guidelines. This consists of CBSE Class 9 Maths Syllabus, Extra Questions and Important Questions Solved.
APT. R 6: LINES AND ANGLESVERY SHORT ANSWER TYPE QUESTIONSQ1. If the complement of an angle is equal to the supplement o. fo. r times the angle, then find the measure. of the a. gle. Q2. In a ∆ABC, LA +LB= 110°, LC+LA=135°. Find LA.Q3. th. given figure, what value of x will.
In the following equations , examine whether x, y and z represents rational or irrational number i) x3 = 27 ii) y2 = 7 iii) z2 = 0.16
Download CBSE NCERT MCQ Questions for Class 9 Maths with Answers Chapterwise PDF for free of cost and take your preparation to next level.
NCERT Solutions Class 9 Maths Chapter 1  CBSE Free PDF Download NCERT Solutions for Class 9 Maths Chapter 1 Number Systems are created by the expert faculty at BYJU'S. The Solutions of NCERT Maths help students in solving problems adroitly and efficiently for the board exams.
Mathematics Assignment for Class IX Ch 1, Number System strictly according to the CBSE syllabus. Extra questions based on the topic Number System.
Free PDF Download of NCERT Maths Book Class 9 in English. Students can also download the NCERT Textbooks Solutions in PDF for Class 9 Maths Book.
All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Quadrilateral Class 9. Students can click on the links below and download all Pdf Assignments for Mathematics Quadrilateral class 9 for free.
CBSE Class 9 Maths Worksheet Chapter 1 Number System  Understand main concepts, their definition, examples and applications. Download free PDF with solutions curated by our experts to practice and prepare yourself for upcoming examination.
NCERT Exemplar Book Class 9 Maths questions are provided by us. Download NCERT Exemplar Book Class 9 Mathematics in PDF format.
Free Resources for CBSE Class 9 offered by Oswaal Books and Learning Pvt Ltd. Get Revision notes, Syllabus, and Self Assessment papers for all subjects.
All Assignments and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Polynomials Class 9. Students can click on the links below and download all Pdf Assignments for Mathematics Polynomials class 9 for free.
Download Class 9 Maths, Chapter 8 Notes, Linear Graphs and Their Applications that contains Solutions of All Exercises, Review Exercises, MCQ's in PDF for free.
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These Assignments for Grade 9 cover all critical topics which can come in your standard 9 class tests and exams. Free printable Assignments for CBSE Class 9, school and class worksheets, and practice test papers have been developed by our experienced class 9 teachers.