case study questions for number system class 9

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CBSE Class 9 Maths Case Study Questions PDF Download

Download Class 9 Maths Case Study Questions to prepare for the upcoming CBSE Class 9 Exams 2023-24. These Case Study and Passage Based questions are published by the experts of CBSE Experts for the students of CBSE Class 9 so that they can score 100% in Exams.

Case study questions play a pivotal role in enhancing students’ problem-solving skills. By presenting real-life scenarios, these questions encourage students to think beyond textbook formulas and apply mathematical concepts to practical situations. This approach not only strengthens their understanding of mathematical concepts but also develops their analytical thinking abilities.

Table of Contents

CBSE Class 9th MATHS: Chapterwise Case Study Questions

Inboard exams, students will find the questions based on assertion and reasoning. Also, there will be a few questions based on case studies. In that, a paragraph will be given, and then the MCQ questions based on it will be asked. For Class 9 Maths Case Study Questions, there would be 5 case-based sub-part questions, wherein a student has to attempt 4 sub-part questions.

Chapterwise Case Study Questions of Class 9 Maths

• Case Study Questions for Chapter 1 Number System
• Case Study Questions for Chapter 2 Polynomials
• Case Study Questions for Chapter 3 Coordinate Geometry
• Case Study Questions for Chapter 4 Linear Equations in Two Variables
• Case Study Questions for Chapter 5 Introduction to Euclid’s Geometry
• Case Study Questions for Chapter 6 Lines and Angles
• Case Study Questions for Chapter 7 Triangles
• Case Study Questions for Chapter 8 Quadrilaterals
• Case Study Questions for Chapter 9 Areas of Parallelograms and Triangles
• Case Study Questions for Chapter 10 Circles
• Case Study Questions for Chapter 11 Constructions
• Case Study Questions for Chapter 12 Heron’s Formula
• Case Study Questions for Chapter 13 Surface Area and Volumes
• Case Study Questions for Chapter 14 Statistics
• Case Study Questions for Chapter 15 Probability

Checkout: Class 9 Science Case Study Questions

And for mathematical calculations, tap Math Calculators which are freely proposed to make use of by calculator-online.net

The above  Class 9 Maths Case Study Question s will help you to boost your scores as Case Study questions have been coming in your examinations. These CBSE Class 9 Maths Case Study Questions have been developed by experienced teachers of cbseexpert.com for the benefit of Class 10 students.

Class 9 Maths Syllabus 2023-24

UNIT I: NUMBER SYSTEMS

1. REAL NUMBERS (18 Periods)

1. Review of representation of natural numbers, integers, and rational numbers on the number line. Rational numbers as recurring/ terminating decimals. Operations on real numbers.

2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number.

3. Definition of nth root of a real number.

4. Rationalization (with precise meaning) of real numbers of the type

(and their combinations) where x and y are natural number and a and b are integers.

5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.)

UNIT II: ALGEBRA

1. POLYNOMIALS (26 Periods)

Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Verification of identities:

RELATED STORIES

and their use in factorization of polynomials.

2. LINEAR EQUATIONS IN TWO VARIABLES (16 Periods)

Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c=0.Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them and showing that they lie on a line.

UNIT III: COORDINATE GEOMETRY COORDINATE GEOMETRY (7 Periods)

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

UNIT IV: GEOMETRY

1. INTRODUCTION TO EUCLID’S GEOMETRY (7 Periods)

History – Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example: (Axiom)

1. Given two distinct points, there exists one and only one line through them. (Theorem)

2. (Prove) Two distinct lines cannot have more than one point in common.

2. LINES AND ANGLES (15 Periods)

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180O and the converse.

2. (Prove) If two lines intersect, vertically opposite angles are equal.

3. (Motivate) Lines which are parallel to a given line are parallel.

3. TRIANGLES (22 Periods)

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).

2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence)

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

4. QUADRILATERALS (13 Periods)

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

5. CIRCLES (17 Periods)

1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.

2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord.

3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.

4. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.

5. (Motivate) Angles in the same segment of a circle are equal.

6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.

7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

UNIT V: MENSURATION 1.

1. AREAS (5 Periods)

Area of a triangle using Heron’s formula (without proof)

2. SURFACE AREAS AND VOLUMES (17 Periods)

Surface areas and volumes of spheres (including hemispheres) and right circular cones.

UNIT VI: STATISTICS & PROBABILITY

STATISTICS (15 Periods)

 Bar graphs, histograms (with varying base lengths), and frequency polygons.

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Benefits of Practicing CBSE Class 9 Maths Case Study Questions

Regular practice of CBSE Class 9 Maths case study questions offers several benefits to students. Some of the key advantages include:

• Deeper Understanding : Case study questions foster a deeper understanding of mathematical concepts by connecting them to real-world scenarios. This improves retention and comprehension.
• Practical Application : Students learn to apply mathematical concepts to practical situations, preparing them for real-life problem-solving beyond the classroom.
• Critical Thinking : Case study questions require students to think critically, analyze data, and devise appropriate solutions. This nurtures their critical thinking abilities, which are valuable in various academic and professional domains.
• Exam Readiness : By practicing case study questions, students become familiar with the question format and gain confidence in their problem-solving abilities. This enhances their readiness for CBSE Class 9 Maths exams.
• Holistic Development: Solving case study questions cultivates not only mathematical skills but also essential life skills like analytical thinking, decision-making, and effective communication.

Tips to Solve CBSE Class 9 Maths Case Study Questions Effectively

Solving case study questions can be challenging, but with the right approach, you can excel. Here are some tips to enhance your problem-solving skills:

• Read the case study thoroughly and understand the problem statement before attempting to solve it.
• Identify the relevant data and extract the necessary information for your solution.
• Break down complex problems into smaller, manageable parts to simplify the solution process.
• Apply the appropriate mathematical concepts and formulas, ensuring a solid understanding of their principles.
• Clearly communicate your solution approach, including the steps followed, calculations made, and reasoning behind your choices.
• Practice regularly to familiarize yourself with different types of case study questions and enhance your problem-solving speed.Class 9 Maths Case Study Questions

Remember, solving case study questions is not just about finding the correct answer but also about demonstrating a logical and systematic approach. Now, let’s explore some resources that can aid your preparation for CBSE Class 9 Maths case study questions.

Q1. Are case study questions included in the Class 9 Maths Case Study Questions syllabus?

Yes, case study questions are an integral part of the CBSE Class 9 Maths syllabus. They are designed to enhance problem-solving skills and encourage the application of mathematical concepts to real-life scenarios.

Q2. How can solving case study questions benefit students ?

Solving case study questions enhances students’ problem-solving skills, analytical thinking, and decision-making abilities. It also bridges the gap between theoretical knowledge and practical application, making mathematics more relevant and engaging.

Q3. How do case study questions help in exam preparation?

Case study questions help in exam preparation by familiarizing students with the question format, improving analytical thinking skills, and developing a systematic approach to problem-solving. Regular practice of case study questions enhances exam readiness and boosts confidence in solving such questions.

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CBSE Class 9 Mathematics Case Study Questions

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If you’re looking for a comprehensive and reliable study resource and case study questions for class 9 CBSE, myCBSEguide is the perfect door to enter. With over 10,000 study notes, solved sample papers and practice questions, it’s got everything you need to ace your exams. Plus, it’s updated regularly to keep you aligned with the latest CBSE syllabus . So why wait? Start your journey to success with myCBSEguide today!

Significance of Mathematics in Class 9

Mathematics is an important subject for students of all ages. It helps students to develop problem-solving and critical-thinking skills, and to think logically and creatively. In addition, mathematics is essential for understanding and using many other subjects, such as science, engineering, and finance.

CBSE Class 9 is an important year for students, as it is the foundation year for the Class 10 board exams. In Class 9, students learn many important concepts in mathematics that will help them to succeed in their board exams and in their future studies. Therefore, it is essential for students to understand and master the concepts taught in Class 9 Mathematics .

Case studies in Class 9 Mathematics

A case study in mathematics is a detailed analysis of a particular mathematical problem or situation. Case studies are often used to examine the relationship between theory and practice, and to explore the connections between different areas of mathematics. Often, a case study will focus on a single problem or situation and will use a variety of methods to examine it. These methods may include algebraic, geometric, and/or statistical analysis.

Example of Case study questions in Class 9 Mathematics

The Central Board of Secondary Education (CBSE) has included case study questions in the Class 9 Mathematics paper. This means that Class 9 Mathematics students will have to solve questions based on real-life scenarios. This is a departure from the usual theoretical questions that are asked in Class 9 Mathematics exams.

The following are some examples of case study questions from Class 9 Mathematics:

Class 9 Mathematics Case study question 1

There is a square park ABCD in the middle of Saket colony in Delhi. Four children Deepak, Ashok, Arjun and Deepa went to play with their balls. The colour of the ball of Ashok, Deepak,  Arjun and Deepa are red, blue, yellow and green respectively. All four children roll their ball from centre point O in the direction of   XOY, X’OY, X’OY’ and XOY’ . Their balls stopped as shown in the above image.

Answer the following questions:

Answer Key:

Class 9 Mathematics Case study question 2

• Now he told Raju to draw another line CD as in the figure
• The teacher told Ajay to mark  ∠ AOD  as 2z
• Suraj was told to mark  ∠ AOC as 4y
• Clive Made and angle  ∠ COE = 60°
• Peter marked  ∠ BOE and  ∠ BOD as y and x respectively

Now answer the following questions:

• 2y + z = 90°
• 2y + z = 180°
• 4y + 2z = 120°
• (a) 2y + z = 90°

Class 9 Mathematics Case study question 3

• (a) 31.6 m²
• (c) 513.3 m³
• (b) 422.4 m²

Class 9 Mathematics Case study question 4

How to Answer Class 9 Mathematics Case study questions

To crack case study questions, Class 9 Mathematics students need to apply their mathematical knowledge to real-life situations. They should first read the question carefully and identify the key information. They should then identify the relevant mathematical concepts that can be applied to solve the question. Once they have done this, they can start solving the Class 9 Mathematics case study question.

Students need to be careful while solving the Class 9 Mathematics case study questions. They should not make any assumptions and should always check their answers. If they are stuck on a question, they should take a break and come back to it later. With some practice, the Class 9 Mathematics students will be able to crack case study questions with ease.

Class 9 Mathematics Curriculum at Glance

At the secondary level, the curriculum focuses on improving students’ ability to use Mathematics to solve real-world problems and to study the subject as a separate discipline. Students are expected to learn how to solve issues using algebraic approaches and how to apply their understanding of simple trigonometry to height and distance problems. Experimenting with numbers and geometric forms, making hypotheses, and validating them with more observations are all part of Math learning at this level.

The suggested curriculum covers number systems, algebra, geometry, trigonometry, mensuration, statistics, graphing, and coordinate geometry, among other topics. Math should be taught through activities that include the use of concrete materials, models, patterns, charts, photographs, posters, and other visual aids.

CBSE Class 9 Mathematics (Code No. 041)

Class 9 Mathematics question paper design

The CBSE Class 9 mathematics question paper design is intended to measure students’ grasp of the subject’s fundamental ideas. The paper will put their problem-solving and analytical skills to the test. Class 9 mathematics students are advised to go through the question paper pattern thoroughly before they start preparing for their examinations. This will help them understand the paper better and enable them to score maximum marks. Refer to the given Class 9 Mathematics question paper design.

QUESTION PAPER DESIGN (CLASS 9 MATHEMATICS)

Mycbseguide: blessing in disguise.

Class 9 is an important milestone in a student’s life. It is the last year of high school and the last chance to score well in the CBSE board exams. myCBSEguide is the perfect platform for students to get started on their preparations for Class 9 Mathematics. myCBSEguide provides comprehensive study material for all subjects, including practice questions, sample papers, case study questions and mock tests. It also offers tips and tricks on how to score well in exams. myCBSEguide is the perfect door to enter for class 9 CBSE preparations.

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14 thoughts on “CBSE Class 9 Mathematics Case Study Questions”

This method is not easy for me

aarti and rashika are two classmates. due to exams approaching in some days both decided to study together. during revision hour both find difficulties and they solved each other’s problems. aarti explains simplification of 2+ ?2 by rationalising the denominator and rashika explains 4+ ?2 simplification of (v10-?5)(v10+ ?5) by using the identity (a – b)(a+b). based on above information, answer the following questions: 1) what is the rationalising factor of the denominator of 2+ ?2 a) 2-?2 b) 2?2 c) 2+ ?2 by rationalising the denominator of aarti got the answer d) a) 4+3?2 b) 3+?2 c) 3-?2 4+ ?2 2+ ?2 d) 2-?3 the identity applied to solve (?10-?5) (v10+ ?5) is a) (a+b)(a – b) = (a – b)² c) (a – b)(a+b) = a² – b² d) (a-b)(a+b)=2(a² + b²) ii) b) (a+b)(a – b) = (a + b

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CBSE Class 9th Maths 2023 : 30 Most Important Case Study Questions with Answers; Download PDF

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CBSE Class 9 Maths exam 2022-23 will have a set of questions based on case studies in the form of MCQs. CBSE Class 9 Maths Question Bank on Case Studies given in this article can be very helpful in understanding the new format of questions.

Each question has five sub-questions, each followed by four options and one correct answer. Students can easily download these questions in PDF format and refer to them for exam preparation.

CBSE Class 9 All Students can also Download here Class 9 Other Study Materials in PDF Format.

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• CBSE Class 9 Science Practice Book 2023 (Released By CBSE) 23 March, 2023, 5:56 pm
• CBSE Class 9th Maths 2023 : 30 Most Important Case Study Questions with Answers; Download PDF 10 February, 2023, 6:20 pm
• CBSE Class 9th Maths 2023 : Important Assertion Reason Question with Solution Download Pdf 9 February, 2023, 12:16 pm
• CBSE Class 9th Exam 2023 : Social Science Most Important Short Notes; Download PDF 16 January, 2023, 4:29 pm
• CBSE Class 9th Mathematics 2023 : Most Important Short Notes with Solutions 27 December, 2022, 6:05 pm
• CBSE Class 9th English 2023 : Chapter-wise Competency-Based Test Items with Answer; Download PDF 21 December, 2022, 5:16 pm
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CBSE Case Study Questions for Class 9 Maths - Pdf PDF Download

Cbse case study questions for class  9 maths.

CBSE Case Study Questions for Class 9 Maths are a type of assessment where students are given a real-world scenario or situation and they need to apply mathematical concepts to solve the problem. These types of questions help students to develop their problem-solving skills and apply their knowledge of mathematics to real-life situations.

Chapter Wise Case Based Questions for Class 9 Maths

The CBSE Class 9 Case Based Questions can be accessed from Chapetrwise Links provided below:

Chapter-wise case-based questions for Class 9 Maths are a set of questions based on specific chapters or topics covered in the maths textbook. These questions are designed to help students apply their understanding of mathematical concepts to real-world situations and events.

Chapter 1: Number System

• Case Based Questions: Number System

Chapter 2: Polynomial

• Case Based Questions: Polynomial

Chapter 3: Coordinate Geometry

• Case Based Questions: Coordinate Geometry

Chapter 4: Linear Equations

• Case Based Questions: Linear Equations - 1
• Case Based Questions: Linear Equations -2

Chapter 5: Introduction to Euclid’s Geometry

• Case Based Questions: Lines and Angles

Chapter 7: Triangles

• Case Based Questions: Triangles

Chapter 8: Quadrilaterals

• Case Based Questions: Quadrilaterals - 1
• Case Based Questions: Quadrilaterals - 2

Chapter 9: Areas of Parallelograms

• Case Based Questions: Circles

Chapter 11: Constructions

• Case Based Questions: Constructions

Chapter 12: Heron’s Formula

• Case Based Questions: Heron’s Formula

Chapter 13: Surface Areas and Volumes

• Case Based Questions: Surface Areas and Volumes

Chapter 14: Statistics

• Case Based Questions: Statistics

Chapter 15: Probability

• Case Based Questions: Probability

Weightage of Case Based Questions in Class 9 Maths

Why are Case Study Questions important in Maths Class  9?

• Enhance critical thinking:  Case study questions require students to analyze a real-life scenario and think critically to identify the problem and come up with possible solutions. This enhances their critical thinking and problem-solving skills.
• Apply theoretical concepts:  Case study questions allow students to apply theoretical concepts that they have learned in the classroom to real-life situations. This helps them to understand the practical application of the concepts and reinforces their learning.
• Develop decision-making skills:  Case study questions challenge students to make decisions based on the information provided in the scenario. This helps them to develop their decision-making skills and learn how to make informed decisions.
• Improve communication skills:  Case study questions often require students to present their findings and recommendations in written or oral form. This helps them to improve their communication skills and learn how to present their ideas effectively.
• Enhance teamwork skills:  Case study questions can also be done in groups, which helps students to develop teamwork skills and learn how to work collaboratively to solve problems.

In summary, case study questions are important in Class 9 because they enhance critical thinking, apply theoretical concepts, develop decision-making skills, improve communication skills, and enhance teamwork skills. They provide a practical and engaging way for students to learn and apply their knowledge and skills to real-life situations.

Class 9 Maths Curriculum at Glance

The Class 9 Maths curriculum in India covers a wide range of topics and concepts. Here is a brief overview of the Maths curriculum at a glance:

• Number Systems:  Students learn about the real number system, irrational numbers, rational numbers, decimal representation of rational numbers, and their properties.
• Algebra:  The Algebra section includes topics such as polynomials, linear equations in two variables, quadratic equations, and their solutions.
• Coordinate Geometry:  Students learn about the coordinate plane, distance formula, section formula, and slope of a line.
• Geometry:  This section includes topics such as Euclid’s geometry, lines and angles, triangles, and circles.
• Trigonometry: Students learn about trigonometric ratios, trigonometric identities, and their applications.
• Mensuration: This section includes topics such as area, volume, surface area, and their applications.
• Statistics and Probability:  Students learn about measures of central tendency, graphical representation of data, and probability.

The Class 9 Maths curriculum is designed to provide a strong foundation in mathematics and prepare students for higher education in the field. The curriculum is structured to develop critical thinking, problem-solving, and analytical skills, and to promote the application of mathematical concepts in real-life situations. The curriculum is also designed to help students prepare for competitive exams and develop a strong mathematical base for future academic and professional pursuits.

Students can also access Case Based Questions of all subjects of CBSE Class 9

• Case Based Questions for Class 9 Science
• Case Based Questions for Class 9 Social Science
• Case Based Questions for Class 9 English
• Case Based Questions for Class 9 Hindi
• Case Based Questions for Class 9 Sanskrit

Frequently Asked Questions (FAQs) on Case Based Questions for Class 9 Maths

What is case-based questions.

Case-Based Questions (CBQs) are open-ended problem solving tasks that require students to draw upon their knowledge of Maths concepts and processes to solve a novel problem. CBQs are often used as formative or summative assessments, as they can provide insights into how students reason through and apply mathematical principles in real-world problems.

What are case-based questions in Maths?

Case-based questions in Maths are problem-solving tasks that require students to apply their mathematical knowledge and skills to real-world situations or scenarios.

What are some common types of case-based questions in class 9 Maths?

Common types of case-based questions in class 9 Maths include word problems, real-world scenarios, and mathematical modeling tasks.

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CBSE Case Study Questions for Class 9 Maths - Pdf Free PDF Download

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Unit 1: Number Systems

Irrational numbers.

• Intro to rational & irrational numbers (Opens a modal)
• Classifying numbers (Opens a modal)
• Classify numbers: rational & irrational Get 5 of 7 questions to level up!
• Classify numbers Get 5 of 7 questions to level up!

Real Numbers and their Decimal Expansion

• Converting a fraction to a repeating decimal (Opens a modal)
• Converting repeating decimals to fractions (part 1 of 2) (Opens a modal)
• Converting repeating decimals to fractions (part 2 of 2) (Opens a modal)
• Writing fractions as repeating decimals Get 5 of 7 questions to level up!
• Converting repeating decimals to fractions Get 5 of 7 questions to level up!
• Converting multi-digit repeating decimals to fractions Get 3 of 4 questions to level up!

Operations on Real Numbers

• Sums and products of irrational numbers (Opens a modal)
• Worked example: rational vs. irrational expressions (Opens a modal)
• Rational vs. irrational expressions Get 3 of 4 questions to level up!

Laws of exponents of real numbers

• Intro to rational exponents (Opens a modal)
• Rewriting roots as rational exponents (Opens a modal)
• Evaluating fractional exponents (Opens a modal)
• Unit-fraction exponents Get 3 of 4 questions to level up!
• Fractional exponents Get 3 of 4 questions to level up!
• Evaluate radical expressions challenge Get 3 of 4 questions to level up!

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CBSE Class 9 Maths Most Important Case Study Based Questions With Solution

Cbse class 9 mathematics case study questions.

In this post I have provided CBSE Class 9 Maths Case Study Based Questions With Solution. These questions are very important for those students who are preparing for their final class 9 maths exam.

All these questions provided in this article are with solution which will help students for solving the problems. Dear students need to practice all these questions carefully with the help of given solutions.

As you know CBSE Class 9 Maths exam will have a set of cased study based questions in the form of MCQs. CBSE Class 9 Maths Question Bank given in this article can be very helpful in understanding the new format of questions for new session.

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Case studies in class 9 mathematics.

The Central Board of Secondary Education (CBSE) has included case study based questions in the Class 9 Mathematics paper in current session. According to new pattern CBSE Class 9 Mathematics students will have to solve case based questions. This is a departure from the usual theoretical conceptual questions that are asked in Class 9 Maths exam in this year.

Each question provided in this post has five sub-questions, each followed by four options and one correct answer. All CBSE Class 9th Maths Students can easily download these questions in PDF form with the help of given download Links and refer for exam preparation.

There is many more free study materials are available at Maths And Physics With Pandey Sir website. For many more books and free study material all of you can visit at this website.

Given Below Are CBSE Class 9th Maths Case Based Questions With Their Respective Download Links.

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Chapter 1 Class 9 Number Systems

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Get solutions of all NCERT Questions of Chapter 1 Class 9 Number System free at teachoo. Answers to all NCERT Exercises and Examples are solved for your reference. Theory of concepts is also made for your easy understanding

In this chapter, we will learn

• Different Types of numbers like Natural Numbers, Whole numbers, Integers, Rational numbers
• How to find rational numbers between two rational numbers
• What is an irrational number
• Checking if number is irrational or not
• And how to draw an irrational number on the number line
• Then, we will study What a real number is
• And find Decimal expansions - Terminating, Non terminating - repeating, Non terminating Non repeating
• Converting non-terminating repeating numbers into p/q form
• Finding irrational numbers between two numbers
• Representing real numbers on the number line (we use magnification)
• We will learn how to add , subtract and multiply numbers with square root (like 5√2 + 3√3 - 8√2)
• We will learn some identities of numbers with square root (like (√a + √b) 2 )
• How to rationalize numbers
• We will also do questions on Law of Exponents (here, the exponents can also be in fractions)

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• NCERT Solutions for Class 9 Maths Chapter 1 - Number Systems
• NCERT Solutions

NCERT Solutions Class 9 Maths Chapter 1 Number System - Free PDF

Class 9 Chapter 1 delves into the principles covered under the topic of the number system. Vedantu offers an expert-curated NCERT answer for CBSE Class 9 Chapter 1. To ace your preparations, get the NCERT solution supplied by our professionals. The freely available pdf offers step-by-step solutions to the NCERT practice problems. The NCERT solutions pdf contains the answers to all of the Class 9 syllabus questions.

If you are a student who is looking for an easy way to summarise the complete chapter, look no further! Start your preparation with the solutions provided by the experts of Vedantu and ace your studies.

Exercises under NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Ncert solutions for class 9 maths chapter 1, "number systems", consists of six exercises, each covering a specific set of questions. below is a detailed explanation of each exercise: exercise 1.1: this exercise covers basic concepts of the number system, such as natural numbers, whole numbers, integers, rational numbers, irrational numbers, etc. the questions in this exercise aim to familiarise students with these concepts and their properties. exercise 1.2: this exercise covers the representation of numbers in decimal form. the questions in this exercise require students to convert fractions into decimals, decimals into fractions, and perform basic operations such as addition, subtraction, multiplication, and division on decimals. exercise 1.3: this exercise deals with the representation of rational numbers on a number line. the questions in this exercise require students to mark the position of given rational numbers on a number line and identify the rational number represented by a given point on the number line. exercise 1.4: this exercise covers the representation of irrational numbers on a number line. the questions in this exercise require students to mark the position of given irrational numbers on a number line and identify the irrational number represented by a given point on the number line. exercise 1.5: this exercise deals with the conversion of recurring decimals into fractions. the questions in this exercise require students to write recurring decimals as fractions and vice versa. exercise 1.6: this exercise covers the comparison of rational numbers. the questions in this exercise require students to compare given rational numbers using the concept of inequality, find rational numbers between two given rational numbers, and represent rational numbers on a number line., ncert solutions class 9 maths chapter 1 number system - free pdf download, exercise (1.1).

1.  Is zero a rational number? Can you write it in the form  $\dfrac{ {p}}{ {q}}$, where $ {p}$ and $ {q}$ are integers and $ {q}\ne  {0}$? Describe it.

Ans: Remember that, according to the definition of rational number,

a rational number is a number that can be expressed in the form of  $\dfrac{p}{q}$, where $p$ and $q$ are integers and  $q\ne \text{0}$. 

Now, notice that zero can be represented as $\dfrac{0}{1},\dfrac{0}{2},\dfrac{0}{3},\dfrac{0}{4},\dfrac{0}{5}.....$

Also, it can be expressed as $\dfrac{0}{-1},\dfrac{0}{-2},\dfrac{0}{-3},\dfrac{0}{-4}.....$

Therefore, it is concluded from here that $0$ can be expressed in the form of $\dfrac{p}{q}$, where $p$ and $q$ are integers.

Hence, zero must be a rational number.

2. Find any six rational numbers between $ {3}$ and $ {4}$. 

Ans: It is known that there are infinitely many rational numbers between any two numbers. Since we need to find $6$ rational numbers between $3$ and $4$, so multiply and divide the numbers by $7$ (or by any number greater than $6$)

Then it gives, 

$ 3=3\times \dfrac{7}{7}=\dfrac{21}{7} $ 

$  4=4\times \dfrac{7}{7}=\dfrac{28}{7} $

Hence, $6$ rational numbers found between $3$ and $4$ are $\dfrac{22}{7},\dfrac{23}{7},\dfrac{24}{7},\dfrac{25}{7},\dfrac{26}{7},\dfrac{27}{7}$.

3. Find any five rational numbers between $\dfrac{ {3}}{ {5}}$ and $\dfrac{ {4}}{ {5}}$.

Ans: It is known that there are infinitely many rational numbers between any two numbers.

Since here we need to find five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$,  so multiply and divide by $6$ (or by any number greater than $5$).

Then it gives,

$\dfrac{3}{5}=\dfrac{3}{5}\times \dfrac{6}{6}=\dfrac{18}{30}$,

$\dfrac{4}{5}=\dfrac{4}{5}\times \dfrac{6}{6}=\dfrac{24}{30}$.

Hence, $5$ rational numbers found between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are

$\dfrac{19}{30},\dfrac{20}{30},\dfrac{21}{30},\dfrac{22}{30},\dfrac{23}{30}$.

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

(i) Every natural number is a whole number. 

Ans: Write the whole numbers and natural numbers in a separate manner.

It is known that the whole number series is $0,1,2,3,4,5.....$. and

the natural number series is $1,2,3,4,5.....$.

Therefore, it is concluded that all the natural numbers lie in the whole number series as represented in the diagram given below.

Thus, it is concluded that every natural number is a whole number.

Hence, the given statement is true.

(ii) Every integer is a whole number.

Ans: Write the integers and whole numbers in a separate manner.

 It is known that integers are those rational numbers that can be expressed in the form of $\dfrac{p}{q}$, where $q=1$.

Now, the series of integers is like $0,\,\pm 1,\,\pm 2,\,\pm 3,\,\pm 4,\,...$.

But the whole numbers are $0,1,2,3,4,...$. 

Therefore, it is seen that all the whole numbers lie within the integer numbers, but the negative integers are not included in the whole number series. 

Thus, it can be concluded from here that every integer is not a whole number.

Hence, the given statement is false.

(iii) Every rational number is a whole number.

Ans: Write the rational numbers and whole numbers in a separate manner. 

It is known that rational numbers are the numbers that can be expressed in the form  $\dfrac{p}{q}$, where $q\ne 0$ and the whole numbers are represented as $0,\,1,\,2,\,3,\,4,\,5,...$

Now, notice that every whole number can be expressed in the form of $\dfrac{p}{q}$

as  \[\dfrac{0}{1},\text{ }\dfrac{1}{1},\text{ }\dfrac{2}{1},\text{ }\dfrac{3}{1},\text{ }\dfrac{4}{1},\text{ }\dfrac{5}{1}\],…

Thus, every whole number is a rational number, but all the rational numbers are not whole numbers. For example,

$\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},...$ are not whole numbers.

Therefore, it is concluded from here that every rational number is not a whole number.

Exercise (1.2)

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

(i) Every irrational number is a real number. 

Ans: Write the irrational numbers and the real numbers in a separate manner.

The irrational numbers are the numbers that cannot be represented in the form $\dfrac{p}{q},$ where $p$ and $q$ are integers and $q\ne 0.$

For example, $\sqrt{2},3\pi ,\text{ }.011011011...$ are all irrational numbers.

The real number is the collection of both rational numbers and irrational numbers.

For example, $0,\,\pm \dfrac{1}{2},\,\pm \sqrt{2}\,,\pm \pi ,...$ are all real numbers.

Thus, it is concluded that every irrational number is a real number.

(ii) Every point on the number line is of the form $\sqrt{m}$, where m is a natural number. 

Ans: Consider points on a number line to represent negative as well as positive numbers.

Observe that, positive numbers on the number line can be expressed as $\sqrt{1,}\sqrt{1.1,}\sqrt{1.2},\sqrt{1.3},\,...$, but any negative number on the number line cannot be expressed as $\sqrt{-1},\sqrt{-1.1},\sqrt{-1.2},\sqrt{-1.3},...$, because these are not real numbers.

Therefore, it is concluded from here that every number point on the number line is not of the form $\sqrt{m}$, where $m$ is a natural number. 

(iii) Every real number is an irrational number. 

Real numbers are the collection of rational numbers (Ex: $\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{5},\dfrac{5}{7},$……) and the irrational numbers (Ex: $\sqrt{2},3\pi ,\text{ }.011011011...$).

Therefore, it can be concluded that every irrational number is a real number, but

every real number cannot be an irrational number.

Hence, the given statement is false. 

2. Are the square roots of all positive integer numbers irrational? If not, provide an example of the square root of a number that is not an irrational number.

Ans: Square root of every positive integer does not give an integer. 

For example: $\sqrt{2},\sqrt{3,}\sqrt{5},\sqrt{6},...$ are not integers, and hence these are irrational numbers. But $\sqrt{4}$ gives $\pm 2$ , these are integers and so, $\sqrt{4}$ is not an irrational number.

Therefore, it is concluded that the square root of every positive integer is not an irrational number.

3. Represent $\sqrt{5}$ on the number line.

Ans: Follow the procedures to get $\sqrt{5}$ on the number line.

Firstly, Draw a line segment $AB$ of $2$ unit on the number line.

Secondly, draw a perpendicular line segment $BC$ at $B$ of $1$ units.

Thirdly, join the points $C$ and $A$, to form a line segment $AC$. 

Fourthly, apply the Pythagoras Theorem as 

$ A{{C}^{2}}=A{{B}^{2}}+B{{C}^{2}} $

$  A{{C}^{2}}={{2}^{2}}+{{1}^{2}} $

$ A{{C}^{2}}=4+1=5 $

$ AC=\sqrt{5} $

Finally, draw the arc $ACD$, to find the number $\sqrt{5}$ on the number line as given in the diagram below.

Exercise (1.3)

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

(i) $\mathbf{\dfrac{ {36}}{ {100}}}$

Ans: Divide $36$ by $100$. 

$\,\,\,\,\,\,\,\,\,\, {0.36}$

$100 {\overline{)\;36\quad}}$

$\underline{\,\,\,\,\,\,\,\,\,-0\quad}$

$\,\,\,\,\,\,\,\,\,\,\,\,\,\,360$

$\underline{\,\,\,\,\,\,\,\,\,\,-300\quad}$

$\;\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,600$

$\underline{\,\,\,\,\,\,\,\,\,\,\,\,\,-600}$

$\underline{\,\,\,\,\,\,\,\,\,\,\,\,\quad 0 \,\,\,\,\,}$

So, $\dfrac{36}{100}=0.36$ and it is a terminating decimal number.

(ii) $\mathbf{\dfrac{ {1}}{ {11}}}$

Ans: Divide $1$ by $11$.

${\,\,\,\,\,\,\,\,0.0909..}$

$11 \, {\overline{)\;1\quad}}$

$\underline{\,\,\,\,\,\,\,-0\quad}$

$\,\,\,\,\,\,\,\,\,\,10$

$\underline{\,\;\;\,\,-0\quad}$

$\;\;\,\,\,\,100$

$\underline{\,\,\,\,\;-99}$

$\,\,\,\,\,\, \quad 10$

$\quad\underline{\;\;-0\quad}$

$\;\;\,\,\,\,\,\,\,\,100$

$\underline{\,\,\,\,\,\,\,\,\;-99}$

$\quad\,\,\,\,\,\,\,1\quad$

It is noticed that while dividing $1$ by $11$, in the quotient $09$ is repeated.

So, $\dfrac{1}{11}=0.0909.....$ or 

$\dfrac{1}{11}=0.\overline{09}$ 

and it is a non-terminating and recurring decimal number.

(iii)  $ \mathbf{{4}\dfrac{ {1}}{ {8}}}$

Ans: $4\dfrac{1}{8}=4+\dfrac{1}{8}=\dfrac{32+1}{8}=\dfrac{33}{8}$

Divide $33$ by $8$.

$\,\,\,\,\,{4.125}$

$8 {\overline{)\;33\quad}}$

$\underline{\,\,\,\,-32\quad}$

$\,\,\,\,\,\,\,\,\,\,\,\,10$

$\underline{\;\;\,\,\,\,-8\quad}$

$\;\;\,\,\,\,\,\,\,\,\,\,\,20$

$\underline{\,\,\,\,\,\,\,\,\,-16}$

$\;\quad\quad\,\,\,\,40$

$\quad\underline{\quad\,\,-40\quad}$

$\quad\underline{\quad\,\, \,\,\,\,0\quad}$

Notice that, after dividing $33$ by $8$, the remainder is found as $0$.

So, $4\dfrac{1}{8}=4.125$ and it is a terminating decimal number.

(iv)  $\mathbf{\dfrac{ {3}}{ {13}}}$

Ans: Divide $3$ by $13$.

$\quad \,\,{0.230769}$

$13 {\overline{)\;3\quad}}$

$\underline{\quad-0\quad}$

$\quad\quad 30$

$\underline{\;\,\quad-26\quad}$

$\;\quad\quad\,\,\,40$

$\underline{\quad\quad\,\,-39\quad}$

$\;\quad\quad\quad\;10$

$\quad\underline{\quad\quad -0\quad}$

$\quad{\quad\quad \quad 100}$

$\quad\quad\underline{\quad \,\, -91\quad}$

$\quad\quad \quad \,\,\,\quad90$

$\quad\quad\underline{\quad\,\,\,\,\,-78\quad}$

$\quad\quad\quad\quad \quad 120$

$\quad \quad\underline{\quad\quad\,\,-117\quad}$

$\quad\quad\underline{\quad \quad\quad\,\, 3\quad}$

It is observed that while dividing $3$ by $13$, the remainder is found as $3$ and that is repeated after each $6$ continuous divisions.

So, $\dfrac{3}{13}=0.230769.......$ or

$\dfrac{3}{13}=0.\overline{230769}$ 

(v)   $\mathbf{\dfrac{ {2}}{ {11}}}$

Ans: Divide $2$ by $11$.

$\quad \,\,{0.1818}$

$11 {\overline{)\;2\quad}}$

$\quad\quad20$

$\underline{\quad\;-11\quad}$

$\quad\quad \;\,90$

$\underline{\quad\,\,\,\, -88\;}$

$\;\quad\quad\;20$

$\quad\underline{\quad-11\quad}$

$\quad{\quad\quad  90}$

$\quad\underline{\,\,\quad -88}$

$\quad\quad\quad\,\,2\quad$

It can be noticed that while dividing $2$ by $11$, the remainder is obtained as $2$ and then $9$, and these two numbers are repeated infinitely as remainders.

So, $\dfrac{2}{11}=0.1818.....$ or 

$\dfrac{2}{11}=0.\overline{18}$ 

(vi) $\mathbf{\dfrac{ {329}}{ {400}}}$

Ans: Divide $329$ by $400$.

$\quad \quad{0.8225}$

$400 {\overline{)\;329\quad}}$

$\underline{\quad\,\,-0\quad}$

$\quad\quad3290$

$\underline{\quad\;-3200\quad}$

$\quad\quad\quad\;900$

$\underline{\quad\quad\quad-800\;}$

$\quad\quad\quad\quad\;1000$

$\quad\underline{\quad\quad\quad-800\quad}$

$\quad{\quad\quad\quad\quad\,\,2000}$

$\quad\underline{\quad\quad\quad\quad-2000\quad}$

$\quad\underline{\quad\quad\quad\quad\,\,\,\,\,\, 0 \quad}$

It can be seen that while dividing $329$ by $400$, the remainder is obtained as $0$.

So, $\dfrac{329}{400}=0.8225$ and is a terminating decimal number.

2. You know that $\dfrac{ {1}}{ {7}} {=0} {.142857}...$. Can you predict what the decimal expansions of $\dfrac{ {2}}{ {7}} {,}\dfrac{ {3}}{ {7}} {,}\dfrac{ {4}}{ {7}} {,}\dfrac{ {5}}{ {7}} {,}\dfrac{ {6}}{ {7}}$  are, without actually doing the long division? If so, how?

$\text{[}$Hint: Study the remainders while finding the value of $\dfrac{ {1}}{ {7}}$ carefully.$\text{]}$

Ans: Note that,  $\dfrac{2}{7},\dfrac{3}{7},\dfrac{4}{7},\dfrac{5}{7}$ and $\dfrac{6}{7}$ can be rewritten as $2\times \dfrac{1}{7},\text{ 3}\times \dfrac{1}{7},\text{ 4}\times \dfrac{1}{7},\text{ 5}\times \dfrac{1}{7},$ and $6\times \dfrac{1}{7}$

Substituting the value of $\dfrac{1}{7}=0.142857$ , gives 

$2 \times \dfrac{1}{7} = 2\times 0.142857...=0.285714...$

$ 3\times \dfrac{1}{7} = 3\times .428571…= .428571...$

\[4\times \dfrac{1}{7}=4\times 0.142857...\]\[\text{=}\,\text{0}\text{.571428}...\]

$5\times \dfrac{1}{7}=5\times 0.71425...$  \[\text{=}\,\text{0}\text{.714285}...\]

$6\times \dfrac{1}{7}=6\times 0.142857...$\[\text{=}\,\text{0}\text{.857142}...\]

So, the values of $\dfrac{2}{7},\text{ }\dfrac{3}{7},\text{ }\dfrac{4}{7},\text{ }\dfrac{5}{7}$ and $\dfrac{6}{7}$ obtained without performing long division are

\[\dfrac{2}{7}=0.\overline{285714}\]

$\dfrac{3}{7}=0.\overline{428571}$

$\dfrac{4}{7}=0.\overline{571428}$

\[\dfrac{5}{7}=0.\overline{714285}\]

$\dfrac{6}{7}=0.\overline{857142}$

3. Express the following in the form \[\dfrac{ {p}}{ {q}}\], where $ {p}$ and $ {q}$ are integers and $ {q}\ne  {0}$.

(i) $\mathbf{ {0} {.}\overline{ {6}}}$

Ans: Let $x=0.\overline{6}$  

 $\Rightarrow x=0.6666$                                                   ….… (1)

 Multiplying both sides of the equation (1) by $10$, gives

$10x=0.6666\times 10$

$10x=6.6666$…..                 …… (2)

Subtracting the equation $\left( 1 \right)$ from $\left( 2 \right)$, gives

$ 10x=6.6666..... $

$ \underline{-x=0.6666.....} $

$  9x=6 $ 

$  9x=6 $

$  x=\dfrac{6}{9}=\dfrac{2}{3} $ 

So, the decimal number becomes

$0.\overline{6}=\dfrac{2}{3}$  and it is in the required  $\dfrac{p}{q}$ form.

(ii) $\mathbf{ {0} {.}\overline{ {47}}}$

Ans: Let  $x=0.\overline{47}$

$\text{   }\Rightarrow x=0.47777.....$                                             ……(a)

Multiplying both sides of the equation (a) by $10$, gives

$10x=4.7777.....$         ……(b)

Subtracting the equation $\left( a \right)$ from $\left( b \right)$, gives

$ 10x=4.7777..... $

$  \underline{-x=0.4777.....} $

$  9x=4.3 $

$x=\dfrac{4.3}{9}\times \dfrac{10}{10} $ 

$ \Rightarrow x=\dfrac{43}{90} $

So, the decimal number becomes 

$0.\overline{47}=\dfrac{43}{90}$  and it is in the required $\dfrac{p}{q}$ form.

(iii) $ \mathbf{{0} {.}\overline{ {001}}}$

Ans: Let $x=0.\overline{001} $           …… (1)

Since the number of recurring decimal number is $3$, so multiplying both sides of the equation (1) by $1000$, gives

$1000\times x=1000\times 0.001001.....$ …… (2)

Subtracting the equation (1) from (2) gives

$ 1000x=1.001001..... $

$  \underline{\text{    }-x=0.001001.....} $

$  999x=1 $

$\Rightarrow x=\dfrac{1}{999}$

Hence, the decimal number becomes 

$0.\overline{001}=\dfrac{1}{999}$ and it is in the $\dfrac{p}{q}$ form.

4. Express $ {0} {.99999}.....$ in the form of $\dfrac{ {p}}{ {q}}$ . Are you surprised by your answer? With your teacher and classmates, discuss why the answer makes sense.

Let $x=0.99999.....$                                                             ....... (a)

Multiplying by $10$ both sides of the equation (a), gives

$10x=9.9999.....$                                                             …… (b)

Now, subtracting the equation (a) from (b), gives

$ 10x=9.99999..... $

$  \underline{\,-x=0.99999.....} $

$  9x=9 $ 

$\Rightarrow x=\dfrac{9}{9}$

$\Rightarrow x=1$.

$0.99999...=\dfrac{1}{1}$ which is in the $\dfrac{p}{q}$ form.

Yes, for a moment we are amazed by our answer, but when we observe that $0.9999.........$ is extending infinitely, then the answer makes sense.

Therefore, there is no difference between $1$ and $0.9999.........$ and hence these two numbers are equal.

5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\dfrac{ {1}}{ {17}}$ ? Perform the division to check your answer.

Ans: Here the number of digits in the recurring block of $\dfrac{1}{17}$ is to be determined. So, let us calculate the long division to obtain the recurring block of $\dfrac{1}{17}$. Dividing $1$ by $17$ gives

$\quad\quad {0.0588235294117646}$

$17{\overline{)\quad1\quad\quad\quad\quad\quad\quad\quad\quad}}$

$\underline{\quad\,\,\,\,-0\quad}\qquad\qquad\qquad$

$\quad \quad \,\,\,10\qquad\qquad\quad\quad$

$\underline{\quad \quad -0\quad}\qquad\qquad\quad$

$\quad \quad \,\,\,\,\,\;100\qquad\qquad\qquad$

$\underline{\quad \quad \,\,-85\;}\qquad\qquad\quad$

$\quad\qquad\,\,\;150\qquad\qquad\quad$

$\quad\underline{\qquad-136\;}\qquad\qquad\quad$

$\quad{\quad\quad\quad 140}\qquad\qquad\;\;$

$\quad\underline{\qquad-136\quad}\qquad\quad$

${\quad \qquad \,\,\quad 40 \quad}\quad$

$\underline{\qquad \,\,\,\quad -34\;\;}\quad$

$\;\qquad \qquad\,\,60$

$\underline{\qquad \qquad-51}$

$\quad\quad \qquad \quad 90$

$\quad\;\;\underline{\quad \qquad-85}$

$\qquad\quad\;\quad\,\,\,\, 50$

$\quad\quad\;\;\underline{\,\,\quad\,\, -34}$

$\quad\quad\qquad \quad 160$

$\qquad\quad\;\underline{\quad-153}$

$\qquad\qquad\quad\;70$

$\qquad\quad\quad\;\;\underline{-68}$

$\quad\,\,\qquad\qquad 20$

$\qquad\qquad\quad\underline{-17}$

$\qquad\qquad\quad\quad\; 130$

$\qquad\qquad\quad\;\;\underline{-119}$

$\qquad\qquad\qquad\quad 110$

$\qquad\qquad\qquad\;\;\underline{-102}$

$\qquad\qquad\qquad\quad\quad\quad 80$

$\qquad\qquad\qquad\qquad\;\underline{-68}$

$\qquad\qquad\qquad\quad\quad\quad\; 120$

$\qquad\qquad\qquad\qquad\;\;\underline{-119}$

$\qquad\qquad\qquad\quad\quad\quad\; 1$

Thus, it is noticed that while dividing $1$ by $17$, we found $16$ number of digits in the

repeating block of decimal expansion that will continue to be $1$ after going through $16$ continuous divisions.

Hence, it is concluded that $\dfrac{1}{17}=0.0588235294117647.....$ or 

 $\dfrac{1}{17}=0.\overline{0588235294117647}$ and it is a recurring and non-terminating decimal number.

6. Look at several examples of rational numbers in the form $\dfrac{ {p}}{ {q}}\left(  {q}\ne  {0} \right)$, 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?

Ans: Let us consider the examples of such rational numbers $\dfrac{5}{2},\dfrac{5}{4},\dfrac{2}{5},\dfrac{2}{10},\dfrac{5}{16}$ of the form $\dfrac{p}{q}$ which have terminating decimal representations.

$ \dfrac{5}{2}=2.5 $

$ \dfrac{5}{4}=1.25 $ 

$ \dfrac{2}{5}=0.4 $

$ \dfrac{2}{10}=0.2 $

$ \dfrac{5}{16}=0.3125 $

In each of the above examples, it can be noticed that the denominators of the rational numbers have powers of $2,5$ or both. 

So, $q$ must satisfy the form either ${{2}^{m}}$, or ${{5}^{n}}$, or  both ${{2}^{m}}\times {{5}^{n}}$ (where $m=0,1,2,3.....$ and $n=0,1,2,3.....$) in the form of $\dfrac{p}{q}$.

7. Write three numbers whose decimal expansions are non-terminating non-recurring.

Ans: All the irrational numbers are non-terminating and non-recurring, because irrational numbers do not have any representations of the form of $\dfrac{p}{q}$ $\left( q\ne 0 \right)$, where $p$ and $q$are integers. For example: 

$\sqrt{2}=1.41421.....$,

$\sqrt{3}=1.73205...$

$\sqrt{7}=2.645751....$

are the numbers whose decimal representations are non-terminating and non-recurring.

8. Find any three irrational numbers between the rational numbers $\dfrac{ {5}}{ {7}}$ and $\dfrac{ {9}}{ {11}}$.

Ans: Converting  $\dfrac{5}{7}$and $\dfrac{9}{11}$ into the decimal form gives

$\dfrac{5}{7}=0.714285.....$ and 

$\dfrac{9}{11}=0.818181.....$

Therefore, $3$ irrational numbers that are contained between $0.714285......$ and $0.818181.....$

$ 0.73073007300073...... $ 

$  0.74074007400074...... $ 

$ 0.76076007600076...... $

Hence, three irrational numbers between the rational numbers $\dfrac{5}{7}$ and $\dfrac{9}{11}$ are

9. Classify the following numbers as rational or irrational:

(i) $\mathbf{\sqrt{ {23}}}$

Ans: The following diagram reminds us of the distinctions among the types of rational and irrational numbers.

After evaluating the square root gives

$\sqrt{23}=4.795831.....$ , which is an irrational number.

(ii) $\mathbf{\sqrt{ {225}}}$

Ans: After evaluating the square root gives

$\sqrt{225}=15$, which is a rational number.

That is, $\sqrt{225}$ is a rational number.

(iii) $ \mathbf{{0} {.3796}}$

Ans: The given number is $0.3796$. It is terminating decimal. 

So, $0.3796$ is a rational number.

(iv) $ \mathbf{{7} {.478478}}$

Ans: The given number is \[7.478478\ldots .\] 

It is a non-terminating and recurring decimal that can be written in the $\dfrac{p}{q}$ form.

Let      $x=7.478478\ldots .$                                   ……(a)

Multiplying the equation (a) both sides by $100$ gives

$\Rightarrow 1000x=7478.478478.....$                                               ……(b)

Subtracting the equation (a) from (b), gives

$ 1000x=7478.478478.... $

$  \underline{\text{    }-x=\text{     }7.478478\ldots .} $

$ 999x=7471 $

$  \text{      }x=\dfrac{7471}{999} $ 

Therefore, $7.478478.....=\dfrac{7471}{999}$, which is in the form of $\dfrac{p}{q}$

So, $7.478478...$ is a rational number.

(v) $ \mathbf{{1} {.101001000100001}.....}$

Ans: The given number is \[1.101001000100001....\]

It can be clearly seen that the number \[1.101001000100001....\] is a non-terminating and non-recurring decimal and it is known that non-terminating non-recurring decimals cannot be written in the form of $\dfrac{p}{q}$.

Hence, the number \[1.101001000100001....\] is an irrational number.

Exercise (1.4)

1.  Visualize \[ {3} {.765}\] on the number line, using successive magnification.

It is clear that the value \[3.765\] lies between the numbers $3$ and $4$.

Also, the number $3.7$ and $3.8$ lie between the numbers $3$ and $4$.

The number $3.76$ and $3.77$ lie between the numbers $3$ and $4$.

Again, the numbers $3.764$ and $3.766$ lie between the numbers $3.76$ and $3.77$.

Thus, the number $3.765$ lies between the numbers $3.764$ and $3.766$.

So, first locate the numbers $3$ and $4$ on the number line, then use the successive magnification as shown in the diagrams below.

Locate the numbers 3 and 4

Apply Magnification Between 3 and 4

Apply Magnification Between 3.7 and 3.8

Apply Magnification Between 3.76 and 3.77

Apply Magnification Between 3.764 and 3.766 and Find 3.765

2. Visualize $ {4} {.}\overline{ {26}}$ on the number line, up to $ {4}$ decimal places.

Ans: The number $4.\overline{26}$ can be represented as $4.262.....$.

Apply successive magnification, after locating the numbers $4$ and $5$ on the number line and visualize the number up to $4$ decimal places as given in the following diagrams.

The number $4.2$ is located between $4$ and $5$ .

The number $4.26$ is located between $4.2$ and $4.3$.

The number $4.262$ is located between $4.26$ and \[4.27\].

The number \[4.2626\] is located between \[4.262\] and \[4.263\].

Exercise (1.5)

1.  Classify the following numbers as rational or irrational:

(i) $ \mathbf{{2-}\sqrt{ {5}}}$

Ans: The given number is $2-\sqrt{5}$.

Here, $\sqrt{5}=2.236.....$ and it is a non-repeating and non-terminating irrational number.

Therefore, substituting the value of $\sqrt{5}$ gives

$2-\sqrt{5}=2-2.236.....$

$=-0.236.....$, which is an irrational number.

So, $2-\sqrt{5}$ is an irrational number.

(ii) $\mathbf{\left(  {3+}\sqrt{ {23}} \right) {-}\left( \sqrt{ {23}} \right)}$

Ans: The given number is $\left( 3+\sqrt{23} \right)-\left( \sqrt{23} \right)$.

The number can be written as

$\left( 3+\sqrt{23} \right)-\sqrt{23}=3+\sqrt{23}-\sqrt{23} $ 

$  =3 $

$=\dfrac{3}{1}$, which is in the $\dfrac{p}{q}$ form and so, it is a rational number.

Hence, the number $\left( 3+\sqrt{23} \right)-\sqrt{23}$ is a rational number.

(iii) $\mathbf{\dfrac{ {2}\sqrt{ {7}}}{ {7}\sqrt{ {7}}}}$

Ans: The given number is $\dfrac{2\sqrt{7}}{7\sqrt{7}}$.

$\dfrac{2\sqrt{7}}{7\sqrt{7}}=\dfrac{2}{7}$, which is in the $\dfrac{p}{q}$  form and so, it is a rational number.

Hence, the number  $\dfrac{2\sqrt{7}}{7\sqrt{7}}$ is a rational number.

(iv) $\mathbf{\dfrac{ {1}}{\sqrt{ {2}}}}$

Ans: The given number is $\dfrac{1}{\sqrt{2}}$.

It is known that, $\sqrt{2}=1.414.....$ and it is a non-repeating and non-terminating irrational number.

Hence, the number $\dfrac{1}{\sqrt{2}}$ is an irrational number.

(v) $ \mathbf{{2\pi }}$

Ans: The given number is $2\pi $.

It is known that, $\pi =3.1415$ and it is an irrational number.

Now remember that, Rational $\times $ Irrational = Irrational.

Hence, $2\pi $ is also an irrational number.

2. Simplify each of the of the following expressions:

(i) $\mathbf{\left(  {3+}\sqrt{ {3}} \right)\left(  {2+}\sqrt{ {2}} \right)}$

Ans: The given number is $\left( 3+\sqrt{3} \right)\left( 2+\sqrt{2} \right)$.

By calculating the multiplication, it can be written as

$\left( 3+\sqrt{3} \right)\left( 2+\sqrt{2} \right)=3\left( 2+\sqrt{2} \right)+\sqrt{3}\left( 2+\sqrt{2} \right)$.

\[= 6 + 4 \sqrt{2} + 2 \sqrt{3}+ \sqrt{6}\]

(ii) $\mathbf{\left(  {3+}\sqrt{ {3}} \right)\left(  {3-}\sqrt{ {3}} \right)}$

Ans: The given number is $\left( 3+\sqrt{3} \right)\left( 3-\sqrt{3} \right)$.

By applying the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$, the number can be written as

$\left( 3+\sqrt{3} \right)\left( 3-\sqrt{3} \right)={{3}^{2}}-{{\left( \sqrt{3} \right)}^{2}}=9-3=6$.

(iii)  $\mathbf{{{\left( \sqrt{ {5}} {+}\sqrt{ {2}} \right)}^{ {2}}}}$

Ans: The given number is ${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}$.

Applying the formula ${{\left( a+b \right)}^{2}}={{a}^{2+}}2ab+{{b}^{2}}$, the number can be written as

${{\left( \sqrt{5}+\sqrt{2} \right)}^{2}}={{\left( \sqrt{5} \right)}^{2}}+2\sqrt{5}\sqrt{2}+{{\left( \sqrt{2} \right)}^{2}}$

 $=5+2\sqrt{10}+2$

 $=7+2\sqrt{10}$.

(iv)  $\mathbf{\left( \sqrt{ {5}}-\sqrt{ {2}} \right)\left( \sqrt{ {5}} {+}\sqrt{ {2}} \right)}$

Ans: The given number is $\left( \sqrt{5}-\sqrt{2} \right)\left( \sqrt{5}+\sqrt{2} \right)$.

Applying the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$, the number can be expressed as

$\left( \sqrt{5}-\sqrt{2} \right)\left( \sqrt{5}+\sqrt{2} \right)={{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}$

$ =3. $ 

3. Recall that, $ {\pi }$ is defined as the ratio of the circumference (say $ {c}$) of a circle 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?

Ans: It is known that, $\pi =\dfrac{22}{7}$, which is a rational number. But, note that this value of $\pi $ is an approximation.

On dividing $22$ by $7$, the quotient $3.14...$ is a non-recurring and non-terminating number. Therefore, it is an irrational number.

In order of increasing accuracy, approximate fractions are

$\dfrac{22}{7}$, $\dfrac{333}{106}$, $\dfrac{355}{113}$, $\dfrac{52163}{16604}$, $\dfrac{103993}{33102}$, and \[\dfrac{245850922}{78256779}\].

Each of the above quotients has the value $3.14...$, which is a non-recurring and non-terminating number.

Thus, $\pi $ is irrational.

So, either circumference $\left( c \right)$ or diameter $\left( d \right)$ or both should be irrational numbers.

Hence, it is concluded that there is no contradiction regarding the value of $\pi $ and it is made out that the value of $\pi $ is irrational.

4. Represent $\sqrt{ {9} {.3}}$ on the number line.

Ans: Follow the procedure given below to represent the number $\sqrt{9.3}$.

First, mark the distance $9.3$ units from a fixed-point $A$ on the number line to get a point $B$. Then $AB=9.3$ units.

Secondly, from the point $B$ mark a distance of $1$ unit and denote the ending point as $C$.

Thirdly, locate the midpoint of $AC$ and denote it as $O$.

Fourthly, draw a semi-circle to the centre $O$ with the radius $OC=5.15$ units. Then 

$ AC=AB+BC $ 

$  =9.3+1 $ 

$  =10.3 $

So, $OC=\dfrac{AC}{2}=\dfrac{10.3}{2}=5.15$.

Finally, draw a perpendicular line at $B$ and draw an arc to the centre $B$ and then let it meet at the semicircle $AC$ at $D$ as given in the diagram below.

5. Rationalize the denominators of the following:

(i) $\mathbf{\dfrac{ {1}}{\sqrt{ {7}}}}$

Ans: The given number is $\dfrac{1}{\sqrt{7}}$.

Multiplying and dividing by $\sqrt{7}$ to the number gives

$\dfrac{1}{\sqrt{7}}\times \dfrac{\sqrt{7}}{\sqrt{7}}=\dfrac{\sqrt{7}}{7}$.

(ii) $\mathbf{\dfrac{ {1}}{\sqrt{ {7}} {-}\sqrt{ {6}}}}$

Ans: The given number is $\dfrac{1}{\sqrt{7}-\sqrt{6}}$.

Multiplying and dividing by $\sqrt{7}+\sqrt{6}$ to the number gives

$\dfrac{1}{\sqrt{7}-\sqrt{6}}\times \dfrac{\sqrt{7}+\sqrt{6}}{\sqrt{7}+\sqrt{6}}=\dfrac{\sqrt{7}+\sqrt{6}}{\left( \sqrt{7}-\sqrt{6} \right)\left( \sqrt{7}+\sqrt{6} \right)}$

Now, applying the formula $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$ to the denominator gives

$ \dfrac{1}{\sqrt{7}-\sqrt{6}}=\dfrac{\sqrt{7}+\sqrt{6}}{{{\left( \sqrt{7} \right)}^{2}}-{{\left( \sqrt{6} \right)}^{2}}} $ 

$ =\dfrac{\sqrt{7}+\sqrt{6}}{7-6} $ 

$  =\dfrac{\sqrt{7}+\sqrt{6}}{1}. $

(iii) $\mathbf{\dfrac{ {1}}{\sqrt{ {5}} {+}\sqrt{ {2}}}}$

Ans: The given number is $\dfrac{1}{\sqrt{5}+\sqrt{2}}$.

Multiplying and dividing by $\sqrt{5}-\sqrt{2}$ to the number gives

$\dfrac{1}{\sqrt{5}+\sqrt{2}}\times \dfrac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}=\dfrac{\sqrt{5}-\sqrt{2}}{\left( \sqrt{5}+\sqrt{2} \right)\left( \sqrt{5}-\sqrt{2} \right)}$

Now, applying the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$  to the denominator gives

$ \dfrac{1}{\sqrt{5}+\sqrt{2}}=\dfrac{\sqrt{5}-\sqrt{2}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{2} \right)}^{2}}} $ 

$ =\dfrac{\sqrt{5}-\sqrt{2}}{5-2} $

$ =\dfrac{\sqrt{5}-\sqrt{2}}{3}. $ 

(iv) $\mathbf{\dfrac{ {1}}{\sqrt{ {7}} {-2}}}$

Ans: The given number is $\dfrac{1}{\sqrt{7}-2}$.

Multiplying and dividing by $\sqrt{7}+2$ to the number gives

$\dfrac{1}{\sqrt{7}-2}=\dfrac{\sqrt{7}+2}{\left( \sqrt{7}-2 \right)\left( \sqrt{7}+2 \right)}\\$.

Now, applying the formula $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ to the denominator gives

$ \dfrac{1}{\sqrt{7}-2}=\dfrac{\sqrt{7}+2}{{{\left( \sqrt{7} \right)}^{2}}-{{\left( 2 \right)}^{2}}} $

$ =\dfrac{\sqrt{7}+2}{7-4} $ 

$  =\dfrac{\sqrt{7}+2}{3}. $

Exercise (1.6)

1. Compute the value of each of the following expressions:

(i) $\mathbf{ {6}{{ {4}}^{\dfrac{ {1}}{ {2}}}}}$

Ans: The given number is \[{{64}^{\dfrac{1}{2}}}\].

By the laws of indices,

${{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$, where$a>0$.

$ {{64}^{\dfrac{1}{2}}}=\sqrt[2]{64} $

$  =\sqrt[2]{8\times \text{8}} $

$  =8. $

Hence, the value of ${{64}^{\dfrac{1}{2}}}$ is $8$.

(ii) $ \mathbf{{3}{{ {2}}^{\dfrac{ {1}}{ {5}}}}}$

Ans: The given number is ${{32}^{\dfrac{1}{5}}}$.

${{a}^{\dfrac{m}{n}}}=\sqrt[m]{{{a}^{m}}}$, where $a>0$

$ {{32}^{\dfrac{1}{5}}}=\sqrt[5]{32}$

$ =\sqrt[5]{2\times 2\times 2\times 2\times 2} $ 

$ =\sqrt[5]{{{2}^{5}}} $

Alternative Method:

By the law of indices ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$, then it gives

$ {{32}^{\dfrac{1}{5}}}={{(2\times 2\times 2\times 2\times 2)}^{\dfrac{1}{5}}}$ 

$ ={{\left( {{2}^{5}} \right)}^{\dfrac{1}{5}}} $

$ ={{2}^{\dfrac{5}{5}}} $

Hence, the value of the expression ${{32}^{\dfrac{1}{5}}}$ is $2$.

(iii) $\mathbf{{12}{{ {5}}^{\dfrac{ {1}}{ {5}}}}}$

Ans: The given number is ${{125}^{\dfrac{1}{3}}}$.

By the laws of indices

${{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$ where$a>0$.

$ {{125}^{\dfrac{1}{3}}}=\sqrt[3]{125} $

$  =\sqrt[3]{5\times 5\times 5} $

$  =5. $

Hence, the value of the expression ${{125}^{\dfrac{1}{3}}}$ is $5$.

2. Compute the value of each of the following expressions:

(i) $\mathbf{{{ {9}}^{\dfrac{ {3}}{ {2}}}}}$

Ans: The given number is ${{9}^{\dfrac{3}{2}}}$.

 ${{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$ where $a>0$.

$ {{9}^{\dfrac{3}{2}}}=\sqrt[2]{{{\left( 9 \right)}^{3}}} $

$  =\sqrt[2]{9\times 9\times 9} $

$ =\sqrt[2]{3\times 3\times 3\times 3\times 3\times 3} $

$=3\times 3\times 3 $

By the laws of indices, ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$, then it gives

$ {{9}^{\dfrac{3}{2}}}={{\left( 3\times 3 \right)}^{\dfrac{3}{2}}}$

$  ={{\left( {{3}^{2}} \right)}^{\dfrac{3}{2}}} $

$  ={{3}^{2\times \dfrac{3}{2}}} $

$ ={{3}^{3}} $

${{9}^{\dfrac{3}{2}}}=27.$

Hence, the value of the expression ${{9}^{\dfrac{3}{2}}}$ is $27$.

(ii) $\mathbf{{3}{{ {2}}^{\dfrac{ {2}}{ {5}}}}}$

Ans: We know that ${{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$ where $a>0$.

We conclude that ${{32}^{\dfrac{2}{5}}}$ can also be written as

$ \sqrt[5]{{{\left( 32 \right)}^{2}}}=\sqrt[5]{\left( 2\times 2\times 2\times 2\times 2 \right)\times \left( 2\times 2\times 2\times 2\times 2 \right)} $ 

$  =2\times 2 $

$ =4 $ 

Therefore, the value of ${{32}^{\dfrac{2}{5}}}$ is $4$.

(iii) $\mathbf{{1}{{ {6}}^{\dfrac{ {3}}{ {4}}}}}$

Ans: The given number is ${{16}^{\dfrac{3}{4}}}$.

By the laws of indices, 

${{a}^{\dfrac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$, where $a>0$.

$ {{16}^{\dfrac{3}{4}}}=\sqrt[4]{{{\left( 16 \right)}^{3}}} $

$  =\sqrt[4]{\left( 2\times 2\times 2\times 2 \right)\times \left( 2\times 2\times 2\times 2 \right)\times \left( 2\times 2\times 2\times 2 \right)} $

$  =2\times 2\times 2 $

Hence, the value of the expression ${{16}^{\dfrac{3}{4}}}$ is $8$.

${{({{a}^{m}})}^{n}}={{a}^{mn}}$, where $a>0$.

$ {{16}^{\dfrac{3}{4}}}={{(4\times 4)}^{\dfrac{3}{4}}} $

$  ={{({{4}^{2}})}^{\dfrac{3}{4}}} $ 

$ ={{(4)}^{2\times \dfrac{3}{4}}} $

$ ={{({{2}^{2}})}^{2\times \dfrac{3}{4}}} $ 

$ ={{2}^{2\times 2\times \dfrac{3}{4}}} $

$ ={{2}^{3}} $

Hence, the value of the expression is ${{16}^{\dfrac{3}{4}}}=8$.

(iv) $\mathbf{{12}{{ {5}}^{ {-}\dfrac{ {1}}{ {3}}}}}$

Ans: The given number is ${{125}^{-\dfrac{1}{3}}}$.

By the laws of indices, it is known that 

${{a}^{-n}}=\dfrac{1}{{{a}^{^{n}}}}$, where $a>0$.

Therefore, 

$ {{125}^{-\dfrac{1}{3}}}=\dfrac{1}{{{125}^{\dfrac{1}{3}}}} $

$  ={{\left( \dfrac{1}{125} \right)}^{\dfrac{1}{3}}} $

$ =\sqrt[3]{\left( \dfrac{1}{125} \right)} $

$ =\sqrt[3]{\left( \dfrac{1}{5}\times \dfrac{1}{5}\times \dfrac{1}{5} \right)} $

$ =\dfrac{1}{5}. $

Hence, the value of the expression ${{125}^{-\dfrac{1}{3}}}$ is  $\dfrac{1}{5}$.

3. Simplify and evaluate each of the expressions:

(i)$\mathbf{{{ {2}}^{\dfrac{ {2}}{ {3}}}} {.}{{ {2}}^{\dfrac{ {1}}{ {5}}}}}$

Ans: The given expression is ${{2}^{\dfrac{2}{3}}}{{.2}^{\dfrac{1}{5}}}$.

By the laws of indices, it is known that

${{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}$, where $a>0$.

 ${{2}^{\dfrac{2}{3}}}{{.2}^{\dfrac{1}{5}}}={{(2)}^{\dfrac{2}{3}+\dfrac{1}{5}}}$

 $ ={{(2)}^{\dfrac{10+3}{15}}} $

 $ ={{2}^{\dfrac{13}{15}}}. $

Hence, the value of the expression ${{2}^{\dfrac{2}{3}}}{{.2}^{\dfrac{1}{5}}}$ is ${{2}^{\dfrac{13}{15}}}$.

(ii) $\mathbf{{{\left( {{ {3}}^{\dfrac{ {1}}{ {3}}}} \right)}^{ {7}}}}$

Ans: The given expression is ${{\left( {{3}^{\dfrac{1}{3}}} \right)}^{7}}$.

It is known by the laws of indices that,

 ${{({{a}^{m}})}^{n}}={{a}^{mn}}$, where $a>0$.

${{\left( {{3}^{\dfrac{1}{3}}} \right)}^{7}}={{3}^{\dfrac{7}{3}}}.$

Hence, the value of the expression ${{\left( {{3}^{\dfrac{1}{3}}} \right)}^{7}}$is  ${{3}^{\dfrac{7}{3}}}$.

(iii) $\dfrac{ {1}{{ {1}}^{\dfrac{ {1}}{ {2}}}}}{ {1}{{ {1}}^{\dfrac{ {1}}{ {4}}}}}$

Ans: The given number is $\dfrac{{{11}^{\dfrac{1}{2}}}}{{{11}^{\dfrac{1}{4}}}}$.

It is known by the Laws of Indices that

 $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$, where $a>0$.

$\dfrac{{{11}^{\dfrac{1}{2}}}}{{{11}^{\dfrac{1}{4}}}}={{11}^{\dfrac{1}{2}-\dfrac{1}{4}}} $

$ ={{11}^{\dfrac{2-1}{4}}} $ 

$  ={{11}^{\dfrac{1}{4}}}. $

Hence, the value of the expression $\dfrac{{{11}^{\dfrac{1}{2}}}}{{{11}^{\dfrac{1}{4}}}}$ is  ${{11}^{\dfrac{1}{4}}}$.

(iv) $\mathbf{{{ {7}}^{\dfrac{ {1}}{ {2}}}} {.}{{ {8}}^{\dfrac{ {1}}{ {2}}}}}$

Ans: The given expression is ${{7}^{\dfrac{1}{2}}}\cdot {{8}^{\dfrac{1}{2}}}$.

${{a}^{m}}\cdot {{b}^{m}}={{(a\cdot b)}^{m}}$, where $a>0$.

$ {{7}^{\dfrac{1}{2}}}\cdot {{8}^{\dfrac{1}{2}}}={{(7\times 8)}^{\dfrac{1}{2}}} $  $={{(56)}^{\dfrac{1}{2}}}. $

Hence, the value of the expression ${{7}^{\dfrac{1}{2}}}\cdot {{8}^{\dfrac{1}{2}}}$ is ${{(56)}^{\dfrac{1}{2}}}$.

You can opt for Chapter 1 - Number System NCERT Solutions for Class 9 Maths PDF for Upcoming Exams and also You can Find the Solutions of All the Maths Chapters below.

NCERT Solutions for Class 9 Maths

Chapter 1 - Number System

Chapter 2 - Polynomials  

Chapter 3 - Coordinate Geometry

Chapter 4 - Linear Equations in Two Variables

Chapter 5 - Introductions to Euclids Geometry

Chapter 6 - Lines and Angles

Chapter 7 - Triangles

Chapter 8 - Quadrilaterals

Chapter 9 - Areas of Parallelogram and Triangles

Chapter 10 - Circles

Chapter 11 - Constructions

Chapter 12 - Herons formula

Chapter 13 - Surface area and Volumes

Chapter 14 - Statistics

Chapter 15 - Probability

NCERT Solutions for Class 9 Maths Chapter 1 All Exercise

Class 9 maths chapter 1 solutions - free pdf download.

Now that we have given you an idea about how important it is to get a base in maths early on, we shall also acknowledge that math is not the easiest subject in the world for everyone. Sometimes, it can be a thing of nightmares, quite literally when you need to stay up all night trying to understand some or another concept for a maths exam the next day. Here’s where the NCERT solutions for class 9 maths ch 1 come in - they can help you out in such situations where you aren’t being able to understand. These NCERT solutions for class 9 maths (chapter 1 especially) are magical for students who dislike or are weak in maths. They provide all the answers to the questions in the back of every chapter in the book so that a student need not incessantly struggle with the same.

NCERT Solutions Class 9 Maths Chapter 1 - Weightage

Maths comes for a total of 100 marks, out of which 20 marks go from the internal assessment and the rest of the 80 marks come from the written final exam. The following is the breakdown of the syllabus and marks weightage of NCERT class 9 maths.

Maths Class 9

The Following is a Breakdown of the Weightage Marks for the Internal Assessment:

The Following is the Weightage Breakdown for the Final Written Exam:

As mentioned in the table, maths class 9 chapter 1, Number Systems, counts for 8 marks out of the total 80 marks for the written exam.

Benefits of NCERT Solutions Class 9 Maths Chapter 1

The NCERT answers for class 9 mathematics chapter 1 are extremely beneficial to students for a variety of reasons. At Vedantu, we strive to make the answers we develop for all students as exact and precise as possible, so that they are usable and beneficial to students. The following are some of the reasons why students should use the class 9 mathematics NCERT answers for chapter 1.

The NCERT solutions by Vedantu are completely free to access - there’s no need to pay for the materials that you need for your studies, and we understand this.

These maths NCERT solutions of class 9 chapter 1 not only help with studying for exams, but they’re also helpful for when students are trying to finish difficult homework questions.

Vedantu’s class 9 maths chapter 1 NCERT solutions are in a PDF format which can be downloaded. This prevents students from unnecessarily wasting time on the internet when the solutions are right there in your files on your PC or even mobile phone or tablet.

The solutions can be printed, and this ensures that screen time for students is reduced.

The solutions have been written by maths teachers who are experienced in the field and, thus, the accuracy of the solutions is ensured.

Key Topics at a Glance

The number system is one of the most important chapters in the Class 9 Mathematics syllabus. The following is a summary of some of the key topics that must be addressed under the number system. We propose that students go through each of these concepts in order to acquire a solid understanding of the entire number system.

The number line

Whole numbers

Natural numbers

Rational numbers

Irrational numbers

Properties of numbers

Divisibility

H.C.F. and L.C.M.

Progressions

Multiplication tables

Squares and square roots

Cubes and cube roots

This concluded the discussion of the NCERT answers for Class 9 Chapter 1. We've seen the answers to every question in Chapter 1's exercises. To ace your exams, download the NCERT answers PDF. We hope we were able to answer your questions.

FAQs on NCERT Solutions for Class 9 Maths Chapter 1 - Number Systems

1. What all Comes Under the Purview of NCERT Maths Class 9 Chapter 1 Number Systems?

The subjects covered in NCERT mathematics class 9 chapter 1 Number Systems include a brief introduction to number systems using number lines, defining rational and irrational numbers using fractions, defining real numbers and declaring their decimal expansions. The chapter then returns to the number line to teach pupils how to express real numbers on it. In addition, the chapter teaches pupils how to add, subtract, multiply, and divide real numbers, or how to perform operations on real numbers. The rules of exponents for real numbers are a part of operations and are the final topic in class 9 mathematics chapter 1.

2. What are the Weightage Marks for Mathematics in Class 9?

The total mathematics paper in class 9 is 100 marks, like any other subject. Out of these 100 marks, 20 marks goes from internal assessments (pen and paper tests, multiple assessments, portfolios/project work and lab practicals for 5 marks each), and the remaining 80 marks are from the written test at the end of the school year. Out of these 80 marks, the chapter Number Systems comes for 8 marks, Algebra for 17 marks,  Coordinate Geometry for 4 marks, Geometry for 28 marks, Mensuration for 13 marks, and Statistics and Probability for 10 marks. All of these chapters’ respective marks total up to a cumulative 80 marks for the written paper.

3. How many sums are there in the NCERT Class 9 Chapter 1 Number System?

There are six exercises in the NCERT Class 9 Chapter 1 Number System. In the first exercise, Ex-1.1, there are 4 sums and in the second exercise, Ex-1.2, there are 3 sums. These first two exercises deal with the basic concepts of the number system, such as identifying the features of a rational number or an irrational number and locating them on the number line. In the third exercise, Ex-1.3, there are 9 sums, and most of them have sub-questions. The fourth exercise, Ex-1.4, comprises 2 sums, that deal with successive magnification for locating a decimal number on the number line. The fifth exercise, Ex-1.5, consists of 5 sums, on the concept of rationalization. The sixth exercise, Ex-1.6, consists of 3 sums, that have sub-questions. The sums in this exercise will require you to find the various roots of numbers.

4. Why should we download NCERT Solutions for Class 9 Maths Chapter 1?

Students should download NCERT Solutions for Class 9 Maths Chapter 1 from Vedantu (vedantu.com) to understand and learn the concepts of the Number System easily. These solutions are available free of cost on Vedantu (vedantu.com). Students must have a solid base of all concepts of Class 9 Maths if they want to score well in their exams. They can download the NCERT Solutions and other study materials such as important questions and revision notes for all subjects of Class 9. You can download these from Vedantu mobile app also.

5. Why are Class 9 Maths NCERT Solutions Chapter 1 important?

Some students find it difficult to study and score good marks in their Maths exam. They get nervous while preparing for it and goof up in their exams. However, if they utilise the best resources for studying, they can do well. This is why the Class 9 Maths NCERT Solutions Chapter 1 is important. The answers to all the questions from the back of each chapter are provided for the reference of students. 

6. Give an overview of concepts present in NCERT Solutions for Class 9 Maths Chapter 1?

The concepts in the NCERT Solutions for Class 9 Maths Chapter 1 include the introduction of number systems, rational and irrational numbers using fractions, defining real numbers, decimal expansions of real numbers, number line, representing real numbers on a number line, addition, subtraction, multiplication and division of real numbers and laws of exponents for real numbers. Chapter 1 of Class 9 Maths has a weightage of 8 marks in the final exam. 

7. Do I Need to Practice all Questions Provided in NCERT Solutions Class 9 Maths Number Systems?

Yes. Students should practice all the questions provided in the NCERT Solutions of the Number Systems chapter of Class 9 Maths, as they have been created with precision and accuracy, by expert faculty, for the students. Students can access them for free and also download them for offline use to reduce their screen time. The solutions are beneficial not only for exams but also for school homework.

8. Where can I get the NCERT Solutions for Class 9 Maths Chapter 1?

Students can download the NCERT Solutions for Class 9 Maths Chapter 1 from NCERT Solutions for Class 9 Maths Chapter 1. These are available free of cost on Vedantu (vedantu.com). These can be downloaded from the Vedantu app as well. The answers to all the questions from the 6 exercises of Chapter 1 Number Systems are provided in the NCERT Solutions. Students would also learn how to solve one question with different techniques if available. This will help them learn how to structure their answers in their Class 9 Maths exam. 

NCERT Solutions for Class 9

• NCERT Solutions
• NCERT Class 9
• NCERT 9 Maths
• 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

Download most important questions for class 9 maths chapter – 1 number systems.

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 2023-24 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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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 n-1 Pn by square root spiral drawing a line segment of unit length perpendicular to OP n-1 . 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 non-terminating non-recurring.

We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring 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 non-terminating and non-recurring 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 non-terminating but recurring, it is a rational number.

(v) 1.101001000100001…

Since the number,1.101001000100001…, is non-terminating non-repeating (non-recurring), 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 non-terminating and non-recurring.

Now, substituting the value of √5 in 2 –√5, we get,

2-√5 = 2-2.2360679… = -0.2360679

Since the number, – 0.2360679…, is non-terminating non-recurring, 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 non-terminating non-recurring, 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 non-terminating non-recurring, 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 = 9-3

(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 ) = 5-2 = 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 semi-circle 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 semi-circle), 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)/√7 2 -√6 2 [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+√6)/(7-6)

= (√7+√6)/1

(iii) 1/(√5+√2)

Multiply and divide 1/(√5+√2) by (√5-√2)

= (√5-√2)/(√5 2 -√2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√5-√2)/(5-2)

= (√5-√2)/3

(iv) 1/(√7-2)

Multiply and divide 1/(√7-2) by (√7+2)

1/(√7-2)×(√7+2)/(√7+2) = (√7+2)/(√7-2)(√7+2)

= (√7+2)/(√7 2 -2 2 ) [denominator is obtained by the property, (a+b)(a-b) = a 2 -b 2 ]

= (√7+2)/(7-4)

Exercise 1.6 Page: 26

64 1/2 = (8×8) 1/2

= 8 1 [⸪2×1/2 = 2/2 =1]

32 1/5 = (2 5 ) 1/5

= 2 1 [⸪5×1/5 = 1]

(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

= 3 3 [⸪2×3/2 = 3]

(ii) 32 2/5

32 2/5 = (2×2×2×2×2) 2/5

= (2 5 ) 2⁄5

= 2 2 [⸪5×2/5= 2]

(iii)16 3/4

16 3/4 = (2×2×2×2) 3/4

= (2 4 ) 3⁄4

= 2 3 [⸪4×3/4 = 3]

(iv) 125 -1/3

125 -1/3 = (5×5×5) -1/3

= (5 3 ) -1⁄3

= 5 -1 [⸪3×-1/3 = -1]

3. Simplify :

(i) 2 2/3 ×2 1/5

2 2/3 ×2 1/5 = 2 (2/3)+(1/5) [⸪Since, a m ×a n =a m+n ____ Laws of exponents]

= 2 13/15 [⸪2/3 + 1/5 = (2×5+3×1)/(3×5) = 13/15]

(ii) (1/3 3 ) 7

(1/3 3 ) 7 = (3 -3 ) 7 [⸪Since,(a m ) n = a m x n ____ Laws of exponents]

(iii) 11 1/2 /11 1/4

11 1/2 /11 1/4 = 11 (1/2)-(1/4)

= 11 1/4 [⸪(1/2) – (1/4) = (1×4-2×1)/(2×4) = 4-2)/8 = 2/8 = ¼ ]

(iv) 7 1/2 ×8 1/2

7 1/2 ×8 1/2 = (7×8) 1/2 [⸪Since, (a m ×b m = (a×b) m ____ Laws of exponents]

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 step-wise 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 problem-solving 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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NCERT Solutions Class 9 Maths Chapter 1 Number Systems

NCERT solutions for class 9 maths chapter 1 number systems consists of an introduction about the number system and the different kinds of numbers in it. The number system has been classified into different types of numbers like natural numbers, whole numbers , integers, rational numbers, irrational numbers , etc. The NCERT solutions class 9 maths chapter 1 covers all the basics of the number system which will be helpful in forming the basic foundation of mathematics.

Class 9 maths chapter 1 number systems will help the students in differentiating between rational and irrational numbers, wherein irrational numbers cannot be expressed in the form of a ratio, and also about real numbers. Class 9 maths NCERT solutions chapter 1 number systems sample exercises can be downloaded from the links below and also you can find some of these in the exercises given below.

• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.1
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.2
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.3
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.4
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.5
• NCERT Solutions Class 9 Maths Chapter 1 Ex 1.6

NCERT Solutions for Class 9 Maths Chapter 1 PDF

These NCERT solutions for class 9 maths involving the important concepts of real numbers , rational and irrational numbers, are available for free pdf download. The questions involving real numbers and their decimal form, the law of exponents are given below:

☛ Download Class 9 Maths NCERT Solutions Chapter 1 Number Systems

NCERT Class 9 Maths Chapter 1   Download PDF

NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

It is advisable for the students to practice the questions in the above links as this will give them better clarity on the kind of numbers and their properties. An exercise-wise detailed analysis of NCERT Solutions Class 9 Maths Chapter 1 number systems is given below for reference.

• Class 9 Maths Chapter 1 Ex 1.1 - 4 Questions
• Class 9 Maths Chapter 1 Ex 1.2 - 4 Questions
• Class 9 Maths Chapter 1 Ex 1.3 - 9 Questions
• Class 9 Maths Chapter 1 Ex 1.4 - 2 Questions
• Class 9 Maths Chapter 1 Ex 1.5 - 5 Questions
• Class 9 Maths Chapter 1 Ex 1.6 - 11 Questions

☛ Download Class 9 Maths Chapter 1 NCERT Book

Topics Covered: The important topics focussed upon are irrational numbers, real numbers, and real numbers when expanded in the decimal form. The class 9 maths NCERT solutions chapter 1 covers the representation of real numbers on a number line, methods to perform operations on real numbers, and laws of exponents when dealing with real numbers.

Total Questions: Class 9 maths chapter 1 Number Systems consists of total 35 questions of which 30 are easy, 2 are moderate and 3 are long answer-type questions.

List of Formulas in NCERT Solutions Class 9 Maths Chapter 1

NCERT solutions class 9 maths chapter 1 covers important facts about the number systems which will help strengthen the math foundation. Like if a number ‘a’ is rational, and ‘b’ represents an irrational number, then ‘a+b’, and ‘a-b’ are irrational numbers, and ‘ab’ and ‘a/b’ are supposed to be irrational numbers, and ‘b’ is not equal to zero. For ‘a’ and ‘b’ positive real numbers the following formula or entities will be true:

• √ab = √a √b
• √(a/b) = √a / √b

Important Questions for Class 9 Maths NCERT Solutions Chapter 1

Video solutions for class 9 maths ncert chapter 1, faqs on ncert solutions class 9 maths chapter 1, do i need to practice all questions provided in ncert solutions class 9 maths number systems.

Practicing the NCERT solutions class 9 maths number systems and exercises on real numbers, rational numbers will help in exploring the number systems in a better way. The NCERT Solutions Class 9 Maths Number Systems will also provide a good insight into the solving of problems.

Why are Class 9 Maths NCERT Solutions Chapter 1 Important?

Since the number systems chapter deals with rational and irrational numbers, real numbers, and their expansion, their decimal form, also covering the law of exponents. Hence, this makes the NCERT solutions class 9 maths important for examinations.

What are the Important Formulas in NCERT Solutions Class 9 Maths Chapter 1?

There are several formulas or entities for positive real numbers which will be helpful in learning mathematics even for higher grades. Like if one wants to rationalize the denominator of 1/ ( √a + b ), then we can multiply and divide by its algebraic conjugate which is √a - b

How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 1 Real Numbers?

The questions in the NCERT Solutions Class 9 Maths Chapter 1 are a great way for learning real numbers. There are around 35 questions dealing with number systems with 25 of them being simple and have straightforward logic, 6 of them are with medium complexity and 4 are elaborative questions.

What are the Important Topics Covered in NCERT Solutions Class 9 Maths Chapter 1?

The NCERT Solutions Class 9 Maths Chapter 1 deal with integers, real numbers, rational and irrational numbers. Apart from these the important topics covered are the real numbers, and what happens when they are expanded in decimal form, the law of exponents in the case of real numbers, how to differentiate between rational and irrational numbers etc.

How CBSE Students can utilize NCERT Solutions Class 9 Maths Chapter 1 effectively?

The students should first practice all the examples to understand the logic and problem solving technique and should try to solve all the exercise questions. The CBSE itself recommends the NCERT Solutions Class 9 Maths for the board exam studies.

Class 9 Maths Case Study Questions of Chapter 1 Real Numbers

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Case study Questions in Class 9 Mathematics Chapter 1  are very important to solve for your exam. Class 9 Maths Chapter 1 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving  Class 9 Maths Case Study Questions  Chapter 1 Real Numbers

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In CBSE Class 9 Maths Paper, Students will have to answer some questions based on Assertion and Reason. There will be a few questions based on case studies and passage-based as well. In that, a paragraph will be given, and then the MCQ questions based on it will be asked.

Real Numbers Case Study Questions With Answers

Here, we have provided case-based/passage-based questions for Class 9 Maths Chapter 1 Real Numbers

Case Study/Passage-Based Questions

Case Study 1: A Mathematics Exhibition is being conducted in your school and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.

Observe the following factor tree and answer the following:

1. What will be the value of x?

Answer: b) 13915

2. What will be the value of y?

Answer: c) 11

3. What will be the value of z?

Answer: b) 23

4. According to the Fundamental Theorem of Arithmetic 13915 is a

a) Composite number

b) Prime number

c) Neither prime nor composite

d) Even number

Answer: a) Composite number

5. The prime factorization of 13915 is

a) 5 × 11 3  × 13 2

b) 5 × 11 3  × 23 2

c) 5 × 11 2  × 23

d) 5 × 11 2  × 13 2

Answer: c) 5 × 112 × 23

Case Study 2: Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.

(i) For what value of n, 4 n  ends in 0?

(a) 10 (b) when n is even (c) when n is odd (d) no value of n

Answer: (d) no value of n3

(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, an is a rational number?

(a) when n is any even integer (b) when n is any odd integer (c) for all n > 1 (d) only when n=0

Answer: (c) for all n > 1

(iii) If x and y are two odd positive integers, then which of the following is true?

(a) x 2 +y 2  is even (b) x 2 +y 2  is not divisible by 4 (c) x 2 +y 2   is odd (d) both (a) and (b)

Answer: (d) both (a) and (b)

(iv) The statement ‘One of every three consecutive positive integers is divisible by 3’ is

(a) always true (b) always false (c) sometimes true (d) None of these

Answer:(a) always true

(v) If n is any odd integer, then n 2 – 1 is divisible by

(a) 22 (b) 55 (c) 88 (d) 8

Answer: (d) 8

Hope the information shed above regarding Case Study and Passage Based Questions for Class 9 Mathematics Chapter 1 Real Numbers with Answers Pdf free download has been useful to an extent. If you have any other queries about CBSE Class 9 Maths Real Numbers Case Study and Passage Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible By Team Study Rate

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CBSE MCQ for Class 9 Maths Chapter 1 Number System Free PDF

Guys, we are working very hard to provide you with TOPIC-WISE MCQs (as listed below). Till then, attached below is the Master PDF having all the topics. Hope you understand. Enjoy your preparation! All the Best! 

CBSE MCQ for Class 9 Maths Chapter 1 Number System PDF

The CBSE MCQ for Class 9 Maths Chapter 1 Number System are provided above, in detailed and free to download PDF format. The solutions are latest , comprehensive , confidence inspiring , with easy to understand explanation . To download NCERT Class 9 Solutions PDF for Free, just click ‘ Download pdf ’.

Other MCQ Questions for Maths Class 9th CBSE

• CBSE MCQ for Class 9 Maths Chapter 2 Polynomials
• CBSE MCQ for Class 9 Maths: Chapter 3 Coordinate Geometry
• CBSE MCQ for Class 9 Maths: Chapter 4 Linear Equations in Two Variables
• CBSE MCQ for Class 9 Maths Chapter 5 Euclid’s Geometry

How should I study for my upcoming exams?

First, learn to sit for at least 2 hours at a stretch

Solve every question of NCERT by hand, without looking at the solution.

Solve NCERT Exemplar (if available)

Sit through chapter wise FULLY INVIGILATED TESTS

Practice MCQ Questions (Very Important)

Practice Assertion Reason & Case Study Based Questions

Sit through FULLY INVIGILATED TESTS involving MCQs. Assertion reason & Case Study Based Questions

After Completing everything mentioned above, Sit for atleast 6 full syllabus TESTS.

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