The Leading Maths - 8

Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Ph. No.: 01-5388827 The Leading MATHS 8 Author Ashok Dangol M.Ed., Maths (TU) ALLIED Prepared on New Curriculum Issued by CDC, Sanothimi, Bhaktapur, Nepal

Allied The Leading MATHS 8 Publisher Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Phone : 01-5378629, 5388827 Written by Ashok Dangol M.Ed., Maths (TU) Special Thanks Nabaraj Pathak Lalbabu Prasad Yadav Jit Bahadur Khanal Ashok Dangol Copyright All rights reserved with the copyright holder. Edition First - 2080 (5000 pcs.) Computer Icon Design House #9849098999 Printed in Nepal

iv Creative Maths -VI Approved by CDC, Nepal PREFACE This Allied The Leading Mathematics - 8 is basically meant for making the teachers and taught active while teaching and learning mathematics. The contents and extent of the series are strictly contained and arranged in accordance with the new vision and mission of the latest New Curriculum 6-8 of Basic Level of CDC, Nepal. This series is basically an outcome of my untiring effort and patience. The long and dedicated service in teaching and popularization of mathematics has been a great asset in preparing this series. It has been designed as a textbook for English medium private and government school students with a new approach. This book provides maximum benefit to both teachers and students because of the following unique features: Unique Features of this Book Ö Arranged especially focusing on child psychology of teaching and learning mathematics which are based on the Areas of Basic Level Curriculum 6-8. Ö Prepared with the firm belief that “Mathematics begins at Home, grows in the Surroundings and takes shape in School (HSS Way)”, and sincere attempts have been made to make the learner, the teacher and reader feel “Mathematics is Fun, Mathematics is Easy and Mathematics is Everywhere (FEE Concept)”. Ö Written the focus of students’ activities and easily perform teaching-learning activities for teachers. Ö Well arranged the four colours of the whole book supports to find easily Units, Chapters, Lessons, Examples, Practices, and other topics from Content. Ö Every Theme begins with its estimated teaching hours (Theory + Practical), competency, learning outcomes, Warm-Up for pre-knowledge. Ö Highlighted the important terms, notes and key points. Ö Included sufficiently all types of Classwork Examples and Home Assessments from simple to complex with suitable figures and reasons. Ö Included Project Works and sample format of Project Work as self-practice to the students at home for some days' activities to memories long time about the entire chapters. Ö Included Mixed Practice and Confidence Level Tests as self-evaluations for Success and Competent by themselves. Ö Available Individual Practical Evaluation Sheet at the end.

PREFACE It is very much hoped that with all the above features, this book will be found really fruitful by teachers and students alike. Thank Allied Publication Pvt. Ltd., Kathmandu, Nepal for taking responsibility for publishing this book, Nariswor Gautam for language editing, Dev Krishna Maharjan for an attractive art of pictures, and Binod Bhandari for an attractive design. I would like to extend my sincere gratitude to the persons whose ideas or creations are directly or indirectly incorporated into the text. I would like to extend my thanks to the teachers and the students who helped me to verify the answers and to check the manuscript of this book. Also, many thanks to the schools that applied this book and suggested it to me. Finally, I heartily welcome criticisms, feedbacks and suggestions from readers so that it may appear with revise from in the coming edition and will be gratefully and thankfully acknowledged and honored.

CONTENTS UNIT - I: SETS 1 Chapter 1: Sets 2 UNIT - II: ARITHMETIC 21 Chapter 2: Real Number System 22 Chapter 3: Ratio and Proportion 59 Chapter 4: Profit and Loss 75 Chapter 5: Unitary Method 83 Chapter 6: Simple Interest 91 UNIT - III: MENSURATION 104 Chapter 7: Area of Plane Shapes 105 UNIT - IV: ALGEBRA 137 Chapter 8: Indices 138 Chapter 9: Algebraic Expression 148 Chapter 10: Equation, Inequality and Graph 183 UNIT - V: GEOMETRY 195 Chapter 11: Lines and Angles 196 Chapter 12: Plane Figures 218 Chapter 13: Congruency and Similarity 251 Chapter 14: Solid Objects 263 Chapter 15: Coordinates 270 Chapter 16: Symmetry and Tessellation 281 Chapter 17: Transformation 287 Chapter 18: Bearing and Scale Drawing 307 UNIT - VI: STATISTICS 324 Chapter 19: Statistics 325 Practice Question Set 341 Specification Grid 343 Sample for Project Work - 1.1 344 Internal Evaluation Sheet 345

SETS 1 A A = Aquatic animal M = Mammals C = Cyan, M = Magenta Y = Yellow K = Black M U Estimation teaching hours - 10 (Th. + Pr.) COMPETENCY  solution of the behaviour problems related to set CHAPTERS 1. Sets LEARNING OUTCOMES separate disjoint and overlapping sets, identify the proper and improper subsets on the given set, construct the proper and improper subsets of the given set. SETS UNIT I M Y C U K

2 The Leading Maths - 8 Red Orange Yellow Green Blue Indigo Violet CHAPTER 1 SETS Lesson Topics Pages 1.1 Review on Sets 3 1.2 Types of Sets 8 8848.86 meters above sea level 414 meters below sea level ” Name the colours of sunlight or rainbow. ” What are the odd numbers less than 8 ? ” Tell the names of the beautiful girls. ” What kinds of girls are called beautiful ? ” Name the boys who have long hairs. ” How many letters are in the word "mathematics"? ” How many water droplets are in the sea ? ” What are the even prime numbers ? ” How many parents does a child have ? ” Tell the name of the highest peak in the world. ” What is the name of the lowest place in the world ? WARM-UP

SETS 3 1.1 Review on Sets At the end of this topic, the student will be able to: ¾ define the set and its different terminology. Learning Objectives ACTIVITY - 1 Observe the adjoining balls of different colours. ” How many colours are there? ” Separate the balls with the same colours. ” How many groups of balls are formed? ” How many balls are there in each group? ” What are the different characteristics in each group? Let us define a set. A set is a well defined collection of objects. The objects that make the set are called its elements or members. By “Well defined” collection we mean a collection in which all the elements are distinct and it is always possible to state clearly whether or not something belongs to or does not belong to the collection that means they have at least one unique character. Notation of a Set Step-1 : A set is usually denoted by an English capital letter A, B, ..... or Z. Step-2 : The elements or members of a set are denoted by small letters a, b, c, ..... and digits 1, 2, 3, ..... or name Ram, mango, Pokhara, ..... or objects/pictures. Step-3 : The elements of the set are separated by commas (,). Step-4 : The elements of the set are enclosed by curly brackets { }. For example: A = {1, 2, 3, 4} is a set because it is possible to state clearly that 1 belongs to the collection but 0 does not. Membership of a Set In the above set A, 1 lies in A but 0 (zero) does not. Here, 1 is the member of A. 0 is not its member.

4 The Leading Maths - 8 The membership of the set is denoted by ∈ (Epsilon), that means “belongs to” and the non-membership by ∉ that means “does not belong to”. Therefore, if A = {1, 2, 3, 4} then, 1 ∈ A and 0 ∉ A. Representation of a Set A set may be defined by descriptive it, by listing it, or by using set-builder relation. For example: (a) By descriptive: A is the set of English vowel letters.. (b) By listing: A = {a, e, i, o, u} (c) By set-builder from: A = {x : x is an English vowel letter} Types of sets Definition Example Finite set A set having countable number of elements Sets of provinces of Nepal, Capital of SAARC, Odd number less than 10 Infinite set A set having uncountable number of elements No. of chlorophyll in a plant, even numbers Null set or void set A set having no elements Set of natural number less than 0, Odd numbers divisible by 2 Singleton set A set having any one element Set of the highest peak, even prime number Equality and Equivalent of Two Sets Two sets having exactly the same elements are equal sets. For example: A = {2, 3, 4, 5} and B = {2, 5, 4, 3} are equal sets, because they have the same elements. It is denoted by A = B. Two sets having the same number of elements are equivalent sets. For example; P = {1, 4, 8} and Q = {a, b, c}. The sets P and Q are equivalent because they have the same number of elements. It is denoted by P ~ Q. For example: If A = {1, 2, 3, 4}, B = {2, 4, 3, 1} and C = {2, 3, 4, 5} Then A = B, A ≠ C and B ≠ C Why? But A ~ C, B ~ C and also A ~ B. Note: All equal sets are also equivalent, but not vice versa.

SETS 5 EXERCISE 1.1 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) Define set with an example. (b) Write any one difference between equal and equivalent sets. (c) Why is finite set different from infinite set? (d) What type of sets are called null set and singleton set. 2. Which of the following is a set or not? (a) A = Set of tall man (b) B = Set of beautiful women (c) C = Set of tall students of class 10 (d) D = Set of counting numbers (e) E = Set of good students (f) F = Set of yellow roses 3. Circle ( ) the correct answer. (a) A set is ................ collection of distinct objects. i. finite ii. infinite iii. well-defined iv. defined (b) What is the name of the set that has no element? i. finite set ii. infinite set iii. singleton set iv. void set (c) Which set is finite set? i. {points in a linesegment} ii. {whole numbers} iii. {numbers between 20 and 40 iv. {3, 6, 9, 12, .......} (d) A set having only one element is called ................ i. null set ii. doubleton set iii. finite set iv. singleton set

6 The Leading Maths - 8 (e) Which is the set of prime numbers less than 10? i. {1, 2, 3, 5, 7} ii. {2, 3, 5, 7} iii. {1, 2, 4, 8} iv. {3, 5, 7} 4. A is the set of natural numbers between 0 and 10 exclusive. (a) List the elements of the set A. (b) Use ∈ or ∉ if 2, 3, 0, 9, 11 belong or do not belong to the set A. (c) Write the set in set-builder notation. (a) List the members of the following sets. A = {odd numbers from 0 to 10} B = {prime numbers between 10 and 24} C = {multiples of 7 between 6 and 50} (b) Which sets are equivalent among A, B and C? 6. Given that P = {0, 1, 3, 5}, Q = {5, 3, 1}, R = {0}, S = {1}, T = φ and S = {3, 1, 5} (a) Write down the number of elements in each set. (b) Which of the sets are equal? (c) Which of the sets have equal number of elements but the sets are not equal? 7. (a) If A = {a, b, c, d} and B = {a, b, c, d, e}, all elements of A are also the elements B, but again A ≠ B. Why? (b) If P = {g, o, d}and Q = {c, u, p}, why are p and Q not equal sets? But P is equivalent to Q. Why? 8. A = {1, 2, 3, 4, 5} is a given set. Write the following subsets of A (i) by listing (ii) by set-builder notation. (a) B = Set of elements of A that are prime numbers (b) C = Set of elements of A that are even numbers (c) D = Set of elements of A that are odd numbers (d) E = Set of elements of A that are even prime numbers (e) F = Set of elements of A that are between 1 and 4 (f) Which of the sets above are equal sets? 5.

SETS 7 9. A set P is defined as P = {whole numbers up to 11}. List the element of the set P. Write the following subsets of P (i) by listing (ii) by set-builder notation. (a) A = Set of elements of P that are factors of 12 (b) B = Set of elements of P that are prime numbers (c) C = Set of elements of P that are multiples of 3 (d) D = Set of elements of P that are exactly divided by 5 (e) E = Set of elements of P that are the numbers up to 6 ANSWERS 4. (a) A = {1, 2, 3, 4, 5, 6, 7, 8, 9} (b) 2∈A, 3∈A, 0∉A, 9∈A, 11∉A (c) A = {x : 0 < x < 10} 5. (a) A = {1, 3, 5, 7, 9}, B = {11, 13, 17, 19, 23}, C = {7, 14, 21, 28, 35, 42, 49} (b) A and B 6. (a) 4, 3, 1, 1, 0, 3 (b) Set Q = Set S (c) R and S 8. (a) (i) B = {2, 3, 5} (ii) B = {x : x is a prime} (b) (i) C = {2, 4} (ii) C = {x : x is an even prime number} (c) (i) D = {1, 3, 5} (ii) D = {x : x is a odd number} (d) (i) E = {2} (ii) E = {x : x is an even prime number} (e) (i) F = {2, 3} (ii) F = {x : x is between 1 and 4.} (f) (i) None sets are equal. 9. P = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} (a) (i) A = {1, 2, 3, 4, 6} (ii) A = {x : x is a factor of 12} (b) (i) B = {2, 3, 5, 7, 11} (ii) B = {x : x is a prime number} (c) (i) C = {3, 6, 9} (ii) C = {x : x is a multiple of 3} (d) (i) D = {5, 10} (ii) D = {x : x is a number exactly divisible by 5} (e) (i) E = {0, 1, 2, 3, 4, 5} (ii) E = {x : x is between 1 and 4.} (f) (i) None sets are equal.

8 The Leading Maths - 8 At the end of this topic, the student will be able to: ¾ define disjoint and overlapping sets. ¾ define subset, proper subset and improper subsets. Learning Objectives I. Introduction Sets may or may not be related to each other. We begin with a simple example. We consider a class C that represets a universal set U consisting of (a) the girl-students: Babita and Chanchali (b) the boy-students: Amar, Dhan and Subindra and (c) the class-teacher: Gangadhar We denote, The set of girl-students by G, The set of boy-students by B, The set of girl-students and the class-teacher by X. The set of boy-students and the class-teacher by Y. and The set of the class-teacher only by T. Here, the universal set U is the set consisting of (a) the girl-students, G = {Babita, Chanchali } or {b, c} (b) the boy-students, B = {Amar, Dhan, Subindra } or {a, d, s} and (c) the class-teacher. T = {Gangadhar} or {g}. Now, what is the relation among the sets G, B, X and Y ? We easily notice that: (a) There is no element common between the sets G and B. (b) There is a common element between the sets X and Y. 1.2 Types of Sets

SETS 9 (c) Every element of the set G is an element of the set X. (d) The set X and the set B have no common element but have the same number of elements. In our consideration, the whole set U consisting of the girl-students, the boy-students and the class-teacher, i.e., the set U = {b, c, a, d, s, b} is the universal set. The various situations described above are shown in the following Venn diagrams. II. Disjoint sets In the adjoining Venn Diagram, we notice that: (a) the set G has no element common with the set B, (b) the set B has no element common with the set T and (c) the set T has no element common with the set G. Pairs of sets like G and B, B and T, and T and G having no common element are called disjoint sets. Similarly, the sets H, T and B have no common elements at all. They are also disjoint sets. The sets that nave no common elements at all, are known as disjoint sets. They are also called non-intersecting or non-overlapping sets. III. Overlapping or intersecting sets In the adjoining Venn Diagram, we notice that the element g is common to both sets X and Y. Why? In fact, 'g' is the teacher of both girlstudents and boy- students. The sets having one or more common elements are known as overlapping or intersecting sets. G T B b c a d s g U b c a d s g U T B X Y G

10 The Leading Maths - 8 IV. Subsets We further note that: (a) the sets G = { b, c} and X = {b, c, g} are overlapping with b and c as common elements, and (b) every element of G is also an element of X. We thus see that G is wholly within X or G is contained in X. We also say that G is a subset of X. In symbol, we write G ⊆ X. In the same way, (a) the sets B = {a, d, s} and Y = { a, d, s, g} are overlapping with a, d and s as common elements and (b) every element of B is also an element of Y. We thus see that B is wholly within Y or B is contained in Y. We also say that B is a subset of Y. In symbols, we write B ⊆ Y. But, the two sets X = {b, c, g} and Y = {a, d, s, g} have one common element ‘g’; and neither X is contained in Y nor Y is contained in X. A set A is said to be a subset of another set B if every element of A is also an element of B. In symbol, we write it as A ⊆ B. Sometimes we call B a superset of A and it is written as, B ⊇ A. In particular, if A = {1, 2, 3}, then its subsets may be { }, {1,}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} or {1, 2, 3}. The sets of all subsets of any non- empty set are called power sets. Here, the number of elements in A, n = 3, the number of subsets of A = 8 Hence, the number of subsets of a set having 'n' elements is 2n . Note, (i) φ is the subsets of every non-empty set. (ii) Any set is the subset itself. V. Types of a subsets a. Proper Subsets A subset that contains at least one element less than that of the super set is called a proper subset. If A is the proper subset of B, it is denoted by A ⊂ B e.g., G X b c g U B Y a d s g U

SETS 11 If A = {1, 2, 3}, then the subsets { } {1}, {2}, {3}, {1, 2}, {2, 3} and {1, 3} are proper subsets of A. Note: (i) φ is the proper subset of every non-empty set. (ii) A set A is not proper subset itself. (iii) φ is not proper subset itself. (iv) The number of proper subsets of a set having 'n' elements is 2n – 1. b. Improper Subsets A subset that contains all the elements of the set itself is called an improper subset. If A is the improper subset of B, it is denoted by A ⊂ B. If A = {1, 2, 3} then the subset {1, 2, 3} or {2, 3, 1} or {3, 2, 1} is the improper subset of A. Note: A set and itself are improper subsets. CLASSWORK EXAMPLES Example : 1 Write down the common element or elements, if any of the following set is: (a) The singleton set {0} and the doubleton set {10, 20} (b) The set of counting numbers less than five and the set of counting numbers greater than three (c) The set R = {r, e, a, p} and P = {p, e, a, r} Solution: (a) Here, Singleton set, S = {0}, Doubleton set, D = {10, 20}. The set S and D have no common element. (b) Suppose, F = Set of counting numbers less than five = {1, 2, 3, 4} T = Set of counting number greater than three = {4, 5, 6, 7, ......}. Here, the element 4 is in common sets F and T. (c) Here, R = {r, e, a, p}, P = {p, e, a, r}. The sets R and P have all common elements. 2 A 1 3 U 2 A B 1 3 U

12 The Leading Maths - 8 Example : 2 State whether the following are disjoint or intersecting sets: (a) A = {123} and B = {321} (b) P = {1, 2, 3} and Q = {3, 2, 1} (c) F = {Amar, Ali} and M = {Gita, Ali} Solution: Why are they disjoint or intersecting sets? Give reasons. a) The sets A and B are disjoint sets because the numbers 123 and 321 are different although they have the same digits. b) The sets P and Q are intersecting sets because both of them have all common elements. (c) The sets F and M are intersecting sets because the element 'son' is in common for the both sets F and M. Example : 3 Is A a subset of B? Give reason. (a) The set A = {t, o, p} and the set B = {s, t, o, p} (b) The set A = {123} and the set B = {321} (c) The set A = {r, e, a, p} and B = {p, e, a, r} Solution: (a) Yes, A is the subset of B because all the elements of A contain in B. (b) No, A is not the subset of B because they are disjoint sets. (c) Yes, A is the subset of B because of all the elements of A are also the elements of B. Example : 4 Write the relation of each of the following pairs of sets in subsets form: (a) M = {m, a, n} and W = {w, o, m, a, n} (b) S = {1, 2, 3} and T = {1, 2} (c) A = {a, e, i, o, u} and B = {i, o, u, e, a} Solution: (a) M ⊂ W (b) T ⊂ S (c) A ⊂ B and B ⊂ A or A ⊂ B Example : 5 Write the proper subsets of the set A = {a, b}. Solution: The proper subsets of the set A = {a, b} are {}, {a} and {b}.

SETS 13 EXERCISE 1.2 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. State whether each of the following is true or false: (a) Two sets are disjoint if they have no common element. (b) Two sets are intersecting if they have no common element. (c) Overlapping sets are also known as intersecting sets. (d) Two sets are disjoint if they have some common elements. (e) Two sets are intersecting if they have at least one common element. 2. Fill in the blanks: (a) If there is no common element in the given sets, they are called ............... sets. (b) If there are at least one element in the given sets, they are called ............. sets. (c) If every element of a set is an element of another set, the later set is called a ............. of the former. (d) If every element of a set is an element of another set, the former set is called a ............. of the later. 3. (a) Write the definition of a superset of a set. (b) Define a subset of a set. (c) Distinguish between disjoint and intersecting sets. 4. Tick mark (or write down) the correct answer: (a) If two sets have different elements, they are said to be i. intersecting ii. disjoint iii. different iv. none of the above (b) If two sets have the same elements and nothing else, they are said to be i. intersecting ii. disjoint iii. different iv. none of the above

14 The Leading Maths - 8 (c) If P = {1, 3. 5} and Q = {1, 2, 3, 4, 5}, then which statement is true? i. P ⊆ Q ii. P ⊃ Q iii. P ⊇ Q iv. P ∼ Q (d) Which is correct notation for "M is proper subset of N"? i. N ⊂ M ii. N ⊆ M iii. M ⊆ N iv. M ⊂ N (e) If A = {1, 2, 3}, B = {1, 3, 5} and C = {1, 2, 3, 4}, then which is the subset of C? i. B ii. A iii. both A and B iv. none of the above 5. State whether the following are disjoint or intersecting: (a) A = {12} and B = {21} (b) { but } and {tub} (c) A = {2, 3} and B = {3, 4} (d) {b, u, t} and {t, u, b} (e) A = {x : x is a counting number less than 3.}and B = {x : x is one of the first two counting numbers.} (f) Points of two sides of a triangle. 6. Write down the common element or elements of the following sets: (a) The singleton set {1} and the set {12, 21} (b) The set of counting numbers less than 10 and the set of counting numbers greater than 5. (c) The set E = {e, a, t} and T = {t, e, a} (d) A = {x : x is an odd number} and B = {x : x is an even number}. 7. Distinguish weather the given pairs of sets are disjoint or intersecting sets. (a) P = {2, 3, 5, 7} and Q = {2, 4, 6, 8} (b) A = {x : x is odd number less than 10} and B = {x : x is even number less than 12} (c) N = {x : x < 5, x ∈ N} and M = {x : – 2 < x ≤ 4, x ∈ Z} (d) X = {x : x is letters in word ‘hiram’} and Y = {x : x is letters in word ‘ashok’}

SETS 15 8. Is A a subset of B for the following sets? (a) The set A = {t, e, a} and the set B = {t, e, a, m} (b) The set T = {12} and the set O = {123} (c) The set L = {l, e, a, p} and P = {p, l, e, a} 9. Write the relation of each of the following pairs of sets in subset form: (a) M = {a, r, t} and W = {c, a, r, t} (b) E = {2, 4, 6, 8} and D = {1, 2, 3, 4, 5, 6, 7, 8, 9}} (c) A = {a, b, c, …, x, y, z} and B = {i, o, u, e, a} 10. M is a set of the English months. (a) Write the elements of M in the listing method. (b) List the following subsets of M; i. the months starting with letter J. ii. the months starting with letter M. iii. the months starting with letter A or D. iv. the months starting with letter M or S or N. 11. Let A = {2, 4, 6, 8, 10} be the given set. List the following subsets of A. Also, write their types. (a) P = Set of prime numbers (b) M = Set of multiples of 2 (c) C = Set of composite numbers (d) F = Set of the prime factors of 4. (e) O = Set of odd numbers (f) S = Set of square number 12. Study the following sets and answer the given questions. (a) The given set is G = {g, o, d}. i. Write all the subsets formed by the elements of the set G. ii. How many subsets of the set G are formed? iii. Establish the relation of the number of the elements of the set G and the number of subsets of G.

16 The Leading Maths - 8 (b) The given set if C = {cube numbers less than 30}. i. Write the set C in listing method. ii. Write all the proper subsets formed by the elements of the set C. iii. How many proper subsets of the set C are formed ? iv. Establish the relation between the number of elements of the given set and the number of proper subsets formed by that set. 13. Given that U = {x:x ≤ 15, x ∈ w}, A = {x:x is an odd number}, B = {x:x is a factor of 12} and C = {x:x is a multiple of 3}. (a) List the elements of the above given sets. (b) Write the elements that have the sets A and B. (c) Write the elements that have all sets A, B and C. (d) Is B ⊇ C? Why? Given reason. (e) Are any two sets equivalent? Why? Given reason. ANSWERS 6. (a) f (b) 6, 7, 8, 9 (c) e, a, t (d) f 7. Overlapping : (a), (c), (d) disjoint : (b) 9. (a) M ⊂ W (b) E ⊂ D (c) B ⊂ A 11. (a) {2}, singleton set (b) {2, 4, 6, 8, 10}, equal sets (c) {4, 6, 8, 10}, finite set (d) {2}, singleton set (e) { }, null set (f) {4}, singleton set 12. (a) (i) { }, {g}, {o}, {d}, {g, o}, {g, d}, {o, d}, {g, o , d} (ii) 8 (b) (i) {1, 8, 27} (ii) { }, {1}, {8}, {27}, {1, 8}, {8, 27}, {1, 27} (iii) 7 13. (a) U = {0, 1, 2, 3, ......., 15}, A = {1, 3, 5, 7, 9, 11, 13, 15}, B = {1, 2, 3, 4, 6, 12}, C = {3, 6, 9, 12, 15} (b) {1, 3) (c) {3} Project Work 1.2 List out the materials in a geometric box of you and your four friends who have the geometric boxes. Let them write in set notation. Answer the following questions using Venn diagram. (a) Who have more materials and who have less ? (b) Who have equal sets and who have equivalent sets? (c) Whose set is subset of others? (d) Whose set is proper or improper subset of others?

SETS 17 Read, Understand, Think and Do 1. Two sets A and B are given below. A = {1, 2, 5} and B = {2, 3, 4} (a) Are A and B disjoint sets? Give reason. (b) Make an improper subset of set A. (c) Write two equal proper subsets from set A and set B. 2. Two sets A and B are given below. A = {even numbers less than 10} and B = {prime number less than 10} (a) A and B are overlapping sets. Give reason. (b) How many proper subsets can be made from set A and set B? (c) Are the proper subsets of A and B equivalent? Justify with reason. 3. The elements of the set P and set Q are shown in the given Venn diagram. (a) Write the set P and set Q in listing method and identify these two sets are either overlapping sets or disjoint sets. (b) How many proper subsets can be made from set P? Write all proper subsets of set P. (c) Write any one of the common proper subsets of set P and set Q. 4. The sets P and C are the subsets of universal set, U = {whole number less than 5} P = {prime number} and C = {composite number} (a) Write the set P and set C in listing method and justify P and C are disjoint sets. (b) How many proper subsets can be made from set C? Write all proper subsets of set C. (c) Is there at least one common proper subset of set P and set C? Justify with reason. P Q m n o U MIXED PRACTICE–I

18 The Leading Maths - 8 5. Elements of set M and set N are shown in the given Venn diagram. (a) Write the common element of set M and set N. (b) Make all proper subsets and improper subset of set M. (c) Are the proper subsets of M and N equivalent? Justify it. 6. The elements of the set K and set L are shown in the given diagram. (a) Write the set K and set L in listing method and identify these two sets are either overlapping sets or disjoint sets. (b) How many proper subsets can be made from set K? (c) Which set is the subset of K and L both ? ANSWERS 1. (a) No (b) {2, 1, 5} or {5, 1, 2} or ....... (c) {2} 2. (a) {2} (b) 15, 15 3. (a) P = {m, n}, Q = {m, o}, overlapping (b) 3, { }, {m}, {n} (c) {n} 4. (a) {2, 3}, {4} (b) 3, { }, {2}, {3} 5. (a) b (b) { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} 6. (a) A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4}, overlapping (b) 31 (c) {1, 3} M N d b a U c f K L 4 8 2 3 1 6 10 5 7 9 U

SETS 19 FM : 21 Time : 40 Min. CONFIDENCE LEVEL TEST I SETS 1. The sets M and N are defined as, M = {prime numbers less than 10} and M = {odd numbers less than 10} (a) Write the type of the relation between the sets M and N. [1] (b) How many improper subsets are made from the set M ? [1] (c) How many proper subsets are made from the set N ? [1] 2. Two sets P and Q are given below. P = A set of even numbers between 2 and 10 and Q = A set of prime number up to 6 (a) Are the sets P and Q disjoint sets? Give reason. [1] (b) Write the proper subsets of the set Q. [1] (c) Are the improper subsets of P and Q equivalent? Justify with reason. [1] 3. The elements of the sets M and set N are shown in the given Venn diagram. (a) Define subset. [1] (b) Make all proper subsets and improper subset of set M. [1] (c) Write the common element of set M and set N. [1] 4. The elements of the set A and set B are shown in the given diagram. (a) Write the set A and the set B in listing method and identify these two sets are either overlapping sets or disjoint sets. [1] (b) How many proper subsets can be made from set A? [1] (c) Which set is the subset of A and B both ? [1] M N a d b U c A B 6 3 11 9 7 4 1 12 2 10 8 U Attempt all the questions.

20 The Leading Maths - 8 5. The sets P and C are the subsets of universal set, U = A set of whole numbers less than 7. P = {prime number} and C = {composite number} (a) Write the set P and set C in listing method and justify P and C are disjoint sets. [1] (b) How many proper subsets can be made from set C? Write all proper subsets of the set C. [1] (c) Is there at least one common proper subset of set P and set C? Justify with reason. [1]

ARITHMETIC 21 COMPETENCY  solution of the behaviour problems related to real number system CHAPTERS 1. Real Number System 3. Ratio and Proportion 4. Profit and Loss 5. Unitary Method 6. Simple Interest LEARNING OUTCOMES introduce binary and quinary number systems, convert decimal numbers into binary and quinary number system, convert binary and quinary numbers into decimal number system, identify the irrational numbers, separate rational and irrational numbers, write the decimal number into scientific notation and scientific notation into decimal number, solve the problems related to ratio and propertion, solve the problems related to profit and loss including discount, solve the problems using direct and indirect variation up to three variables, state the concept of simple interest, solve the problems related to simple interest. ARITHMETIC UNIT II Estimation Teaching Hours - 49 (Th. + Pr.)

22 The Leading Maths - 8 CHAPTER 2 REAL NUMBER SYSTEM Lesson Topics Pages 2.1 Quinary Number System 23 2.2 Binary Number System 30 2.3 Rational Numbers 36 2.4 Irrational Numbers 47 2.5 Scientific Notation 52 ” How many legs do two boys and two girls have? ” How many wheels do five bicycles and four motor cycles have? ” How many fingers are there in two hands and one leg? ” How many weeks are there in a week? ” How many weeks are there in a year? WARM-UP

ARITHMETIC 23 2.1 Quinary Number System At the end of this topic, the student will be able to: ¾ introduce quinary number system, ¾ express the decimal numbers into quinary numbers and vice versa. Learning Objectives I. Review on Hindu-Arabic Number System The numbers are represented by the numerals of figures or digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 which are similar to us as our Hindu-Arabic Numerals. Using these numerals, we can write any number we like to. The same digit has different value according to its place in Hindu-Arabic Numeration system based in place value of 10, it is termed as the decimal system of numeration. It is also called denary numeral system. ACTIVITY - 1 How many ticks () are there below? As we count one, two, three, .......... etc. But to write the numbers we use the figures of Hindu-Arabic Numerals. To write the numbers we first make groups of ten as follows: This means that these are two groups of tens and 3 units (ones) in Hindu-Arabic Numeration. Units are placed in the first place from the right, groups of tens are placed in the second place from the right, groups of tens of ten (hundreds) are placed in the third place from the right and so on. So, the given number is written as, 0 2 3 or 0 0 2 3 or, simply 2 3 = 20 + 3 = 2 × 10 + 3 × 1 = 2 × 101 + 3 × 100 Which means that there are 3 units and two groups of tens, no groups of hundreds and no groups of tens hundreds (thousand). We can however think of other numeration system that is not based in base 10.

24 The Leading Maths - 8 II. Number in Bases Other than 10 Hindu-Arabic Numeration System is based on the group of 10 and uses ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 in uniting numerals. However, we can think of the numeration system that counts based on grouping other than 10. i.e group of 5 (data format), group of 2 (computing system), group of 7 (week), group of 8 (generation computing system, 32 bit, 64 bit), group of 12 (clock), etc. III. Quinary number system ACTIVITY - 2 If the above units are arranged in the groups of five, how many groups of 5 and units are there? The groups are as follow: In the base five system, we use just 5 symbols or figures as 0, 1, 2 ,3 and 4. In this system, we place the units in the first from the right, the groups of 5 are placed in the second from the right, the groups of five are in the third place and so on. We use the above digits to write the numerals as follows: 4 × 5 + 3 × 1 = 0 4 3 or, simply 4 3 Which means there are 3 units and 4 groups of 5. We write this numeral in base five as, 435 The base 5 numeral system is called quinary number system. Based on the above discussion, now we have the following place value table for the Quinary system of Numeration. Quinary System Digits 7th 6th 5th 4th 3rd 2nd 1st Place Value 56 55 54 53 52 51 50 Quinary Number (2410432) 2 4 1 0 4 3 2 Numerical value is base 10 15625 3125 625 125 25 5 1

ARITHMETIC 25 Decimal Number (44492) 2×15625 = 31250 4×3125 = 12500 1×625 = 625 0×125 = 0 4×25 = 100 3×5 = 15 2×1 = 2 31250 + 12500 + 625 + 0 + 100 + 15 + 2 = 44492 IV. Conversion Quinary Number and Decimal Number Using the place value chart we can convert number in one system to the number in other system. Look at the following example: a. Converting Decimal Number into Quinary Number Convert 142 into base 5 system: 53 52 51 50 125 25 5 1 1 0 3 2 In 142 there is one 125. Record 1 below 125. The reminder is 17. In 17 there are 3 fives (3 × 5=15) and the remainder is 2. Record 3 below 5. In 2 there are two ones. Therefore, record 2 below 1. In 17 there is no 25. Record 0 below 17. The remainder is 17. Therefor,142 = 1032(5) In shortcut we have 5 142 5 28 + 2 5 5 + 3 5 1 + 0 0 + 1 Remainder Therefore, 142 = 1032(5) Steps for converting decimal number to quinary number Step-1: Divide the given decimal number by 5 and keep the remainder in the right. Step-2: Divide continuously by 5 till the quotient becomes zero (0). Step-3: Write the remainders in the reverse order from bottom to top, which is the required quinary number.

26 The Leading Maths - 8 b. Converting of quinary number into decimal number Convert 1243(5) into the base 10 system. Inserting 1243(5) into base-5 place value chart, we get 53 52 51 50 125 25 5 1 1 2 4 3 1 × 125 = 125 2 × 25 = 50 4 × 5 = 20 3 × 1 = 3 198 Sum = 198 Therefore, 1243(5) = 198 (The base in base 10 is self understood) Alternatively In horizontal addition method, we have 1245(5) = 1 × (53 ) + 2 × (52 ) + 4 × (51 ) + 3 × (50 ) = 1 × 125 + 2 × 25 + 4 × 5 + 3 × 1 = 125 + 50 + 20 + 3 = 198, without base, because the base 10 is self understood in many cases. Steps for converting quinary number to decimal number Step-1: Multiply each digit from right by increasing power of 5 starting from 0. Step-2: Calculate as in decimal number system. CLASSWORK EXAMPLES Example : 1 Write 1101(5) in its expanded form and write its equivalent in base 10-numeration system. Solution: The numeration in expanded form is, 1101(5) = 1 × 53 + 1 × 52 + 0 × 51 + 1× 50

ARITHMETIC 27 To change this numeral in base 10 system, we have 1101(5) = 1 × 53 + 1 × 52 + 0 × 51 + 1× 50 = 1 × 125 + 1 × 25 + 0 × 5 + 1× 1 = 125 + 25 + 0 + 1 = 151 Example : 2 Convert 674 in base 5-numeration system. Solution: Here, the given decimal number is 675. 5 674 Remainder 5 134 → 4 5 26 → 4 5 5 → 1 5 1 → 0 0 → 1 ∴ The required quinary number is 10144(5). EXERCISE 2.1 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) What is quinary number ? Define it. (b) How many digits are there in the quinary number system? What are they? (c) Why is quinary number system different from denary number system? Give reason. (d) What is the decimal number of 105 ? 2. Circle ( ) the correct answer. (a) Which is quinary number system? i. Base 2 ii. Base 3 iii. Base 5 iv. Base 10

28 The Leading Maths - 8 (b) Which number is in quinary number system? i. 1024 ii. 35710 iii. 10112 iv. 3145 (c) How many digits are there in the quinary number system? i. 2 ii. 3 iii. 4 iv. 5 (d) Which digits are in the quinary number system? i. 0, 1 ii. 0, 1, 2, 3 iii. 0, 1, 2, 3, 4 iv. 0, 1, 2, 3, 4,..., 9 (e) What is decimal number of 115 ? i. 2 ii. 6 iii. 11 iv. 30 (f) What is quinary number of 25? i. 1005 ii. 1015 iii. 3005 iv. 2005 3. Write the following quinary numbers in its expanded form and write its equivalent into base 10-numeration system. (a) 4(5) (b) 14(5) (c) 20(5) (d) 102(5) (e) 134(5) (f) 142(5) (g) 1101(5) (h) 1204(5) (i) 23103(5) (j) 34240(5) (k) 41301(5) (l) 344210(5) (m) 3411302(5) (n) 4034241(5) (o) 23403432(5) 4. Convert the following quinary numbers into decimal numbers. (a) 12(5) (b) 34(5) (c) 123(5) (d) 234(5) (e) 432(5) (f) 4033(5) (g) 1243(5) (h) 4314(5) (i) 43024(5) (j) 12434(5) (k) 12443(5) (l) 320422(5) (m) 432114(5) (n) 4315424(5) (o) 1241034(5)

ARITHMETIC 29 5. Convert the following decimal numbers into quinary (base 5) numbers. (a) 36 (b) 45 (c) 63 (d) 75 (e) 99 (f) 255 (g) 205 (h) 315 (i) 480 (j) 565 (k) 6951 (l) 10253 (m) 3553 (n) 4804 (o) 5757 ANSWERS 3. (a) 4 (b) 9 (c) 10 (d) 27 (e) 44 (f) 47 (g) 151 (h) 179 (i) 1653 (j) 2445 (k) 2701 (l) 12430 (m) 60202 (n) 64946 (o) 216117 4. (a) 7 (b) 19 (c) 38 (d) 69 (e) 117 (f) 518 (g) 198 (h) 584 (i) 2889 (j) 994 (k) 998 (l) 10737 (m) 14659 (n) 73239 (o) 24519 5. (a) 121(5) (b) 140(5) (c) 223(5) (d) 300(5) (e) 344(5) (f) 2010(5) (g) 1310(5) (h) 2230(5) (i) 3410(5) (j) 4230(5) (k) 210301(5) (l) 212003(5) (m) 103203(5) (n) 123204(5) (o) 141012(5)

30 The Leading Maths - 8 2.2 Binary Number System At the end of this topic, the student will be able to: ¾ introduce binary number system. ¾ convert the decimal numbers into binary numbers and vice versa. Learning Objectives I. Introduction Hindu-Arabic Numeration System is based on the group of 10 and uses ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 in uniting numerals. However, we can think of the numeration system that counts based on grouping other than 10. For example, In the above lesson if the ticks () are arranged in the groups of two’s, how many groups of 2 and unit are there? The groups are as follows: In the base two system, we use just 2 symbols or figures as 0 and 1. In this system, we place the units in the first from the right, the groups of 2 are placed in the second from the right, the groups of two of 2 are in the third place and so on. We arrange the above groups of two’s as follows again, There is one group of 8 two’s, i.e., a group of 16; no group of 4 two's i.e. one group of 8; one group of 2 two’s, i.e., a group of 4; one group of two’s and one group of one. 1 × 16 + 0 × 8 + 1 × 4 + 1 × 2 + 1 × 1 = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20 We use the above digits to write the numerals as follows: 1 0 1 1 1 We write this numeral in base two as,101112 . The base 2 numeral system is called binary number system. We know that 16 = 24 , 4 = 22 2 = 21 1 = 20

ARITHMETIC 31 Based on the above discussion, now we have the following place value table for the binary system of numeration. Binary System Position of Digits 7th 6th 5th 4th 3rd 2nd 1st Place Value 26 25 24 23 22 21 20 Binary Number (1010111) 1 0 1 0 1 1 1 Numerical value in base 10 64 32 16 8 4 2 1 Decimal Number (87) 1×64 = 64 0×32 = 32 1×16 = 16 0×8 = 0 1×4 = 4 1×2 = 2 1×1 = 1 64 + 0 + 16 + 0 + 4 + 2 + 1 = 87 II. Conversion of Binary Number and Decimal Number We can convert number in one system to the number in other system using the place value chart. Look at the following example: a. Converting decimal number into buinary number Convert 69 into base 2. Hence, 69(10) = 1000101(2) 26 25 24 23 22 21 20 64 32 16 8 4 2 1 1 0 0 0 1 0 1 In 69, there is one 64. Record 1 below 64. The reminder is 5. In 5, there is one 4. Record 1 below 4. The reminder is 1. In 5, there is no 32. Record 0 below 32. In 5, there is no 16. Record 0 below 16. In 5, there is no 8. Record 0 below 8. In 1, there is no 2. Record 0 below 2. In 1, there is one 1. Record 1 below 1. 2 69 Remainder 2 34 + 1 2 17 + 0 2 8 + 1 2 4 + 0 2 2 + 0 2 1 + 0 0 + 1 ∴ Hence, 69(10) = 1000101(2) “Alternatively”

32 The Leading Maths - 8 Steps of converting decimal number to binary number Step-1 : Divide the given decimal number by 2 and keep the remainder in the right. Step-2 : Divide continuously by 2 till the quotient becomes zero (0). Step-3 : Write the remainders in the reverse order from bottom to top, which is the required binary number. b. Converting quinary number into decimal number Convert 1011011(2) into decimal system. Inserting 1011011(2) in base 2 place value chart, we get 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 1 0 1 1 0 1 1 1 × 64 = 64 0 × 32 = 0 ∴ 1011011(2) = 91 1 × 16 = 16 1 × 8 = 8 0 × 4 = 0 1 × 2 = 2 1 × 1 = 1 Sum = 91 “Alternatively” 1 × (26 ) + 0 × (25 ) + 1 × (24 ) + 1 × (23 ) + 0 × (22 ) + 1 × (21 ) + 1 × (20 ) = 1 × 64 + 0 × 32 + 1 × 16 + 1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 64 + 16 + 8 + 2 + 1 = 91, without base, because the base 10 is self understood in many cases. Steps for converting binary number to decimal number Step-1:Multiply each digit from right by increasing power of 2 starting from 0. Step-2: Calculate as in decimal number system.

ARITHMETIC 33 CLASSWORK EXAMPLES Example : 1 Change the numeral 110012 into the decimal numeral. Solution: Here, 110012 = 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 = 16 + 8 + 0 + 1 = 25 Example : 2 Convert 674 in base 2 numeration system. Solution: Here, the given decimal number is 675. 2 113 Remainder 2 56 → 1 2 28 → 0 2 14 → 0 2 7 → 0 2 3 → 1 2 1 → 1 0 → 1 ∴ The required binary number is 11100012 . Example : 3 Express 3425 in binary number system. Solution: Here, 3425 = 3 × 52 + 4 × 51 + 2 × 50 = 3 × 25 + 4 × 5 + 2 × 1 = 75 + 20 + 2 = 97 Again, converting the decimal number 97 into binary number system, we get 2 97 Remainder 2 48 → 1 2 24 → 0 2 12 → 0 2 6 → 0 2 3 → 0 2 1 → 1 0 → 1 ∴ The required binary number is 11000012 .

34 The Leading Maths - 8 EXERCISE 2.2 Your mastery depends on practice. Practice like you play. Read, Understand, Think and Do Keeping Skill Sharp 1. (a) Define binary number system. (b) Write any one difference between binary and denary number system. (c) What is the binary number of 9? (d) What is the decimal number of 1012 ? 2. Circle ( ) the correct answer. (a) What is the base of the binary number system? i. 1 ii. 2 iii. 5 iv. 10 (b) The number system having base 2 is called .................. i. quinary ii. denary iii. binary iv. octonary (c) Which digits are used in the binary number system? i. 0, 1 ii. 0, 1, 2, 3 iii. 0, 1, 2, iv. 0, 1, 2, ......., 9 (d) Which is the decimal number of 1012 ? i. 4 ii. 6 iii. 7 iv. 5 (e) The binary number of 9 is .................... i. 10112 ii. 11012 iii. 1012 iv. 1112 3. Write the following binary numbers in expanded form and write their equivalent in base 10-numeration system: (a) 102 (b) 112 (c) 1012 (d) 1102 (e) 11012 (f) 101012 (g) 110112 (h) 111102 (i) 110102 (j) 101012 (k) 111112 (l) 1011002 (m) 11001102 (n) 10001112 (o) 10011002 4. Convert the following binary numbers into decimal numbers: (a) 1102 (b) 1112 (c) 10112 (d) 101012 (e) 110112 (f) 100112 (g) 1101102 (h) 1100112 (i) 11001102 (j) 11000112 (k) 11111112 (l) 10000012

ARITHMETIC 35 5. Change the following decimal numbers into binary numbers: (a) 15 (b) 32 (c) 36 (d) 72 (e) 121 (f) 357 (g) 465 (h) 512 (i) 526 (j) 599 6. Express the following quinary numbers into binary numbers: (a) 145 (b) 215 (c) 405 (d) 435 (e) 1235 (f) 2345 (g) 3005 (h) 40035 (i) 234125 (j) 104425 7. Convert the following binary numbers into quinary numbers: (a) 112 (b) 1112 (c) 10112 (d) 110012 (e) 11000112 (f) 11000002 (g) 10001112 (h) 11110112 8. (a) If you weight is 48 kg, answer the following questions: i. Convert your weight in decimal system. ii. What is the weight of your body in quinary system? Find it. (b) The shorter two edges of a right triangular garden are 5 m and 12 m. i. Convert all the edges of the garden in binary numbers. ii. Convert all the edges of the garden in quinary numbers. ANSWERS 3. (a) 2 (b) 3 (c) 5 (d) 6 (e) 13 (f) 21 (g) 27 (h) 30 (i) 26 (j) 21 (k) 31 (l) 44 (m) 102 (n) 71 (o) 76 4. (a) 6 (b) 5 (c) 7 (d) 11 (e) 13 (f) 9 (g) 21 (h) 27 (i) 19 (j) 54 (k) 51 (l) 102 (m) 99 (n) 127 (o) 65 5. (a) 11112 (b) 1000002 (c) 1001002 (d) 10010002 (e) 11110012 (f) 1011001012 (g) 1110100012 (h) 10000000002 (i) 10000011102 (j) 10010101112 6. (a) 10012 (b) 10112 (c) 101002 (d) 101112 (e) 1001102 (f) 10001012 (g) 10010112 (h) 1111101112 (i) 110110000102 (j) 10111010112 7. (a) 35 (b) 125 (c) 215 (d) 1005 (e) 3445 (f) 3415 (g) 2415 (h) 4435 8. (a) i. 1100002 ii. 1435 (b) i. 1012 , 11002 , 11012 ii. 105 , 225 , 235 Project Work 2.2 Ask the price of any ten goods at your nearby shop. Convert all the prices into binary and quinary number systems.

36 The Leading Maths - 8 3.8 WARM-UP I. Review on Integers Before we do some arithmetic of the base-two and base-five systems of numeration, we need to revisit the arithmetic of various operations based on the decimal system. We consider counting numbers denoted by the Hindu-Arabic numerals: 1, 2, 3, 4, 5, 6, 7, 8, 9, ............... The counting numbers (also called positive integers) together with the number zero “ 0 ” form the set of whole numbers. When a whole number is added to a whole number it becomes a whole number. So, addition is meaningful in the set of whole numbers. But taking away or removing something or subtracting a whole number from another whole number may or may not be meaningful. For examples, 3 – 2 = 1 is meaningful but 2 – 3 is not meaningful. To make 2 – 3 meaningful, a new number – 1 was invented. And so was the case with the invention of the new numbers – 2, – 3, – 4, …. . These later numbers were called negative integers. The positive integers, zero and the negative integers taken together form the set of integers. ” What is your weight and height ? ” How much length of the scale has sunk into the water of the given glass? ” How much length of the scale has not sunk? ” What represents the water level, numbers in water and above water? ” How many parts of the scale are sunk in water? 2.3 Rational Numbers At the end of this topic, the student will be able to: ¾ find the rational numbers between any two rational numbers. Learning Objectives

ARITHMETIC 37 They are …, – 3, – 2, – 1, 0, 1, 2, 3, …. Thus, in the set of integers when “An integer is added to or subtracted from another, the result is again an integer ”. Since multiplication is just a repetition of addition, multiplication has a meaning in the set of integers. II. Introduction to rational number We use number lines to give the geometrical representation of integers. We define a unit length on the number line, then the other integers are represented by the unit distance from the origin. –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 If the integer is positive, it lies to the right side of the number zero. If the integer is negative, it lies to the left side of the number zero. For example, what happens when + 1 is divided by + 2 ? i.e., where does + 1 2 lie on the number line? This means to divide the unit length 0 to 1 into two equal parts and one part to represent 1 2 (called half). –2 –2 –1 –2 –2 0 +1 +2 1 4 1 2 + + Hence + 1 2 lies exactly half way between 0 and 1. Similarly + 1 4 lies exactly half way 0 between 0 and + 1 2. In this way, while working with integers, the product of two integers can be represented by a point on the number line. Consider for example, (+ 1 + + 3) × – 2 = + 4 × (– 2) = – 8 – 8 can be represented by a point 1 unit away to the left from 0. Let us think, what happens when the sum of two integers is divided by an integer? Can it be represented by a point on the number line?

38 The Leading Maths - 8 Consider for example, + 1 + + 2 +2 , i.e., the sum of + 1 and + 2 divided by + 2 is + 1 + + 2 + 2 = + 2 + + 1 + 2 + + 2 + 2 = + 1 + 2 = 1 + + 1 + 2 = + 1 1 2 –2 –2 –1 –2 –2 0 +1 +2 1 2 +1 The number + 1 1 2 is one and a half unit away to the right side from the origin (0). Actually , + 1 + + 2 is an integer sum +3 , hence + 3 divided by + 2 and this is the best question of dividing an integer by an integer. A number obtained by dividing an integer by a non-zero integer is a rational number. Symbolically, Q = {n: n = a b, where a, b are integers and b ≠ 0 } is the set of rational number. Note: (a) Every integer is a rational number because it has denominator 1. i.e., 4 = 4 1, –3 = – 3 1, 0 = 0 1 , etc. So, every natural number and whole numbers are also rational numbers. (b) Decimal numbers are also rational numbers. i.e., 0.4 = 4 10 = 2 5, 0.467 = 467 1000 , etc. (c) All fractions are rational numbers. But all the rational number are not fractions. Fractions are always positive. a. Rational number is a terminating decimal number. Let us take a rational number as 3 8. Now, convert the rational number into decimal number, 3 8 = 0.375, which has no more digits after 5 except 0. Therefore, it is a terminating decimal number. 8 ) 3 0 ( 0.375 – 24 6 0 – 56 4 0 – 40 0

ARITHMETIC 39 b. Rational number is a non-terminating recurring decimal number. Let us take a rational number as 25 11. Now, convert the rational number into decimal number, 25 11 = 2.272727..... = 2.27, which has many more 27 after 27. Therefore, it is a non-termination repeating decimal number. c. Non-terminating and non-repeating decimal number is not rational number. Let us take a number as 15. Now, evaluate the square root of 15, 15 = 3.8729......... , which has many more digits after 9 but no repeated of the same digit or pattern of the digits. Therefore, it is a non-repeating and nonterminating decimal number. III. Properties of Rational Numbers We will learn some useful properties of rational numbers. Property 1: If a b is a rational number and m is a non-zero integer, then a b = a × m b × m is also a rational number. In other words, a rational number remains unchanged, if we multiply its numerator and denominator by the same non-zero integer. Property 2: If a b is a rational number and m is a common divisor of a and b, then a b = a ÷ m b ÷ m is also a rational number. In other words, if we divide the numerator and denominator of a rational number by a common divisor of both, the rational number remains unchanged. 11) 2 5 ( 2.2727 – 22 3 0 – 22 8 0 – 77 3 0 – 22 8 0 – 27 3 3 3 15 – 9 3.8729 68 + 8 600 544 767 + 7 5600 5369 7742 + 2 23100 15484 77449 + 9 761600 697041 77458 64559

40 The Leading Maths - 8 Property 3: Let a b and c d be two rational numbers. Then a b = c d ⇔ a × d = b × c. Note: (a) Except zero, every rational number is either positive or negative. (b) Every pair of rational numbers can be compared. Property 4: For each rational number m, exactly one of the following is true: (a) m > 0 (b) m = 0 (c) m < 0 eg, 2 3 > 0, 0 3 = 0 and – 2 3 < 0. Property 5: For any two rational numbers a and b, exactly one of the following is true: (a) a > b (b) a = b (c) a < b Property 6: If a, b and c are rational numbers such that a > b and b > c, then a > c. Let a, b, and c be rational numbers, variables, or algebraic expressions. SN Property Example 1. Commutative Property of Addition : a + b = b + a 2 + 3 = 3 + 2 2. Commutative Property of Multiplication a × b = b × a 2 × ( 3 ) = 3 × ( 2 ) 3. Associative Property of Addition a + ( b + c ) = ( a + b ) + c 2 + ( 3 + 4 ) = ( 2 + 3 ) + 4 4. Associative Property of Multiplication a × ( b × c ) = ( a × b ) × c 2 × ( 3 × 4 ) = ( 2 × 3 ) × 4 5. Distributive Property a × ( b + c ) = a × b + a × c 2 × ( 3 + 4 ) = 2 × 3 + 2 × 4 6. Additive Identity Property a + 0 = a 3 + 0 = 3 7. Multiplicative Identity Property a × 1 = a 3 × 1 = 3 8. Additive Inverse Property a + (–a) = 0 3 + (–3) = 0

ARITHMETIC 41 SN Property Example 9. Multiplicative Inverse Property a . 1 a = 1 Note: a cannot = 0 3. 1 3 = 1 10. Zero Property a × 0 = 0 5 × 0 = 0 11. Closure Property of Addition a + b is a real number 10 + 5 = 15 12. Closure Property of Multiplication a × b is a real number. 10 × 5 = 50 13. Addition Property of Equality If a = b, then a + c = b + c. If x = 10, then x + 3 = 10 + 3 14. Subtraction Property of Equality If a = b, then a – c = b – c. If x = 10, then x – 3 = 10 – 3 15. Multiplication Property of Equality If a = b, then a × c = b × c. If x = 10, then x × 3 = 10 × 3 16. Division Property of Equality If a = b, then a / c = b / c, assuming c ≠ 0. If x = 10, then x 3 = x 3 17. Substitution Property If a = b, then a may be substituted for b, or conversely. If x = 5, and x + y = z, then 5 + y = z. 18. Reflexive (or Identity) Property of Equality a = a 12 = 12 19. Symmetric Property of Equality If a = b, then b = a. If 3.5 = 31 2, then 31 2 = 3.5. 20. Transitive Property of Equality If a = b and b = c, then a = c. If 2a = 10 and 10 = 4b, then 2a = 4b. 21. Law of Trichotomy Exactly ONE of the following holds: a < b, a = b, a > b If 8 > 6, then 8 ≠ 6 and 8 is not < 6.

42 The Leading Maths - 8 IV. Rational Numbers between Any Two Integers We locate the rational numbers between any two integers using number line as follows: To locate the rational number 3 5 on the number line. Draw a number line. Construct a line l1 through 0 as shown in the given figure. Also, construct another line l2 through + 1 and parallel to the line l1 . Q1 Q2 P1 Q3 P2 Q4 P3 Q5 P4 E P5 l 2 l 1 0 +1 +2 A B C D Construct 5 equal division with a pencil compass along l1 and l2 . Let P1 , P2 , P3 , P4 and P5 be the points on l1 dividing into 5 equal parts and Q1 , Q2 , Q3 , Q4 and Q5 be the points dividing l2 into 5 equal parts. Now, join O to Q1 , P1 to Q2 , P2 to Q3 , P3 to Q4 , P4 to Q5 , and P5 to +1 or E. Then A, B, C, D, E divide the length of 1 unit between 0 and 1 into 5 equal parts. The point C is at 3 5 parts of 1 unit from 0. Hence, C is the required point. Similarly, to locate 2 3 5 on the number line repeat the process as above constructing two parallel lines from 2 and 3. Construct 5 equal divisions along these parallel lines.

ARITHMETIC 43 0 +1 +2 +3 Q1 Q2 P1 Q3 P2 Q4 P3 Q5 P4 E P5 l 2 l 1 X is the point indicating 2 3 5. Classification of Real Number System We classify the real number system as follows: Real Number (R) Rational Number (Q) Fraction (F) Integer (Z) Negative Integers (Z) Zero (0) Odd Number (O) Even Number (E) Natural Number (N/Z+ ) Whole Number (W) Irrational Number (Q) Real Number System In Set R Q Q W Z O N E

44 The Leading Maths - 8 CLASSWORK EXAMPLES Example : 1 Insert three rational numbers between 2 and 3. Solution: One of the rational numbers that lies half way between 2 and 3 is the arithmetic average of 2 and 3. Therefore, one of the rational numbers is 1 2 (2 + 3) = 5 2 = 2 1 2. The other rational number may lie half way between 2 and 21 2; which is the arithmetic average of 2 and 21 2. We have, 1 2 2 + 2 1 2 = 1 2 2 + 5 2 = 1 2 4 + 5 2 = 1 2 × 9 2 = 9 4 = 2 1 4. And the next rational number may be lying half way between 21 2 and 3, which is again the arithmetic average of 21 2 and 3 Now, 1 2 2 1 2 + 3 = 1 2 5 2 + 3 = 1 2 5 + 6 2 = 1 2 × 11 2 = 11 4 = 23 4. Hence, the required rational numbers between 2 and 3 are 9 4, 5 2 and 11 4 as 21 4 , 21 2 and 23 4. 0 1 2 3 4 2 1 4 2 1 2 2 3 4 In fact, in between two numbers there lie infinite rational numbers. Therefore, there is not unique solution for this type of problem. Example : 2 Find the 5 rational numbers between 11 4 and 11 3 . Solution: Here, the given two rational numbers are 11 4 and 11 3 . The LCM of the denominators of 4 and 3 is 12. Convert the denominators to 12, 11 4 = 11 × 3 4 ×3 = 33 12 and 11 3 = 11 × 4 3 × 4 = 44 12. ∴ The 5 rational numbers between 11 4 and 11 3 are 34 12, 36 12, 39 12, 42 12 and 43 12.