Excel in Mathematics - BOOK 8 Final (2080)_compressed

Book 8 Author Hukum Pd. Dahal Editor Tara Bahadur Magar MATHEMATICS MATHEMATICS vedanta Approved by the Government of Nepal, Ministry of Education, Science and Technology, Curriculum Development Centre, Sanothimi, Bhaktapur as an Additional Learning Material Vedanta Publication (P) Ltd. j]bfGt klAns];g k|f= ln= Vanasthali, Kathmandu, Nepal +977-01-4982404, 01-4962082 [email protected] www.vedantapublication.com.np vedanta Excel in

Book 8 Published by: Vedanta Publication (P) Ltd. j]bfGt klAns];g k|f= ln= Vanasthali, Kathmandu, Nepal +977-01-4982404, 01-4962082 [email protected] www.vedantapublication.com.np All rights reserved. No part of this publication may be reproduced, copied or transmitted in any way, without the prior written permission of the publisher. MATHEMATICS MATHEMATICS vedanta Excel in First Edition: B.S. 2077 (2020 A. D.) Second Edition: B.S. 2078 (2021 A. D.) Third Edition: B.S. 2079 (2022 A. D.) Fourth Revised and Updated Edition : B.S. 2080 (2023 A. D.) Price: Rs 487.00 Illustration: Prakash Sameer Printed Copies: 40,000

Preface The series of 'Excel in Mathematics' is completely based on the contemporary pedagogical teaching learning activities and methodologies extracted from Teachers' training, workshops, seminars, and symposia. It is an innovative and unique series in the sense that the contents of each textbooks of the series are written and designed to fulfill the need of integrated teaching learning approaches. The series has focused on promoting the fundamental mathematical skills: Social skill, Logical skill, Computational skill and Problem solving skill of the learners. Excel in Mathematics is an absolutely modified and revised edition of my three previous series: 'Elementary mathematics' (B.S. 2053), 'Maths in Action (B. S. 2059)', and 'Speedy Maths' (B. S. 2066). Excel in Mathematics has incorporated applied constructivism. Every lesson of the whole series is written and designed in such a manner, that makes the classes automatically constructive and the learners actively participate in the learning process to construct knowledge themselves, rather than just receiving ready made information from their instructors. Even the teachers will be able to get enough opportunities to play the role of facilitators and guides shifting themselves from the traditional methods of imposing instructions. Each unit of Excel in Mathematics series is provided with many more worked out examples. Worked out examples are arranged in the order of the learning objectives and they are reflective to the corresponding exercises. Therefore, each textbook of the series itself plays the role of a ‘Text Tutor’. There is a proper balance between the verities of problems and their numbers in each exercise of the textbooks in the series. Clear and effective visualization of diagrammatic illustrations in the contents of each and every unit in grades 1 to 5, and most of the units in the higher grades as per need, will be able to integrate mathematics lab and activities with the regular processes of teaching learning mathematics connecting to real life situations. The learner friendly instructions given in each and every learning content and activity during regular learning processes will promote collaborative learning and help to develop learner-centred classroom atmosphere. In grades 6 to 10, the provision of ‘General section’, ‘Creative section - A’, and ‘Creative section - B’ fulfill the coverage of overall learning objectives. For example, the problems in ‘General section’ are based on the knowledge, understanding, and skill (as per the need of the respective unit) whereas the ‘Creative sections’ include the Higher ability problems. Furthermore, the evaluations of all levels of learning achievements (knowledge, understanding, application and higher ability) based on a single problem are incorporated in every exercise, according to the revised and updated specification grid developed by CDC, Government of Nepal.

The provision of ‘Classwork’ from grades 1 to 5 promotes learners in constructing knowledge, understanding and skill themselves with the help of the effective roles of teacher as a facilitator and a guide. Besides, the teacher will have enough opportunities to judge the learning progress and learning difficulties of the learners immediately inside the classroom. These classworks prepare learners to achieve higher abilities in problem solving. Of course, the commencement of every unit with 'Classwork-Exercise' plays a significant role as a 'Textual-Instructor'. The 'project works' given at the end of most of the exercises in the textbooks of each grade provide some ideas to connect the learning of mathematics to the real life situations. The provision of ‘Section A’ and ‘Section B’ in grades 4 and 5 provides significant opportunities to integrate mental maths and manual maths simultaneously. Moreover, the problems in ‘Section A’ judge the level of achievement of knowledge and understanding, and diagnose the learning difficulties of the learners. The provision of ‘Looking back’ at the beginning of each unit in grades 1 to 8 plays an important role of ‘placement evaluation’ which is in fact used by a teacher to judge the level of prior knowledge and understanding of every learner to select their teaching learning strategies. The socially communicative approach by language and literature in every textbook, especially in primary level of the series, plays a vital role as a ‘textual-parents’ to the young learners and helps them overcome maths anxiety. Furthermore, as per the need of ICT in teaching and learning mathematics, we have included 'Vedanta ICT Corner' to fulfil this requirement in all classes. The Excel in Mathematics series is completely based on the latest curriculum of mathematics, designed and developed by the Curriculum Development Centre (CDC), the Government of Nepal. I do hope the students, teachers, and even the parents will be highly benefited from the ‘Excel in Mathematics’ series. Constructive comments and suggestions for the further improvements of the series from the concerned are highly appreciated.

Acknowledgment In making effective modification and revision in the Excel in Mathematics series from the previous series, I am highly grateful to the Principals, HODs, Mathematics teachers and experts, PABSON, NPABSAN, PETSAN, ISAN, EMBOCS, NISAN, UNIPS and independent clusters of many other Schools of Nepal, for providing me with opportunities to participate in workshops, Seminars, Teachers’ training, Interaction programme, and symposia as the resource person. Such programmes helped me a lot to investigate the teaching-learning problems and to research the possible remedies and reflect to the series. I’m very proud of my wife Rita Rai Dahal who always encourages me to write the texts in a more effective way so that the texts stand as useful and unique in all respects. I’m equally grateful to my son Bishwant Dahal and my daughter Sunayana Dahal for their important roles as my ‘Model Learners’. I am extremely grateful to Dr. Ruth Green, a retired professor from Leeds University, United Kingdom who provided me with very valuable suggestions about the effective methods of teaching-learning mathematics and many reference materials. I would like to express my deepest appreciation to Mr. Tara Bahadur Magar (the editor of the series) and Mr. Bir Bahadur Chalaune (Veer) for their praiseworthy efforts in the preparation of ICT Materials, which are presented in the Vedanta ICT Corner. I am thankful to Dr. Komal Phuyal for editing the language of the series. Moreover, I gratefully acknowledge all Mathematics Teachers throughout the country who encouraged me and provided me with the necessary feedback during the workshops/interactions and teachers’ training programmes in order to prepare the series in this shape. I am profoundly grateful to the Vedanta Publication (P) Ltd. for publishing this series. I would like to thank Chairperson Mr. Suresh Kumar Regmi, Managing Director Mr. Jiwan Shrestha, and Marketing Director Mr. Manoj Kumar Regmi for their invaluable suggestions and support during the preparation of the series. Also, I am heartily thankful to Mr. Pradeep Kandel, the Computer and Designing Senior Officer of the publication house for his skill in designing the series in such an attractive form. -Author

S.N Unit Page 1. Sets 11-31 1.1 Set - Looking back, 1.2 Set - A collection of well-defined objects, 1.3 Notation of set, 1.4 Methods of describing sets, 1.5 Cardinal number of sets, 1.6 Types of sets, 1.7 Relationships between sets, 1.8 Representation of disjoint and overlapping sets in Venn-diagrams, 1.9 Universal set, 1.10 Subset and super set, 1.11 Proper and improper subsets, 1.12 Number of subsets of a given set 2. Number System in Different Bases 32-38 2.1 Denary, binary, and quinary numbers - Looking back, 2.2 Decimal numeration system, 2.3 Binary number system, 2.4 Formation of binary number system, 2.5 Expanded form of binary number, 2.6 Conversion of decimal numbers to binary numbers, 2.7 Conversion of binary numbers to decimal numbers, 2.8 Quinary number system, 2.9 Conversion of decimal numbers to quinary numbers, 2.10 Conversion of quinary numbers to decimal numbers 3. Real Numbers 39-55 3.1 Types of numbers - Looking back, 3.2 Natural numbers and whole numbers - Looking back, 3.3 Integers , 3.4 Sign rules of addition of integers, 3.5 Sign rules of multiplication and division of integers, 3.6 Rational numbers - review, 3.7 Terminating and non-terminating recurring decimals, 3.8 Non-terminating and non-recurring decimals, 3.9 Irrational numbers, 3.10 Comparison between rational and irrational numbers, 3.11 Real numbers, 3.12 Scientific notation of numbers 4. Ratio and Proportion 57-73 4.1 Ratio – Looking back, 4.2 Ratio - A comparision of quantities of the same kind, 4.3 Compounded ratio, 4.4 Proportion, 4.5 Types of proportions, 5. Unitary Method 74-83 5.1 Unit quantity and unit value - Looking back, 5.2 Unitary method, 6. Simple Interest 84-90 6.1 Principal and Interest, 6.2 Simple Interest, 6.3 Rate of interest, 6.4 Formula of simple interest 7. Profit and Loss 91-105 7.1 Profit and loss - Looking back, 7.2 Profit and loss percent, 7.3 To find S.P. when C.P. and profit or loss percents are given, 7.4 To find C.P. when S.P. and profit or loss percent are given, 7.5 Discount and discount percent, 7.6 Value Added Tax (VAT) 8. Laws of Indices 107-114 8.1 Indices - Looking back, 8.2 Laws of Indices - Review 9. Algebraic Expressions 115-139 9.1 Algebraic terms and expressions - Looking back , 9.2 Polynomials and degree of polynomials, 9.3 Evaluation of algebraic expressions, 9.4 Special products and formulae, 9.5 Factors and factorisation – Looking back, 9.6 Highest Common Factor (H.C.F) , 9.7 H.C.F of polynomial expressions, 9.8 Lowest Common Multiple (L.C.M) 10. Rational Expressions 140-147 10.1 Rational expressions, 10.2 Reduction of rational expressions to their lowest terms, 10.3 Multiplication of rational expressions, 10.4 Division of rational expressions, 10.5 Addition and subtraction of rational expressions Assessment - I Assessment - II Contents

S.N Unit Page 11. Equation and Graph 148-165 11.1 Equation - Looking back, 11.2 Linear equations with one variable, 11.3 Solution of linear equation, 11.4 Linear equation with two variables, 11.5 Graph of linear equation, 11.6 Simultaneous equations, 11.7 Methods of solving simultaneous equations, 11.8 Application of simultaneous equations, 11.9 Quadratic equation – Introduction, 11.10 Solution of quadratic equations, 11.11 Solving quadratic equations by factorisation method 12. Transformation 167-182 12.1 Transformation –Review, 12.2 Reflection, 12.3 Reflection of geometrical figure using coordinates, 12.4 Rotation, 12.5 Rotation of geometrical figures using coordinates, 12.6 Displacement, 12.7 Displacement of geometrical figures using coordinates 13. Geometry: Angles 183-194 13.1 Different pairs of angles – Looking back, 13.2 Experimental verifications of pair of angles formed by two intersecting lines, 13.3 Different pairs of angles made by a transversal with two straight lines, 13.4 Relation between pairs of angles made by a transversal with parallel lines 14. Geometry: Triangle 195-215 14.1 Triangle - Looking back, 14.2 Experimental verification of properties of triangles, 14.3 Congruent triangles, 14.4 Experiment on the conditions of congruency of triangles by construction, 14.5 Conditions of congruency of triangles, 14.6 Similar triangles, 14.7 Conditions of similarity of triangles 15. Geometry: Quadrilateral and Regular Polygon 216-237 15.1 Quadrilaterals - Looking back, 15.2 Some special types of quadrilaterals, 15.3 Construction of parallelogram, 15.4 Construction of rectangle, 15.5 Construction of square, 15.6 Regular polygons, 15.7 Sum of the interior angles of a polygon, 15.8 Sum of the exterior angles of a polygon 16. Coordinates 238-248 16.1 Coordinates - Looking back, 16.2 Pythagoras Theorem, 16.3 Pythagorean Triples, 16.4 Distance between two points, 17. Area and Volume 249-278 17.1 Area of plane figures - Looking back , 17.2 Area of triangle, 17.3 Area of quadrilateral, 17.4 Perimeter of circle - Looking back, 17.5 Area of circle, 17.6 Solids and their nets, 17.7 Area of solids, 17.8 Volume of solids 18. Symmetry, Design and Tessellation 279-283 18.1 Symmetrical and asymmetrical shapes, 18.2 Line or axis of symmetry, 18.3 Tessellations, 18.4 Types of tessellations 19. Bearing and Scale Drawing 284-291 19.1 Bearing – Looking back, 19.2 Scale drawing – Review 20. Statistics 293-313 20.1 Review, 20.2 Collection of data, 20.3 Frequency table, 20.4 Grouped and continuous data, 20.5 Cumulative frequency table, 20.6 Graphical representation of data, 20.7 Measures of central tendency, 20.8 Arithmetic mean, 20.9 Median, 20.10 Quartiles, 20.11 Mode, 20.12 Range Answers 315-329 Model Question Set 330-332 Assessment - III Assessment - IV Assessment - V

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 11 Vedanta Excel in Mathematics - Book 8 1.1 Set - Looking back Classwork - Exercise Let’s say and write the answers of the following questions. 1. The height of a few students of class 8 in a school are given below. Rajesh - 4 ft 5 in, Alina - 4 ft 3 in, Kishan - 5 ft 2 in, Bikash - 4 ft 9 in Parwati - 4ft 1 in, Pinkey - 4 ft 6 in, Tenzin - 5 ft 1 in, Fatima - 4 ft 8 in a) Can you make a set of 5 'tall students'? Give reason. ........................................................................................................................... b) Let's write a set (P) of 5 tall students whose heights are more than 4 ft 5 in. ........................................................................................................................... c) Let's write a set (Q) of students whose height is less than 4 ft 3 in. ........................................................................................................................... d) Let's write a set (R) of students whose height is more than 5 ft 2 in. ........................................................................................................................... e) Let's categorise the sets P, Q, and R into empty set, unit set or infinite set. Set P is ......................., Set Q is ......................., Set R is ....................... 2. The factor-cards of the factors of 8 and 12 are shown below: 1 2 4 8 1 2 3 4 6 12 a) Let's write the set of factors of 8 (F8 ) in listing method. ........................................................................................................................... b) Let's write the set of factors of 12 (F12) in set-builder method. ........................................................................................................................... c) Let's show these two sets in the given Venn-diagram. d) Are the sets F8 and F12 disjoint or overlapping? Give reason. F8 F12 Unit 1 Sets

Vedanta Excel in Mathematics - Book 8 12 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 1.2 Set - A collection of well-defined objects Let's discuss about the following two types of collections of objects. Collection - A Collection - B 1. A collection of today's delicious snacks in school canteen. 1. A collection of today's snacks in school canteen. 2. A collection of four favourite activities conducted in the last 'Annual School Day' 2. A collection of four activities conducted in the last 'Annual School Day' a) Can you make sets from the collection - A? Discuss the reason with your friends and present in the class. b) Can you make sets from the collection - B? Discuss the reason with your friends and present in the class. Furthermore, let's take a collection of even numbers less than 10. The members of this collection are definitely 2, 4, 6, 8. These members are distinct and distiguishable objects. Because the statement 'even number less than 10' clearly defines the members which are to be included in the collection. Therefore, it is a well-defined collection. A well-defined collection of objects is called a set. On the other hand, a collection of today's delicious snacks in school canteen is not welldefined. Because delicious snacks for you may not be delicious for others. Therefore, it is not possible to include distinct and distinguishable members in this collection. Hence, it is not a set. However, a collection of today's snacks in school canteen is a set. 1.3 Notation of set We usually denote sets by capital letters. The members or elements of a set are enclosed inside the braces { } and the members are separated with commas. The table given below shows a summary of the symbols which are used in the notation of sets. Symbol Name Example Explanation { } Set W = {0, 1, 2, 3, 4} O = {1, 3, 5, 7, 9} The members of the sets are enclosed inside braces { } and separated with commas. Membership 1 W, 4 W, 5 O, 9 O The symbol '' denotes the membership of an element of the given set. Non-membership 5 W, 6 W, 2 0, 4 0 The symbol '' denotes the non-membership of an element to the given set. 2 6 8 4 Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 13 Vedanta Excel in Mathematics - Book 8 1.4 Methods of describing sets We usually use three methods to describe a set. These methods are description, listing (or roster), and set-builder (or rule) methods. Method Example Explanation Description M is a set of multiples of 3 less than 15. The common properties of elements of a set are described by words. Listing (or roster) M = {3, 6, 9, 12} The elements of a set are listed inside braces (or curly brackets) { }. Set-builder (or rule) M = {x : x multiples of 3, x < 15} A variable such as ‘x’ is used to describe the common properties of the elements of set by using symbols. 1.5 Cardinal number of sets Let's say and write the members of the following sets. a) Set of Mathematics Teachers of Basic Level in your school. = ........................................................................................................................ How many members are there in this set? ................................... b) Set of students of your class whose roll number is an odd number less than 10. = ...................................................................................................................... How many members are there in this set? ................................... Thus, the number of members (or elements) contained by a set is called the cardinal number of the set. For example: S = {Nepal, India, Bangladesh, Pakistan, Bhutan, Sri Lanka, Afghanistan, Maldives} is a set of SAARC countries. Here, the number of members of the set S is 8. Therefore, the cardinal number of the set S written as n(S) = 8. Similarly, P = {2, 3, 5, 7} is a set of prime numbers less than 10. In this set, there are 4 members. Therefore, n(P) = 4. Symbolically, we write the cardinal number of set A as n(A), set B as n(B), set X as n(X), and so on. Sets

Vedanta Excel in Mathematics - Book 8 14 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 1.6 Types of sets According to the cardinal number of sets, there are four types of sets. Types of sets Examples Explanation Null or empty set The set of odd numbers between 5 and 7. A = { } or A = φ It does not contain any element. It is denoted by empty braces { } or by φ (phi) Unit or singleton set The set of prime numbers between 5 and 9. P = {7} It contains only one element. Finite set The set of natural numbers less than 50. N = {1, 2, 3, 4, 5, … 49} It contains finite numbers of elements. It means counting of elements can be ended. Infinite set The set of natural numbers. N = {1, 2, 3, 4, 5, …} It contains infinite number of elements. It means, counting of elements is never ended. Facts to remember 1. A collection of any 4 games is a well-defined collection. However, a collection of any 4 favourite games is not a well-defined collection. 2. Set is a well-defined collection of objects. 3. The membership of objects in a set is denoted by the symbol and the non-membership is denoted by . 4. Description, listing (or roster) and set-builder (or rule) are the three main methods of describing sets. 5. The number of elements contained by a set is called its cardinal number. 6. A null or empty set does not contain any element and it is represented by { } or f (phi). Its cardinal number is 0 (zero). 7. A unit or singleton set contains only one element, a finite set contains finite number of elements and an infinite set contains infinite number of elements. Worked-out Examples Example 1: A collection of 'bigger even numbers less than 10.' a) Is it a well-defined collection? Give reason. b) Is it a set? Given reason. c) Let's express it as a well-defined collection. Solution: a) It is not a well-defined collection. Because it does not include distinct and distinguishable members. b) It is not a set. Because it is not a well-defined collection. c) A collection of even numbers less than 10 is a well-defined collection. Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 15 Vedanta Excel in Mathematics - Book 8 Example 2: The first five multiples of 3 and 4 are shown in the following numbered cards. 3 6 9 12 15 4 8 12 16 20 a) Let's write the set of the first five multiples of 3 in listing method. b) Let's write the set of the first five multiples of 4 in set-builder method. Solution: a) M3 = {3, 6, 9, 12, 15} b) M4 = {x : x is a multiple of 4, x ≤ 20} Example 3: Let F4 = {x : x is a factor of 4} and F6 = {y : y is a factor of 6} a) List the elements of the sets F4 and F6 separately and write the cardinal number of each set. b) List the common elements of F4 and F6 in a separate set (A) and write its cardinal number. c) List the common prime factors of F4 and F6 in a separate set (B). Is the set B a unit set? Give reason. Solution: a) F4 = {1, 2, 4} and n(F4 ) = 3 F6 = {1, 2, 3, 6} and n(F6 ) = 4 b) A = {1, 2} and n(A) = 2 c) B = {2} It is a unit or singleton set because it has only one element. EXERCISE 1.1 General Section - Classwork 1. Let's tick (√) the well-defined collections. a) A collection of Nepali movies relased in 2079 B.S. b) A collection of favourite Nepali movies released in 2079 B.S. c) A collection of smaller prime numbers less than 10. d) A collection of prime numbers less than 10. 2. If A = { 2, 4, 6, 8, ...} , let's say and write ‘true’ or ‘false’ as quickly as possible. a) 6 A ......................... b) 12 A ......................... c) 5 A ......................... d) 10 A ......................... e) 15 A ......................... f) 18 A ......................... www.geogebra.org/classroom/jadjcxpu Classroom code: JADJ CXPU Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Sets

Vedanta Excel in Mathematics - Book 8 16 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 3. Let's say and write whether the following sets are empty, unit, finite or infinite. a) A = { 1, 3, 5, 7, ...} ............................... b) B = { 2, 4, 6, 8, ... 100} ............................... c) C = { x : x natural numbers and x < 1 } ............................... d) D = { even numbers between 7 and 10} ............................... Creative Section - A 4. Let's take a collection of any three 'high mountains of Nepal' and answer the following questions. a) Is it a well-defined collection? Give reason. b) Is it a set? Give reason. c) Let's express it as a well-defined collection. 5. Let's take a collection of 'smaller odd numbers less than 10'. Answer the following questions. a) Is it a well-defined collection? Give reason. b) Is it a set? Give reason. c) Let's express it as a well-defined collection and list the member of the set so formed. d) What is the cardinal number of this set? 6. The factor-cards given below show the all possible factors of 12 and 18. 1 2 3 4 6 12 1 2 3 6 9 18 a) Let's write the set of factors of 12 (F12) in listing method. b) Let's write the set of factors of 18 (F18) in set-builder method. c) List the common elements of F12 and F18 in a separate set P and write its cardinal number. 7. The multiple-cards given below show the first five multiples of 4 and 7. 4 8 12 16 20 7 14 21 28 35 a) Let's write the set of the first five multiples of 4 (M4 ) and 7 (M7 ) in listng method. b) Rewrite the sets M4 and M7 ub set builder method. c) Let's write a set A taking the common elements of M4 and M7 . d) What type of set is the set A? Give reason. 8. Let P = {x : x is a prime number, x < 10} and Q = {y : y is an even number, y < 10}. a) List the elements of the sets P and Q separately and write the cardinal number of each set. Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 17 Vedanta Excel in Mathematics - Book 8 b) List the common elements of the sets P and Q in a separate set C and write its cardinal number. c) What type of set is the set C? Give reason. 9. a) A set of composite numbers less than 12. Express it in listing and set-builder methods. b) B = {the first five multiples of 2}. Express it in roster method and rewrite in set-builder method. c) Z = { x : x integers – 2 < x < 2}. List the elements of this set and also express it in description method. d) P = {2, 3, 5, 7}, express it in description method and in rule method. e) If A = {x : x is a letter of the word ‘SCHOOL’}, list the members and find n (A). f) If B = {p : p is a factor of 12}, list the elements of this set and find n (B). It's your time - Project Work and Activity Sections 10. a) Let's write any two collections of your own choice which are not well-defined. Is it possible to make these collections well-defined? b) Let's observe around your classroom and select any five objects as the members of a set. Then, express the set in description, listing and set-builder methods. c) Let's collect the names of your classmates who have the following roll numbers. odd numbers less than 10 even numbers between 10 and 20 prime numbers less than 15 multiples of 5 less than 40 (i) Write four separate sets of your classmates in roster method. (ii) Rewrite these four sets in rule method 1.7 Relationships between sets Let's study the following sets and discuss on the questions given below. 1. A = {a, e, i, o, u} and B = {u, i, o, a, e} 2. P = {2, 3, 5, 7} and Q = {2, 4, 6, 8} 3. M = {Ram, Shiva, Pinky} and N = {Bijash, Laxmi, Dorje, Sarada} 4. X = {Hari, Sita, Rekha, Badri} and Y = {Rekha, Bishwant, Sunayana} a) Do the sets A and B have exactly the same elements? What type of sets are A and B? b) Do the sets P and Q have exactly the same elements? Do they have the equal cardinal number? What type of sets are P and Q? c) Do the sets M and N have any common member? What type of sets are M and N? d) Do the sets X and Y have any common member? What type of sets are X and Y? Sets

Vedanta Excel in Mathematics - Book 8 18 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Thus, according to the types of elements or the number of elements contained by two or more sets, we can define various types of relationships between the sets: 1. Equal sets 2. Equivalent sets 3. Disjoint sets 4. Overlapping sets 1. Equal sets Equal sets have exactly the same elements. So, they always have the equal cardinal number. For example: A = {s, v, u, 3, ª} and B = {u, ª, v, s, 3} are the equal sets. Also, n(A) = n(B) = 5. We write equal sets A and B as A = B. 2. Equivalents sets Equivalent sets do not have exactly the same elements. However, they always have the equal cardinal number. For example: C = {c, cf, O, O{} and D = {c, d], l/, sf} are the equivalent sets because they have equal cardinal number, i.e. n(c)= n(D) = 4. 3. Disjoint sets: Let's study the following two separate sets of students of class VIII who like basket ball and cricket. B = {Ramesh, Sangita, Yelgina, Dipesh, Rahul} C = {Kopila, Paras, Sachin, Bhurashi} Is there are any student in sets B and C who like basketball as well as cricket? Is there any common member of the sets B and C? Here, the sets B and C do not have any common member (or element). Therefore, the sets B and C are disjoint sets. Thus, two or more than two sets are said to be disjoint sets if they do not have any common element. Similarly, P = {1, 3, 5, 7, 9} and Q = {2, 4, 6, 8, 10} are also disjoint sets. 4. Overlapping sets: Again, let's study the following sets of students whose roll numbers are even numbers (E) or prime numbers (P) less than 10. E = {Preeti, Sekhar, Dinesh, Maya} P = {Preeti, Mohan, Saru, Bikash} Now, let's discuss on the questions given below. a) Are the sets E and P disjoint sets? Why? b) Do the sets E and P have any common member? c) Who is the common member of the sets E and P? The common member of the sets E and P is 'Preeti'. Therefore, the sets E and P are overlapping sets. Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 19 Vedanta Excel in Mathematics - Book 8 Here, the sets E and P are not disjoint sets because they have a common member. The common member of the sets E and P is 'Preeti'. Therefore, the sets E and P are overlapping sets. Thus, two or more than two sets are said to be overlapping sets if they contain at least one common member (or element). Similarly, A = {1, 3, 5, 7, 9} and B = {3, 6, 9, 12} are also overlapping sets. Because they have the common elements 3 and 9. 1.8 Representation of disjoint and overlapping sets in Venn-diagrams We can use different types of diagrams to represent sets and relationships between the sets. The idea of such representations was first introduced by Swiss Mathematician Euler and it was further developed by the British Mathematician John Venn. So, the diagrams are famous as Venn Euler diagrams or simply Venn-diagrams. Usually, Venn-diagrams are of the shape of rectangle, circle or oval. Let's study the following illustrations and learn to represent disjoint and overlapping sets by using Venn-diagrams. A A and B are disjoint sets. B P and Q are overlapping sets. P Q X and Y are overlapping sets. X Y (i) A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10} are two disjoint sets. The non-intersecting circles represent the disjoint sets. (ii) P = {3, 6, 9, 12, 15, 18} and Q = {4, 8, 12, 16} are two overlapping sets. The intersecting circles represent the overlapping sets. The common element 12 is written in the common shaded region. www.geogebra.org/classroom/ss3fbud8 Classroom code: SS3F BUD8 Vedanta ICT Corner Please! Scan this QR code or browse the link given below: A 1 5 9 7 3 B 4 8 10 6 2 3 P Q 6 9 15 18 12 4 8 16 Sets

Vedanta Excel in Mathematics - Book 8 20 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Facts to remember 1. Equal sets have exactly the same elements. E.g. A = {a, e, i, o, u} and B = {u, o, i, e, a} are equal sets. 2. Equivalent set have equal cardinal numbers. E.g. P = {2, 4, 6, 8} and Q = {1, 3, 5, 7} are equivalent sets. 3. Disjoint sets do not have any common element. E.g. M = {s, v, u, 3} and N = {c, cf, O, O{} are disjoint sets. 4. Overlapping sets have at least one common element. E.g. X = {1, 2, 3, 4, 6, 12} and Y = {1, 3, 5, 15} are overlapping sets. Worked-out Examples Example 1: The table given below shows the name of students of class VIII who are participating in various Extra Curricular Activities (ECA) of an annual school's day. ECA Name of students Sports Yogesh, Rupa, Raju, Beena, Prasant Music Beena, Shusil, Gopal, Rahim Dance Rupa, Bikash, Keshav, Rita a) Write the separate sets of students who are participating in sports (S), music (M) and dance (D) in listing method. b) Are the sets M and D equal or equivalent? Give reason. c) Are the sets S and M disjoint or overlapping ? Give reason. d) Combine the members of the sets S and D in a new set and show it in a Venn-diagram. Solution: a) S = {Yogesh, Rupa, Raju, Beena, Prasant} M = {Beena, Shusil, Gopal, Rahim} D = {Rupa, Bikash, Keshav, Rita} b) The sets M and D are equivalent sets because n(M) = n(D) = 4. But they don't have exactly the same members. c) The sets S and M are overlapping sets because they have a common member which is Beena. d) Combination of the sets S and D = {Yogesh, Rupa, Raju, Beena, Prasant, Bikash, Keshav, Rita} Yogesh Bikash Keshav Rita Raju Beena Prasant Rupa S P Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 21 Vedanta Excel in Mathematics - Book 8 Example 2: List the elements of the sets given below. Then, answer the following questions. N = {natural numbers less than 6} W = {x : x is a whole number, 0<x<6} M = {y : y is a multiple of 6, y ≤ 30} F = {p : p is a factor of 12} a) Are the sets N and W equal or equivalent? Give reason. b) Are the sets M and F disjoint or overlapping? Give reason. c) Show the overlapping sets N and F in a Venn-diagram. Solution: a) N = {1, 2, 3, 4, 5} and W = {1, 2, 3, 4, 5} Since the sets N and W have exactly the same elements, they are equal sets. b) M = {6, 12, 18, 24, 30} and F = {1, 2, 3, 4, 6, 12} The sets M and F are overlapping sets as they contain some common elements which are 6 and 12. c) N = {1, 2, 3, 4, 5} F = {1, 2, 3, 4, 6, 12} Example 3: The members of the overlapping sets A and B are shown in the adjoining Venn-diagram. a) List the members of the sets A and B. b) Write the cardinal numbers of the sets A and B. c) List the members which belong the sets A as well as B. d) List the members which belong to only A. e) List the members which belong to only B. Solution: a) A = {4, 6, 8, 9, 10} and B = {1, 4, 9, 16, 25 } b) n(A) = n(B) = 5 c) The set of members which belong to A as well as B = {4, 9} d) Only A = {6, 8, 10} e) Only B = {1, 16, 25} EXERCISE 1.2 General Section - Classwork 1. Let's say whether the following pairs of sets are equal or equivalent. Write A = B or A ~ B in the blank spaces. a) A = {Ram, Sita, Gita, Hari} and B = {Hari, Laxmi, Gita, Shiva} ......................... b) A = {2, 3, 5, 7, 11} and B = {11, 7, 5, 3, 2} ......................... 5 N F 1 2 3 4 6 12 6 A B 8 10 4 9 1 16 25 Sets

Vedanta Excel in Mathematics - Book 8 22 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2. Let's say and write whether the following pairs of sets are disjoint or overlapping sets. a) P = {3, 6, 9, 12} and Q = {5, 10, 15}, P and Q are ...................................... b) X = {1, 3, 5, 7, 9} and Y = {1, 2, 3, 4, 5}, X and Y are ...................................... 3. Let's write 'disjoint' or 'overlapping' below the Venn-diagrams of the given pairs of sets. a) b) c) Creative Section - A 4. a) Define equal sets and equivalent sets with appropriate examples. b) Define disjoint sets and overlapping sets with appropriate examples. 5. Let A = {x : x is a factor of 15, x > 1} and B = {y : y is a factor of 16, y > 1} are two sets. a) List the members of the sets A and B. b) State whether the sets A and B are disjoint or overlapping sets. Give reason. c) Show the members of the sets A and B in Venn-diagram. 6. Let, P = {p : p ∈ multiples of 2, p < 10} and Q = {q : q ∈ multiples of 3, q < 10} are two sets. a) List the elements of the sets P and Q. b) State whether the sets P and Q are disjoint or overlapping. Give reason. c) Show the elements of the sets P and Q in Venn-diagram. 7. Let's list the members of the following pairs of overlapping sets shown in the Venn-diagrams. a) b) c) .................................... .................................... P Q N W .................................... A B Dinesh Pinky Rosy Gopal Arjun Ram A B 3 1 2 4 8 6 12 16 24 P X Y Q 0 6 7 8 9 10 1 2 4 5 3 Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 23 Vedanta Excel in Mathematics - Book 8 Creative Section - B 8. The table given below shows the name of students of class VIII who are participating in various. Extra Curricular Activities (ECA) of an annual school's day. ECA Name of students Singing Sanju, Binita, Harka, Kishan Dancing Binita, Rekha, Kishan, Gopal, Tashi Sports Hari, Shaswat, Anamol, Rekha, Tashi a) Write separate sets of students who are participating in singing (A), dancing (B) and sports (C). b) Are the sets B and C equal or equivalent? Give reason. c) Are the sets A and C disjoint or overlapping? Give reason. d) Are the sets A and B disjoint or overlapping? Give reason. e) Combine the members of the sets B and C in a new set D and show it in a Venn-diagram. 9. List the elements of the sets given below and answer the following questions. E = {x : x is an even number, x < 10} O = {y : y is an odd number, y < 10} P = {p : p is an prime number p < 10} a) Write a pair of equal sets. Why are they equal sets? b) Write a pair of disjoint sets. Why are they called disjoint sets? c) Write two pairs of overlapping sets. Why are they called overlapping sets? d) Combine the elements of the sets E and P and list in a new set. Then, show it in a Venn-diagram. e) Combine the elements of the sets O and P in a new set and show it in a Venndiagram. 10. The Venn-diagrams given below show the farmers who are involving in crops farming (C), vegetable farming (V) and animal farming (A) in a 'Model Village' of Sunsari district. Bimala Manoj Raju Prem Hari Shiva Kamala Gopal Lakhan C V Bimala Manoj Raju Harka Rahim Tshiring Prem Gopal Lakhan C A Raju Harka Prem Gopal Lakhan Hari Rahim Shiva Tshiring A V Kamala a) List the members of the sets C, V and A separately. b) Write the cardinal number of each set and identify the equivalent sets. c) List the members who belong to (i) C as well as V (ii) C as well as A and (iii) A as well as V in three separate sets. Sets

Vedanta Excel in Mathematics - Book 8 24 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 11. The elements of the overlapping sets P and Q are shown in the adjoining Venn-diagram. a) List the elements of the sets P and Q. b) Write the cardinal numbers of the sets P and Q. c) List the elements which belong to the sets P as well as Q. d) List the elements which belong to only P. e) List the elements which belong to only Q. It's your time - Project Work and Activity Section 12. a) Let's make groups of 10 classmates. Conduct a survey in your group (including yourself) and list the names in the following sets. (i) Sets of classmates who like football (F) and cricket (C) (ii) Are the sets F and C disjoint or overlapping? (iii) Combine the members of the sets F and C and show in Venn-diagram. (iv) If F and C are overlapping sets, list the members which belong to the sets F as well as C, only F and only C. b) Let's take the natural numbers from 1 to 20 and list the elements to form the following sets. (i) Sets of multiples of 2 (M2 ) and multiples of 3 (M3 ) upto 20. (ii) Combine the elements of the sets M2 and M3 and show in Venn-diagram. (iii) Sets of prime numbers (P) and odd numbers (O) les than 20. (iv) Combine the elements of the sets P and O and show in Venn-diagram. 1.9 Universal set Let's take a set of whole numbers upto 20, W = {0, 1, 2, 3, ... 20} Now, let's select certain elements from this set and make a few other sets. N = {natural numbers less than 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9} P = {prime numbers between 10 and 20} = {11, 13, 17, 19} E = {x : x is an even number, x ≤ 10} = {2, 4, 6, 8, 10} M = {y : y is a multiple of 3, y < 20} = {3, 6, 9, 12, 15, 18} W 0 N 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 5 3 W P 1 0 17 19 2 3 4 5 6 7 8 9 10 12 14 15 1816 20 11 13 0 W E 1 6 8 10 3 5 7 911 1213 14 15 16 17 18 19 20 2 4 W M 1 9 15 18 2 4 5 7 8 1011 12 13 14 16 17 19 20 3 6 0 Here, the set of whole numbers upto 20 is a universal set. The sets, N, P, E and M are the subsets of the universal set. 2 4 8 6 16 20 12 10 P Q Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 25 Vedanta Excel in Mathematics - Book 8 Thus, in a given situation, the set of all the elements being considered from which many other subsets can be formed is called a universal set. A set of students of a school is also a universal set. From this universal set, the subsets like the set of girls, the sets of boys, the set of eight class students, etc. can be formed. A universal set is denoted by U or {xi} 1.10 Subset and super set Let's take any two sets: A = {1, 2, 3, ... 10} and B = {2, 4, 6, 8, 10}. Here, every elements of set B is contained by the set A. In other words, the set B is contained by the set A. Therefore, set B is called the subset of A. B is subset of A is denoted as B ⊂ A. Similarly, A is said to be the super set of B. It is denoted as B ⊃ A. 1.11 Proper and improper subsets Proper subset Let's take a set of natural numbers upto 10. N = {set of natural numbers upto 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Now, let's discuss on the following questions. a) Is set A = {1, 3, 5, 7, 9} a subset of the set N? b) Are the sets N and A equal set? Here, A is a subset of the set N. However, the subset A is not equal to its super set N. Therefore, A is a proper subset of N. Thus, between a super set A and its subset B, the st B is said to be a proper subset of A if it contains at least one element less than the set A. It is denoted B ⊂ A. Improper subset Let's take a set of the first five letters of Nepali alphabets. P = {s, v, u, 3, ª} Now, let's discuss on the following questions. 1. Is set Q = {s, v, u, 3, ª} a subset of the set P? 2. Are the sets P and Q equal sets? Here, Q is a subset of the set P and the subset Q is equal to the super set P. Therefore, Q is the improper subset of P. It is denoted as Q ⊆ P. (Q ⊆ P means, Q ⊂ P and Q = P). Thus, between a super set A and its subset B, the set B is said to be an improper subset of A if B is equal to A. A B 1 3 5 7 9 2 4 8 10 6 Sets

Vedanta Excel in Mathematics - Book 8 26 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 1.12 Number of subsets of a given set Let's study the table given below and investigate the rule to find number of subsets of a given set. Set Subsets Cardinal number No. of subsets A = { } { } (or f ) n(A) = 0 1 ← 20 B = {a} {a}, f n(B) = 1 2 ← 21 C = {a, n} {a}, {n}, {a, n}, f n(C) = 2 4 ← 22 D = (a, n, t) {a}, {n}, {t}, {a, n}, {a, t}, {n, t}, {a, n, t}, f n(D) = 3 8 ← 23 From the above table, we can generalise a rule to find the number of subsets of a given set as 2n, where n is the cardinal number of given set. Facts to remember 1. A set under the consideration from which many other subsets can be formed is called a universal set (U). 2. Between the sets A and B, if B is contained by A, B is said to be a subset of A. However, if A is contained by B, A is the subset of B. A B B ⊂ A B A A ⊂ B 3. Between a given set and its subset, if the subset contains at least one element less than the set, it is a proper subset. It is denoted by the symbol '⊂'. 4. Between a given set and its subset, if the subset is equal to the given set, it is an improper subset. It is denoted by the symbol '⊆'. 5. An empty (or null) set f is always a subset of every given set. 6. Every given set is an improper subset itself. 7. The number of possible subsets of any given set = 2n, where 'n' is the cardinal number of the given set. Worked-out Examples Example 1: Let A = {1, 2, 3, 4, 5, ...10} and B = {1, 3, 5, 7, 9} a) Which one is the universal set? b) Which are a subset of the universal set? Give reason. c) Show the universal set and its subset in venn-diagram. Solution: a) Here, A = {1, 2, 3, 4, 5, ... 10} is the universal set. Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 27 Vedanta Excel in Mathematics - Book 8 b) B = {1, 3, 5, 7, 9} is a subset of the universal set A. Because every element of B is contained by A. c) Example 2: Let P = {x : x is multiple of 2, x ≤ 20} Q = {y : y is an even number, y ≤ 10} and R = {z : z is a factor of 16, z > 1} a) List the elements of the sets P, Q, R and identify the universal set and its subsets. b) Are the subsets proper or improper? Give reason. c) Combine the elements of Q and R and show in Venn-diagram along with the universal set. Solution: a) P = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}, Q = {2, 4, 6, 8, 10}, R = {2, 4, 8, 16} P is the universal set. Q and R are the subsets of P. b) The subsets Q and R are the proper subsets. Because the subsets Q and R are not equal to the set P. c) The combined elements of Q and R = {2, 4, 6, 8, 10, 16} Example 3: Let A = {s, e, t} is a set of letters of the word 'set' a) Write the formula to find the number of subsets of the set A. b) Find how many subsets of the set A are possible? c) Write all possible subsets of the set A. d) Which one is the improper subset? Give reason. Solution: a) The required formula is 2n, where n is the cardinal number. b) Here, n(A) = 3. So, the number of students of A = 2n = 23 = 8 c) All possible subsets of the set A are: {s}, {e}, {t}, {s, e}, {s, t}, {e, t}, {s, e, t} and f d) {s, e, t} is the improper subset. Because this subset is equal to the given set A. A B 2 4 6 8 10 1 3 7 9 5 6 4 8 16 10 Q R 2 1 3 5 7 9 11 12 13 14 15 17 18 19 20 P Sets

Vedanta Excel in Mathematics - Book 8 28 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Example 4: There are 30 students in a class. 15 of them are participating in sports, 12 are participating in music and 5 are participating in sports as well as music. a) Represent the universal set by U, the subsets by S (sports) and M (music). Then, write the cardinal numbers of U, S and M. b) Write the cardinal number of the members who belong to the sets S as well as M. c) Show the universal set and its subsets in Venn-diagram. d) From the Venn-diagram, find (i) How many students participate only sports? (ii) How many students participate only music? (iii) How many students do not participate any of two activities? Solution: a) n(U) = 30, n(S) = 15 and n(M) = 12 b) n(S as well as M) = 5 c) d) (i) 10 students participate only sports. (ii) 7 students participate only music (iii) 8 students do not participate in any of two activities. EXERCISE 1.3 General section - Classwork 1. Let's say and write the name of the sets, which are universal set or its subset. a) In A = {mathematics teachers of a school} and B = {teachers of the school} Universal set is ......................... and its subset is ......................... b) W = {whole number less than 100} and N = {natural numbers less than 100} Universal set is ......................... and its subset is ......................... c) W = {whole numbers less than 10} and Z = {integers less than 10} Universal set is ......................... and its subset is ......................... 2. Let's say and write the name of proper and improper subsets of the given universal sets. a) U = {Pooja, Dinesh, Harka, Pinky, Ganesh}, A = {Pooja, Pinky} B = {Ganesh, Pinky, Harka, Dinesh, Pooja} Proper subset is ....................... and improper subset is ....................... 10 5 7 S M 8 U Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 29 Vedanta Excel in Mathematics - Book 8 b) U = {–4, –3, –2, –1, 0, 1, 2, 3, 4}, W = {0, 1, 2, 3, 4}, Z= {integers between –5 and +5} Proper subset is ....................... and improper subset is ....................... 3. Let's say and write the number of subsets of the following sets. a) A = f, the number of subset is ....................... b) B = {c}, the number of subsets are ....................... c) C = {c, o}, the number of subsets are ....................... d) D = {c, o, w}, the number of subsets are ....................... 4. Let's write all possible subsets of the following sets. a) P = f, the subset is ....................................................................................... b) Q = {h}, the subset are ....................................................................................... c) R = {h, e}, the subset is ....................................................................................... Creative section - A 5. What can be the universal sets from which the following subsets can be formed? a) The set of cricket players of class VIII. b) The set of cricket players of the school. c) The set of odd numbers less than 10. 6. Let's answer the following questions. a) Is A = {1, 4, 9, 16} a subset of B = {1, 2, 3, 4, 5, ... 15}? Give reason. b) Is E = {2, 4, 6, 8} a subset of W = {0, 1, 2, 3, 4, 5, ... 10}? Give reason. c) Is P = {1, 2, 4, 5, 10, 20} a proper or improper subset of Q = {x : x is a factor of 20}? Write with reason. 7. a) F = {mango, apple} is a given set of fruits. (i) Find the number of subsets of the set F by using formula. (ii) Write all possible subsets of the set F. (iii) Among these subsets, which one is the improper subset? Write with reason. b) Let X = {b, a, l} is a set of letters of the word 'BALL'. (i) Find the number of subsets formed from the set X by using formula. (ii) Write all possible subsets of the set X. (iii) Among the all subsets, which one is the improper subset? Why? c) P = {x : x is a prime number, x < 10} is a given set. (i) List the members of the set P. (ii) Find the number of subsets of P using formula. (iii) Write the subsets of containing one member. Sets

Vedanta Excel in Mathematics - Book 8 30 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur (iv) Write the subsets containing two members. (v) Write the subsets containing three members. (vi) Write the subsets containing four members. 8. Let A = {1, 2, 3, 4, 5, ... 10} and B = {2, 4, 6, 8} are any two sets. a) Which one is the universal set? b) Which one is the subset of the universal set? Give reason. c) Show the universal set and its subset in Venn-diagram. 9. Let P = {x : x is a natural number, x ≤ 20}, Q = {y : y ∈ odd number, y < 10} and R = {z : z is a multiple of 3, z < 20} are any three sets. a) List the elements of the sets, P, Q and R. Identify the universal set and its subsets. b) Are the subsets proper or improper? Give reason. c) Combine the elements of Q and R and show in Venn-diagram along with the universal set P. Creative section - B 10. From the following Venn-diagrams, let's express the universal set and its subsets in listing method, description method and set-builder method. Then, answer the following questions. a) b) c) d) If p be the set of the common members of the sets A and B, list the members of P. e) If Q be the set of the common members of the sets C and D, list the members of Q. f) Are the sets E and F disjoint or overlapping? Write with reason. 11. In the given Venn-diagram, U is the set of all farmers in a village. A is the set of farmers involving in animal farming and V is the set of farmers involving in vegetable farming, a) Write the cardinal number of the universal set U. b) List the members of the sets A and V separately. Mahesh Shiva Goma Bikram Keshav Mina Kedar Laxmi Gopal Mohan Sita Dipak A V Hem Sundar Maya U 9 18 12 4 1 2 3 6 A 5 7 8 10 11 B 13 14 15 16 17 U 9 3 10 15 2 4 8 6 12 C 5 1 7 11 D 13 14 U 9 1 3 5 10 7 2 4 6 8 E 11 12 F 13 14 15 U Sets

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 31 Vedanta Excel in Mathematics - Book 8 c) List the members who belong to the sets A as well as V. d) List the members who belong to only set A. e) List the members who belong to only set V. f) List the members who do not belong to neither A nor V. 12. There are 36 students in a class. 18 of them are participating in sports, 14 are participating in music and 7 are participating in sports as well as music. a) Represent the universal set by U, the subsets by S (sport) and M (music). Write the cardinal numbers of U, S and M. b) Write the cardinal number of the members who belong to S as well as M. c) Show the universal sets and its subsets in Venn-diagram. d) From the Venn-diagram, find: (i) How many students participate only sports? (ii) How many students participate only music? (iii) How many students do not participate in any of two activities? It's your time - Project Work and Activity Section 13. a) Let's take a set of natural numbers upto 20 as a universal set U. Then, write a few subsets of the universal set, such as set of odd numbers, etc. b) Let's take a set of students of your class as a universal set. Then, write a few subsets of this universal set. c) Let's write a set of your own choice containing 4 members. Then, write all possible subsets of the set. Count and find, how many subsets are formed? Also, find the number of subsets by using formula. How many proper and improper subsets are formed? 14. Let's make a group of your at least 10 classmates (including yourself). Conduct a survey to collect the data and complete the table given below. Name of classmates Names who like mango Names who like orange Names who like mango as well as orange a) Let's draw a Venn-diagram to show the above data. b) From the Venn-diagram, find the following numbers. (i) How many students like mango in your group? (ii) How many students like orange in your group? (iii) How many of them like mango as well as orange? (iv) How many students like only mango? (v) How many students like only orange? (vi) If there are some students who like none of these two fruits, find their number. Sets

Vedanta Excel in Mathematics - Book 8 32 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2.1 Denary, binary, and quinary numbers - Looking back Classwork - Exercise 1. Let's say and write the short forms of the given bases of numbers. a) The decimal number of 4 × 103 + 6 × 101 + 8 × 10° is ........................... b) The binary number of 1 × 24 + 1 × 22 + 1 × 21 + 1 × 2° is ........................... c) The quinary number of 3 × 53 + 4 × 52 + 2 × 51 + 1 × 5° is ........................... 2. Let's say and write the answers as quickly as possible. a) The digits in denary system are ...................................................................... e) The digits in binary system are ...................................................................... f) The digits in quinary system are ......................................................................... 3. Let's say and tick the correct value. a) Value of 3 in binary number is (i) 102 (ii) 112 (iii) 1012 b) Value of 9 in quinary number is (i) 145 (ii) 1045 (iii) 415 c) Value of 1012 in denary number is (i) 3 (ii) 6 (iii) 5 d) Value of 135 in denary number is (i) 7 (ii) 8 (iii) 9 2.2 Decimal numeration system Let’s take 18 blocks of cubes and regroup them into the group of 10 blocks. 18 = 10 + 8 = 1 × 101 + 8 × 10° Let’s take 36 pencils and regroup them into the group of 10 blocks. 36 = 30 + 6 = 3 × 101 + 6 × 10° Similarly, 342 = 300 + 40 + 2 = 3 × 102 + 4 × 101 + 2 × 100 . In this way, whole numbers can be regrouped into the base of 10 with some power of 10. It is called the decimal numeration system or denary system. Unit 2 Number System in Different Bases

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 33 Vedanta Excel in Mathematics - Book 8 Number System in different Bases Example 1: Express 472510 in the expanded form. Solution: 472510 = 4 × 1000 + 7 × 100 + 2 × 10 + 5 = 4× 103 + 7 ×102 + 2 × 101 + 5 × 100 . 2.3 Binary number system Decimal or denary number system is base 10 system. Besides this number system, there are four alternative base systems that are most useful in computer applications. These are binary (base two), quinary (base five), octal (base eight), and hexadecimal (base sixteen) systems. Computers and handheld calculators use the binary system for their internal calculations. The system consists of only two digits 0 and 1. So, all numbers can then be represented by electronic switches of one kind or another, where “on” indicates 1 and “off” indicates 0. 2.4 Formation of binary number system Again, let’s take 15 blocks of cubes and regroup them into the group of 2 blocks. 7 pairs blocks of cube and 1 cube Now, let’s arrange the groups of 2 blocks into the base of 2 with the maximum possible powers. 8 4 2 1 23 22 21 20 So, 15 = 1 × 23 + 1 × 22 + 1 × 21 + 1 × 2° = 11112 In this way, the denary number 15 can be expressed in binary number as 11112 . The system of numeration in base 2 with some power of 2 is called the binary numeration system. In this system, we use only two digits 0 and 1. 2.5 Expanded form of binary number Let's take a few binary numbers 12 , 112 and 101, and express them in the expanded forms. Here, 12 = 1 × 2° 112 = 1 × 21 + 1 × 2° 1012 = 1 × 22 + 0 × 21 + 1 × 2° 22 21 20 1 1 1 1 0 1

Vedanta Excel in Mathematics - Book 8 34 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2.6 Conversion of decimal numbers to binary numbers Let's take any two denary numbers 18 and 29. Now, let's expand these numbers with some exponents of 2. Here, 18 = 16 + 2 = 1 × 24 + 1 × 21 = 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 = 100102 Also, 29 = 16 + 8 + 5 = 16 + 8 + 4 + 1 = 1 × 24 + 1 × 23 + 1 × 22 + 1 × 20 = 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 111012 In this way, we can convert a decimal number into binary number by expanding the decimal number with the appropriate exponents of 2. Then, using the place value table of binary number, we get the required binary number. Alternatively, we convert a decimal number into binary number, dividing the given number successively by 2 until the quotient becomes 0. The remainder obtained in each successive division is listed in a separate column. The remainders arranging in the reverse order is the required binary number. This process is useful to convert bigger decimal numbers into binary numbers. For example: Example 2: Convert 185 into binary system. Solution ∴ 185 = 101110012 2.7 Conversion of binary numbers to decimal numbers We convert a binary number into decimal number, just by expanding the given binary number in the power of 2. Then, by simplifying the expanded form of the binary number, we obtain the required decimal number. For example, Example 3: Convert 1101102 into decimal system. Solution: 1101102 = 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 = 32 + 16 + 4 + 2 + 0 = 54 16 8 4 2 1 24 23 22 21 20 1 0 0 1 0 16 8 4 2 1 24 23 22 21 20 1 1 1 0 1 www.geogebra.org/classroom/xjeajf4k Classroom code: XJEA JF4K Vedanta ICT Corner Please! Scan this QR code or browse the link given below: 2 2 2 2 2 2 2 185 92 46 23 11 5 2 1 Remainder Arranging the remainders from the bottom up: 111011012 1 0 0 1 1 1 0 1×28 1×27 1×26 1×25 1×24 1×23 1×22 1×21 1×20 256 128 64 32 16 8 4 2 1 185 = 128 + 0 + 32 + 16 + 8 + 0 + 0 + 1 = 1×27 + 0 + 1×25 + 1×24 + 1×23 + 0 + 0 + 1×10 = 101110012 Alternative process: By using place value table of binary number. Number System in different Bases

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 35 Vedanta Excel in Mathematics - Book 8 EXERCISE 2.1 General Section - Classwork 1. Let's say and tick the short form of these numbers as quickly as possible. a) 3 × 102 + 7 × 101 + 5 × 100 (i) 3075 (ii) 375 (iii) 370 b) 2 × 104 + 8 × 102 + 4 × 101 + 9 × 100 (i) 20849 (ii) 28049 (iii) 2849 c) 1 × 24 + 1 ×23 + 1 × 22 + 1 × 21 (i) 110112 (ii) 111012 (iii) 111102 d) 1 × 25 + 1 × 23 + 1 × 21 + 1 × 20 (i) 1010112 (ii) 1100112 (iii) 1010102 2. Let's say and write the values of these decimal numbers in binary numbers as quickly as possible. a) 1= ............. b) 2 = ............. c) 3 = ............. d) 4 = ............. e) 5 = ............. f) 6 = ............. g) 7 = ............. h) 8 = ............. 3. Let's say and write the values of these binary numbers in decimal numbers as quickly as possible. a) 12 = ............. b) 102 = ............. c) 112 = ............. d) 1002 = ............. e) 1012 = ............. f) 1102 = ............. g) 1112 = ............. h) 10002= ............. Creative section 4. A mathematic teacher writes the expanded form of a decimal number in the board as: 2 × 102 + 4 ×101 + 3 × 100 . a) Write this decimal number in short form. b) Convert this decimal number into the binary numbers. 5. Manoj write the expanded form of a binary number as: 1 × 23 + 1 ×22 + 1 ×21 + 1 ×20 . a) Write this binary number in the short form. b) Convert this binary number into the decimal number. 6. Expand the following decimal numbers in power of 10s. a) 276 b) 3816 c) 54027 d) 809005 e) 7000409 7. Convert the following decimal numbers into binary numbers: a) 9 b) 10 c) 11 d) 12 e) 18 f) 55 g) 124 h) 216 i) 361 j) 490 8. Convert the following binary numbers into decimal numbers: a) 10012 b) 10102 c) 101112 d) 1010102 e) 1111012 f) 10101112 g) 11001112 h) 110010112 i) 111011102 j) 1101110112 It's your time - Project work and Activity section 9. a) Let's take a few number of matchsticks and represent the following binary numbers by pairing the required number of matchsticks. (i) 102 (ii) 112 (iii) 1012 (iv) 1112 (v) 1102 (vi) 11102 (vii) 11112 Number System in different Bases

Vedanta Excel in Mathematics - Book 8 36 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur b) Let's draw the required number of pairs of dots in the table to convert the given denary numbers into binary numbers. Denary numbers 24 (16) 23 (8) 22 (4) 21 (2) 20 (1) 5 13 18 2.8 Quinary number system Let's take two separate sets of 18 and 34 toothpicks. Make the groups of 5 toothpicks with as highest exponent of 5 as possible from 18 and 34 toothpicks separately. Keep the remaining toothpicks in a separate group. 18 = 3 × 5 + 3 = 3 × 51 + 3 × 50 = 335 34 = = 1 × 52 + 1 × 51 + 4 × 50 = 1145 In this way, quinary number system is the base five number system. In the quinary place system, five digits 0, 1, 2, 3, and 4 are used to represent any real number. The numbers in the quinary system can be expressed as the product of digits and powers of 5. For example, 245 = 2 × 51 + 4 × 5°, 4325 = 4 × 52 + 3 × 51 + 2 × 5° and so on. 2.9 Conversion of decimal numbers to quinary numbers To convert a decimal number into quinary number, we should divide the given number successively by 5 until the quotient becomes zero. The remainder obtained in each successive division is listed in a separate column. Then, the remainders arranging in the reverse order is the required quinary number. For example: Example 1: Convert 728 into quinary number. Solution: 5 5 5 5 5 728 145 29 5 1 0 Remainder Arranging the remainders from, the bottom up, 104035 \ 728 = 104035 3 0 4 0 1 Alternative process: By using place value table of quinary number. 54 53 52 51 50 625 125 25 5 1 728 = 625 + 0 + 4 × 25 + 0 + 3 = 1×54 + 0 + 4×52 + 0 + 3×50 = 104035 www.geogebra.org/classroom/fazej48a Classroom code: FAZE J48A Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Number System in different Bases

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 37 Vedanta Excel in Mathematics - Book 8 2.10 Conversion of quinary numbers to decimal numbers To convert a quinary number into decimal number, it is expanded in the power of 5. Then, by simplifying the expanded form of the quinary number, we get the required decimal number. For example, Example 2 : Convert 31045 into decimal number. Solution: 31045 = 3 × 53 + 1 × 52 + 0 × 51 + 4 × 5° = 3 × 125 + 1 × 25 + 4 = 375 + 25 + 4 = 404 EXERCISE 2.2 General Section 1. Let's say and and tick the short form of the quinary numbers. a) 2 × 53 + 4 × 51 + 1 × 50 (i) 2415 (ii) 24105 (iii) 20415 b) 4 × 54 + 2 × 53 + 3 × 52 + 1 × 51 (i) 423105 (ii) 423015 (iii) 42315 c) 3 × 54 + 1 × 52 + 2 × 50 (i) 31205 (ii) 301025 (iii) 301205 2. Let's say and write the quinary numbers as quickly as possible. a) 5 = ............ b) 6 = ............ c) 7 = ............ d) 8 = ........... e) 9 = ............ f) 10 =............ g) 15 = ............ h) 25 = ............ 3. Let's say and write the decimal numbers as quickly as possible. a) 105 = ............ b) 115 = ............ c) 205 = ............ d) 235 = ............ e) 305 = ............ f) 405 = ............ g) 1005 = ............ h) 1105 = ............ Creative section - A 4. Let's convert the following decimal numbers into quinary numbers. a) 8 b) 10 c) 27 d) 72 e) 136 f) 257 g) 444 h) 516 i) 817 j) 1539 5. Let's convert the following quinary numbers into decimal numbers. a) 145 b) 235 c) 435 d) 2145 e) 4315 f) 14235 g) 21305 h) 43215 i) 120435 j) 302425 Number System in different Bases

Vedanta Excel in Mathematics - Book 8 38 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Creative section - B 6. Kalpana writes a binary number in the expanded form as : 1 × 24 + 0 × 23 + 1 × 22 + 0 ×21 + 1 × 20 . a) Write the short form of this binary number. b) Convert this binary number into decimal number. c) Convert this decimal number into quinary number. 7. Mr. Yadav is a mathematic teacher. He writes a quinary number in the expanded form as: 3 × 53 + 2 × 52 + 4 × 51 + 1 × 50 . a) Write the short form of this quinary number. b) Convert this quinary number into decimal number. c) Convert this decimal number into binary number. 8. Let's convert binary into quinary or quinary into binary. a) 11102 b) 110112 c) 1111012 d) 135 e) 2415 f) 30245 9. a) If the value of a number in decimal system is 955, find its value in quinary system and in binary system. b) Which one is greater between 11110112 and 1435 ? Also, find their difference in decimal system. c) Which one is greater 110112 or 3225 ? Also, find their difference in denary system. 10. Three students Ramila, Shashwat, and Arabi were asked to convert 2215 into binary number. Ramila got the answer 1110112 , Shashwat got the answer 1111012 , and Arabi got the answer 1101102 . Who got the correct answer? Find by calculation. It's your time - Project work and Activity section 11. a) Write the number of your family members and convert it into binary as well as quinary systems. b) Take the total number of students of your class. Then, convert it into binary as well to as to quinary systems. 12. Let's draw the required number of groups of 5 dots in the table to convert given denary numbers into quinary numbers. Denary numbers 52 51 50 6 9 10 12 16 27 30 Number System in different Bases

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 39 Vedanta Excel in Mathematics - Book 8 3.1 Types of numbers - Looking back Classwork- Exercise 1. Let's say and write 'true' or 'false' for the following statements. a) Zero is the least whole number. ................ b) Set of whole number is the subset of natural number. ................ c) {..., –1, 0, 1, …} is the set of integers. ................ d) Set of integers is the subset of whole numbers. ................ e) The least integer is infinite. ................ f) 0 is less than –1. ................ g) The difference of any two whole numbers is always a whole number. ................ h) The difference of any two integers is always an integer. ................ 2 Let's say and write the following sets of integers. a) integers between - 3 and 3, Z = { .....................................................................} b) integers between –5 and 1, Z = { .....................................................................} 3.2 Natural numbers and whole numbers - Looking back N = {1, 2, 3, 4, 5, ...} is the set of natural numbers which are also called the counting numbers. The operations of addition and multiplication of any two natural numbers always give natural numbers. For example, 3 + 4 = 7 (7 is also a natural number) 3 × 4 = 12 (12 is also a natural number) Similarly, in 4 – 3 = 1, 1 is a natural number. However, in 3 – 3 = 0, 0 is not a natural number. It demands another set of numbers that should include 0 (zero) also. So, 0 is included in the set of natural numbers to develop a new set of numbers that we call the set of whole numbers. It is denoted by W. Thus, W = {0, 1, 2, 3, 4, 5, ...} is the set of whole numbers. Unit 3 Real numbers

Vedanta Excel in Mathematics - Book 8 40 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 3.3 Integers Now, let’s take any two whole numbers 4 and 5. Say and write the answers as quickly as possible. 4 + 5 = 9, is 9 a whole number? ......................................................... 5 – 4 = 1, is 1 a whole number? ......................................................... 5 × 4 = 20, is 20 a whole number? ......................................................... 4 – 5 = –1, is –1 a whole number? ......................................................... We know that –1 is not the member of the set of whole numbers. It must be the member of another set of numbers that we call the set of integers. So, –1 is an integer. Thus, the set of all numbers both positive and negative numbers including zero (0) is called the set of integers. The set of integers is denoted by the letter ‘Z’. Z = {..., –3, –2, –1, 0, 1, 2, 3, ...} is the set of integers. Z+ = {+1, +2, +3, ...} is the set of positive integers. Z– = {–1, –2, –3, ...} is the set of negative integers. The integers can be shown in the number line. The positive integers are written right to the zero in increasing order whereas the negative integers are written left to the zero in decreasing order. 3.4 Sign rules of addition of integers (i) The positive integers are always added and the sum holds the positive (+) sign. For example, (+2) + (3) = 2 + 3 = + 5 or 5 (ii) The negative integers are always added and the sum holds the negative (–) sign. For example, (–2) + (–5) = –2 – 5 = – 7 (iii) The positive and negative integers are always subtracted and the difference holds the sign of the bigger digit. For example, –10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 decreasing order zero increasing order -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Real numbers

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 41 Vedanta Excel in Mathematics - Book 8 (+6) + (–4) = 6 – 4 = 2 [Positive digit (+6) is bigger] (+2) + (–7) = 2 – 7 = –5 [Negative digit (–7) is bigger] -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Thus, the integers with the same sign are always added and the integers with the different sign are always subtracted. 3.5 Sign rules of multiplication and division of integers (i) The product or quotient of two positive integers is always positive. For example, (+4) × (+2) = 4 × 2 = + 4, (+8) ÷ (+4) = 8 ÷ 4 = +2, and so on. (ii) The product or quotient of two negative integers is always positive. For example, (–3) × (–2) = + 6, (–12) ÷ (–4) = + 3, and so on. Let’s investigate the idea of this fact from the following pattern of operations. 2 × (–4) = –8 1 × (–4) = –4 0 × (–4) = 0 (–1) × (–4) = 4 (–2) × (–4) = 8 products are increasing multiplicand are decreasing (iii) The product or quotient of two positive and negative integers is always negative. For example: (+3) × (–2) = –6, (–4) × (+5) = –20, (+18) ÷ (–3) = –6, and so on. EXERCISE 3.1 General Section - Classwork 1. Let's write the operations of integers and find the sums from the number lines given below. a) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ............+............ = ............ b) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ............+............ = ............ c) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ............+............ = ............ d) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ............+............ = ............ (+3) (+4) (+7) Real numbers

Vedanta Excel in Mathematics - Book 8 42 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 2. Let's show the operations of addition and multiplication of integers in the given number lines. Then, find the sums or products. a) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 (+3) + (+5) = ............... b) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 (+2) + (–6) = ............... c) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 (+7) + (–4) = ............... d) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 2 × (+4) = ............... e) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 3 × (–2) = ............... f) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 (–2) × (–3) = ............... Creative section 3. In a winter morning, the temperature of Jomsom was measured to be –2° C and in the mid-day it raised to 10° C. a) Show this information in a number line. b) By how many degree celsius did the temperature raise? Find it from the number line. 4. On a winter day, the temperature at Pathibhara temple situated in Taplejung district was 13°C and it dropped to –3° C at mid-night. a) Show this information in a number line. b) By how many degree celsius did the temperature change? Find it from the number line. 5. A dolphin was swimming 10 feet below the surface of an ocean. Suddenly it jumped up to the height of 10 feet from the surface of the ocean, and sank back to the depth of 5 feet below the surface. a) Show this information in a number line. b) How many feet did it jump up from its original position? c) How many feet did it fall while sinking back? Real numbers

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 43 Vedanta Excel in Mathematics - Book 8 It's your time - Project Work and Activity Section 6. Let's draw separate number lines and use them to verify the following statements with at least two illustrations. a) The sums of positive integers are always positive. b) The sums of negative integers are always negative. c) The sum of a positive and a negative integer may be positive or negative. d) The products of positive integers are always positive. e) The product of a positive and a negative integer is always negative. 3.6 Rational numbers - review Let's review the sets of natural numbers, whole numbers and integers once again. N = {1, 2, 3, 4, 5, ...} is the set of natural numbers. W = {0, 1, 2, 3, 4, 5, ...}is the set of whole numbers. Z = {..., –2, –1, 0, 1, 2, 3, 4, 5, ...} is the set of integers. Now, let's answer the following questions. (i) Is the set of natural numbers a proper subset of whole numbers? (ii) Is the set of whole numbers a proper subset of integers? Thus, set of natural numbers and whole numbers are the proper subsets of the set of integers. Again, let's discuss the answers of the following questions. (i) Are the sums of integers always integers? (ii) Are the differences of integers always integers? (iii) Are the products of integers always integers? (iv) When an integer is divided by another integer, is the quotient always an integer? Further more, let's take any two integers 3 and 6. Then, answer the following questions. (i) In 3 + 6 = 9, is 9 an integer? (ii) In 3 – 6 = –3, is –3 an integer? (iii) In 3 × 6 = 18, is 18 an integer? (iv) In 3 ÷6 = 3 6 = 1 2 , is 1 2 an integer? Of course, 1 2 is not an integer. Thus, when an integer is divided by another integer, the quotient is not always an integer. Therefore, there must be another set of numbers that includes such quotients which are not integers. The set of such numbers is called the set of rational numbers. Z W N I got it! 2 ÷ 4 = 1 2 and 1 2 is not an integer! We also understood! 5 ÷ 2 = 5 2 is not an integer! It is a rational number. Real numbers

Vedanta Excel in Mathematics - Book 8 44 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Any numbers which can be expressed in the form a b , where a and b are integers and b ≠ 0 are called rational numbers. The set of rational number is denoted by ‘Q’. ∴ Q = {... – 3, – 5 2 , – 2, – 3 2 , – 1, – 1 2 , 0, , 1, 3 2 , 2, 5 2 , 3, ...} In this way, the sets of natural numbers, whole numbers and integers are the subsets of the set of rational numbers. It means every natural number, whole number and integer is a rational number. For example: 0 1 = 0, 7 1 = 7, – 9 1 = –9, …, where 0, 7, –9, … are all rational numbers. 3.7 Terminating and non-terminating recurring decimals When we express a rational number into decimal, it may be terminating or non-terminating decimal. If the decimal is non-terminating, a digit or block of digits in the decimal part repeat after a certain interval. It is called non-terminating recurring decimal. For example, The non-terminating recurring decimals can be indicated by putting dots just above the beginning and end of the repeated digit or block of digits. 1 2 2 5 3 4 = 0.5 (Terminating decimal) = 0.4 (Terminating decimal) = 0.75 (Terminating decimal) 1 3 5 6 8 11 = 0.3 ....... (Non-terminating decimal) = 1.83 ....... (Non-terminating decimal) = 0.72 ....... (Non-terminating decimal) 3.8 Non-terminating and non-recurring decimals On the other hand, let's find the square roots of some of the following numbers. 2 = 1.4142135623731... (Non-terminating non-recurring decimal) 3 = 1.73205080756888... (Non-terminating non-recurring decimal) 5 = 2.23606797749979... (Non-terminating non-recurring decimal) 8 = 2.82842712474619... (Non-terminating non-recurring decimal) Thus, the square roots of 2, 3, 5, 6, 7, 8, 10, etc. are non-terminating and nonrecurring decimals. Therefore, 2, 3, 5, etc. are not rational numbers. 3.9 Irrational numbers Let’s take a square of unit length (say 1 cm). OB is the diagonal of the square. D OAB is the right-angled triangle right-angled at O. So, OA is the base, AB is the perpendicular and OB is the hypotenuse of the right-angled triangle OAB. O C 1 unit A B 1 unit Real numbers

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 45 Vedanta Excel in Mathematics - Book 8 By using Pythagorus theorem to find the length of OB, OB = OA2 + AB2 = 12 + 12 = 2 unit Here, 2 does not belong to the set of rational numbers. Therefore, besides rational numbers, another set of numbers also must exist in the number system and that we call the set of irrational numbers. Now, the question is whether it is possible to show 2 in the number line. Let’s take a point P of the coordinates (1, 1) in the coordinate axes of the graph paper and join it with the origin O (0, 0). Now, by using distance formula between the points O (0, 0) and P (1, 1), OP = (x2 – x1 ) 2 + (y2 – y1 ) 2 = (1 – 0)2 + (1 – 0)2 = 12 + 12 = 2 units Let’s draw a circle with the radius OP and centre at O. The circumference of the circle intersect OX (the number line) at Q. Here, radius of the circle OP = OQ = 2 units. In this way, 2 can also be shown in the number line. It means irrational numbers can also be shown in the number line. Again, let’s consider the number 2 . Here, 2 = 1.414213562... Thus, when 2 is expressed into decimal, it is non-terminating and non-recurring. Therefore, it is not a rational number. It is an irrational number. 3, 5 , 6 , 7 , 8 , etc. are a few examples of irrational numbers. On the other hand, 4 is not an irrational number because the square root of 4 is 2 which is a rational number. 3.10 Comparison between rational and irrational numbers The table given below shows some of the important differences between rational and irrational numbers. 1 2 2 Q P (1, 1) O 2 www.geogebra.org/classroom/bx62pjm6 Classroom code: BX62 PJM6 Vedanta ICT Corner Please! Scan this QR code or browse the link given below: Real numbers

Vedanta Excel in Mathematics - Book 8 46 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Rational Numbers Irrational Numbers 1. Rational numbers are either terminating decimals or nonterminating recurring decimals. 1. Irrational numbers are always nonterminating and non-recurring decimals. 2. Rational numbers can be expressed in the form of a b where b ≠ 0 and a and b are integers. 2. Irrational numbers cannot be expressed in such form. 3. Rational numbers are closed under the operation of addition, subtraction, multiplication and division. 3. Irrational numbers are not closed under the operation of multiplication and division. 3.11 Real numbers We have already learned the following facts about the number system. (i) A set of natural numbers is the subset of the set of whole numbers. (ii) A set of whole numbers is the subset of the set of integers. (iii) A set of integers is the subset of the set of rational numbers. (iv) The set of numbers which are not the rational numbers are the irrational numbers. Now, the set of numbers that contains both the rational and irrational numbers and can be shown in the number line is called the set of real numbers. It is denoted by R. ∴ R = {x : x ∈ Q and x ∈ Q } Where Q is the set of rational numbers and Q is the set of irrational numbers. Study the following diagram to know the relationship of different sets of numbers to the real number system. Real numbers Negative integers –4, –3, –2, –1, … Positive integers 0, 1, 2, 3, 4, … Natural numbers 1, 2, 3, 4, … Whole number 0, 1, 2, 3, 4, … Terminating decimals 1 2, 1 4, 3 5, 7 8, … Non-terminating recurring decimals 1 3, 5 6, 6 7, … Rational numbers 0, 1, 2, –1, –5, 1 2, 2 3, … Irrational numbers 2, 3, 5, … Fractional numbers 1 2, 2 3, 1 4, 5 7, … Integers –2, –1, 0, 1, 2, 3, … R Q Q Z W N Real numbers

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 47 Vedanta Excel in Mathematics - Book 8 Worked-out Examples Example 1: 4 and 8 are two real numbers. a) Which one is the rational number? Give reason. b) Which one is the irrational number? Give reason. Solution: a) 4 is the rational number. Because 4 is a perfect square number and its square root is 2 which is a rational number. b) 8 is the irrational number. Because 8 is not a perfect square. So, its square root is obtained as non-terminating and non-recurring decimal which is an irrational number. Example 2: 3 and 10 are two real numbers. a) Identify the rational and irrational number between these two real numbers. b) Compare these two real numbers. Solution: a) 3 is a rational number and 10 is an irrational numbers. b) Here, 32 = 3 ×3 = 9 ( 10) 2 = 10 × 10 = 10 Since, 9 < 10, so, 3< 10. Example 3: 2 and 2 are two real numbers. a) Is the sum of 2 + 2 is rational or irrational number? Give reason. b) Is the product of 2 × 2 rational or irrational number? Give reasons. Solution: a) The sum of 2 + 2 is irrational number. Because when a rational number is added to an irrational number, the decimal value of the sum is always non-terminating and non-recurring decimal. b) The product of 2 × 2 is irrational number. Because when an irrational number is multiplied by a rational number, the decimal value of the product is always nonterminating and non-recurring decimal. Example 4: 5 and 2 5 are two irrational numbers. a) Find the sum of 5 + 2 5. b) Find the difference of 2 5 – 5. c) Find the product of 5 × 2 5. d) Find the quotient of 5 ÷ 2 5. Real numbers

Vedanta Excel in Mathematics - Book 8 48 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur Solution: a) 5 + 2 5 = 3 5 b) 2 5 – 5 = 5 c) 5 × 2 5 = 2 × ( 5) 2 = 2 × 5 = 10 d) 5 ÷ 2 5 = 5 2 5 = 1 2 Example 5: Simplify a) 3 + 3 3 – 2 3 b) 4 2 × 2 3 c) 10 6 ÷ 5 2 Solution: a) 3+ 3 3 – 2 3 = 4 3 – 2 3 = 2 3 b) 4 2 × 2 3 = 4 × 2 2 × 3 = 8 6 c) 10 6 ÷5 2 = (10 ÷ 5) 6 ÷ 2 = 2 3 EXERCISE 3.2 General Section - Classwork 1. Let's list the rational and irrational numbers separately. 2 , 2 5 , 22 7 , p, 6.1428, 25 , 9 16 , 36 40 , – 4, 27 3 , 27 2. Let's list fractions separately which are of terminating decimals or non-terminating recurring decimals. 1 2 , 1 3 , 3 4 , 2 7 , 3 5 , 5 6 , 5 8 , 9 10, 4 9 , 7 16 3. Let's read the statements carefully and write ‘T' for the true statements and ‘F' for the false statements. a) Set of whole number is the universal set of natural numbers. We got it! As like x + 2x = 3x, 5 + 2 5 = 3 5 As like 2x – x = x, 2 5 – 5 = 5 We also understood! We can compare 3 + 3 3 – 2 3 with x + 3x – 2x. So, 3 + 3 3 – 2 3 = 4 3 – 2 3 = 2 3 Rational numbers Irrational numbers Terminating Decimal numbers Non-terminating Decimal numbers Real numbers

Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 49 Vedanta Excel in Mathematics - Book 8 b) Set of integers is the subset of whole numbers. c) The set of rational numbers is the universal set of integers. d) The set of rational numbers is the improper subset of the set of real numbers. e) The set of irrational numbers is the subset of the set of real numbers. f) If R is the set of real numbers, Q is the set of rational numbers and Q is the set of irrational numbers, then R = Q ∪ Q. Creative Section - A 4. A Mathematics Teacher wrote three numbers 3 5, 5 3 and 6 on the white board during the discussion about real numbers. Answers the following questions. a) Among these numbers list out the rational numbers with reason. i) Which rational number has terminating decimal? ii) Which rational number has non-terminating recurring decimal? b) Among the above three numbers list out the irrational number with reason. c) Name the universal set of numbers that contains the sets of rational and irrational numbers as its subsets. 5. Using the standard form of rational number, justify that 0 is also a rational number. 6. A number is separated by an expression x + 1 x – 1 , where x = 1. Is this a rational number? Write with reason. 7. Is the square root of any rational number a rational number? Justify your answer with an example. 8. 9 and 10 are two real numbers. a) Which one is the rational number? Give reason. b) Which one is the irrational number? Give reason. 9. p and 22 7 are two real numbers. a) Which one is the rational number? Give reason. b) Which one is the irrational number? Give reason. 10. 4 and 15 are two real numbers. a) Identify the rational and irrational numbers between these two numbers. b) Compare these two real numbers. 11. 27 3 and 3 are two real numbers. a) Identify the rational and irrational numbers between these two numbers. b) Compare these two real numbers. Real numbers

Vedanta Excel in Mathematics - Book 8 50 Approved by Curriculum Development Centre, Sanothimi, Bhaktapur 12. 3 and 3 are two real numbers. a) Is the sum of 3 + 3 rational or irrational number? Give reason. b) Is the product of 3 × 3 rational or irrational number? Give reason. 13. 2 and 2 2 are two irrational numbers. a) Is the difference of 2 2 – 2 a rational or irrational number? Give reason. b) Is the product of 2 × 2 2 a rational or irrational number? Give reason. c) Is the quotient of 2 ÷ 2 2 a rational or irrational number? Give reason. Creative Section - B 14. Let's find the sum or difference of the following irrational numbers. a) 2 + 2 b) 2 2 + 3 2 c) 2 3 + 4 3 d) 5 6 + 6 e) 4 2 – 2 2 f) 7 3 – 3 3 g) 8 10 – 9 10 h) 3 7 – 6 7 15. Let's simplify. a) 3 2 +2 2 – 4 2 b) 2 3 - 3 + 5 3 c) 10 5 –3 5 – 4 5 d) 10 – 5 10 - 2 10 + 3 10 16. Let's find the product or quotient of the operations of following irrational numbers. a) 2 × 3 b) 2 3× 5 c) 3 7 × 2 2 d) 4 10 × 5 3 e) 6 ÷ 2 f) 15 ÷ 3 g) 8 12 ÷ 2 6 h) 6 14 ÷ 3 7 17. The figure alongside is a white-board in the shape of a square inside a classroom. a) Find the perimeter of the board. b) Is the number obtained as the perimeter of the board a rational or an irrational number? Give reason. c) Find the area of the board. d) Is the number obtained as the area of the board a rational or an irrational number? Give reason. It's your time - Project work and Activity section 18. a) Let's draw a Venn-diagram in a chart paper to show the relationship between the sets of a real numbers (R), rational numbers (Q) and irrational numbers (Q). b) Let's draw a Venn-diagram in a chart paper to show the relationship between the sets of real numbers (R), rational numbers (Q), integers (Z), whole numbers (W), natural numbers (N) and irrational numbers (Q). 2 2 m Real numbers