Grade 9 Mathematics Textbook PDF

MATHEMATICS MATHEMATICS MATHEMATICS STUDENT’S TEXTBOOK GRADE 9 STUDENT’S TEXTBOOK...

MATHEMATICS MATHEMATICS MATHEMATICS STUDENT’S TEXTBOOK GRADE 9 STUDENT’S TEXTBOOK GRADE 9 STUDENT’ S TEXTBOOK GRADE 9 FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA FDRE MINISTRY OF EDUCATION MoE MINISTRY OF EDUCATION  This textbook is the property of your school. Take good care not to damage or lose it. Here are 10 ideas to help take care of the book: Cover the book with protective material, such as plastic, old newspapers or magazines. Always keep the book in a clean dry place. Be sure your hands are clean when you use the book. Do not write on the cover or inside pages. Use a piece of paper or cardboard as a bookmark. Never tear or cut out any pictures or pages. Repair any torn pages with paste or tape. Pack the book carefully when you place it in your school bag. Handle the book with care when passing it to another person. When using a new book for the first time, lay it on its back. Open only a few pages at a time. 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This will keep the cover in good condition.  MATHEMATICS STUDENT TEXTBOOK GRADE 9 Authors: Gurju Awgichew Zergaw (PhD) Adem Mohammed Ahmed (PhD) Editors: Mohammed Yiha Dawud (PhD) (Content Editor) Akalu Chaka Mekuria (MA) (Curriculum Editor) Endalfer Melese Moges (MA) (Language Editor) Illustrator: Bahiru Chane Tamiru (MSc) Designer: Aknaw H/mariam Habte (MSc) Evaluators: Matebie Alemayehu Wasihun (MED) Mustefa Kedir Edao (BED) Dawit Ayalneh Tebkew (MSc) Tesfaye Sileshi Chala (MA) FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA HAWASSA UNIVERSITY MINISTRY OF EDUCATION First Published August 2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education, under the General Education Quality Improvement Program for Equity (GEQIP- E) supported by the World Bank, UK’s Department for International Development/DFID- now merged with the Foreign, Common wealth and Development Office/FCDO, Finland Ministry for Foreign Affairs, the Royal Norwegian Embassy, United Nations Children’s Fund/UNICEF), the Global Partnership for Education (GPE), and Danish Ministry of Foreign Affairs, through a Multi Donor Trust Fund. © 2023 by the Federal Democratic Republic of Ethiopia, Ministry of Education. All rights reserved. The moral rights of the author have been asserted. No part of this textbook reproduced, copied in a retrieval system or transmitted in any form or by any means including electronic, mechanical, magnetic, photocopying, recording or otherwise, without the prior written permission of the Ministry of Education or licensing in accordance with the Federal Democratic Republic of Ethiopia as expressed in the Federal Negarit Gazeta, Proclamation No. 410/2004 - Copyright and Neighboring Rights Protection. The Ministry of Education wishes to thank the many individuals, groups and other bodies involved – directly or indirectly – in publishing this Textbook. Special thanks are due to Hawassa University for their huge contribution in the development of this textbook in collaboration with Addis Ababa University, Bahir Dar University, Jimma University and JICA MUST project. Copyrighted materials used by permission of their owners. If you are the owner of copyrighted material not cited or improperly cited, please contact the Ministry of Education, Head Office, Arat Kilo, (P.O.Box 1367), Addis Ababa Ethiopia. Printed by: GRAVITY GROUP IND LLC P.O.Box 13TH Industrial Area, Sharjah UNITED ARAB EMIRATES Under Ministry of Education Contract no. MOE/GEQIP-E/LICB/G-01/23 ISBN: 978-99990-0-024-6 Welcoming Message to Students. Dear grade 9 students, you are welcome to the first grade of secondary level education. This is a golden stage in your academic career. Joining secondary school is a new experience and transition from primary school Mathematics education. In this stage, you are going to get new knowledge and experiences which can help you learn and advance your academic, personal, and social career in the field of Mathematics. Enjoy it! Introduction on Students’ Textbook. Dear students, this textbook has 9 units namely: Further on sets, the number system, Solving Equations, Solving Inequalities, Introduction to Trigonometry, Regular Polygons, Congruency and Similarity, Vectors in two Dimensions and Statistics and Probability respectively. Each of the units is composed of introduction, objectives, lessons, key terms, summary, and review exercises. Each unit is basically unitized, and lesson based. Structurally, each lesson has four components: Activity, Definition, Examples, and Exercises (ADEE). The most important part in this process is to practice problems by yourself based on what your teacher shows and explains. Your teacher will also give you feedback, assistance and facilitate further learning. In such a way you will be able to not only acquire new knowledge and skills but also advance them further. Basically, the four steps of each of the lessons are: Activity, Definition/Theorem/Note, Example and Exercises. Activity This part of the lesson demands you to revise what you have learnt or activate your background knowledge on the topic. The activity also introduces you what you are going to learn in new lesson topic. Definition/Theorem/Note This part presents and explains new concepts to you. However, every lesson may not begin with definition, especially when the lesson is a continuation of the previous one. Example and Solution Here, your teacher will give you specific examples to improve your understanding of the new content. In this part, you need to listen to your teacher’s explanation carefully and participate actively. Note that your teacher may not discuss all the examples in the class. In this case, you need to attempt and internalize the examples by yourself. Exercise Under this part of the material, you will solve the exercise and questions individually, in pairs or groups to practice what you learnt in the examples. When you are doing the exercise in the classroom either in pairs or groups, you are expected to share your opinions with your friends, listen to others’ ideas carefully and compare yours with others. Note that you will have the opportunity of cross checking your answers to the questions given in the class with the answers of your teacher. However, for the exercises not covered in the class, you will be given as a homework, assignment, or project. In this case, you are expected to communicate your teacher for the solutions. This symbol indicates that you need some time to remember what you have learnt before or used to enclose steps that you may be encouraged to perform mentally. This can help you connect your previous lessons with what it will come in the next discussions.  Contents Unit 1 FURTHER ON SETS….……………………… 1 1.1 Sets and Elements_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 1.2 Set Description_ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ 4 1.3 The Notion of Sets_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 7 1.4 Operations on Sets_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 12 1.5 Application_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ __ _ _ 20 Summary_ _ _ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _ __ _ 22 Review Exercise_ _ _ _ _ _ __ _ _ _ _ _ __ _ _ _ 25 Unit 2 THE NUMBER SYSTEM……………………… 27 2.1 Revision on Natural Numbers and Integers _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ 28 2.2 Rational Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 45 2.3 Irrational Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 51 2.4 Real Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 61 2.5 Application_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ 95 Summary_ _ _ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _ __ _ 99 Review Exercise_ _ _ _ _ _ __ _ _ _ _ _ __ _ _ _ 102 Unit 3 SOLVING EQUATIONS ……………………… 105 3.1 Revision on Linear Equation in One Variable_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ 106 3.2 Systems of Linear Equations in Two Variables _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 109 3.3 Solving Non-linear Equations _ _ _ _ _ _ _ _ 123 3.4 Applications of Equations _ _ _ _ _ _ _ _ _ _ 143 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 147 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ 149 Unit 4 SOLVING INEQUALITIES …………………… 151 4.1 Revision on Linear Inequalities in One Variable _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ 152 4.2 Systems of Linear Inequalities in Two Variables _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 157 4.3 Inequalities Involving Absolute Value _ _ 166 4.4 Quadratic Inequalities _ _ _ _ _ _ _ _ _ _ _ _ 171 4.5 Applications of Inequalities_ _ __ _ _ _ _ _ 176 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 182 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ 183 Unit 5 INTRODUCTION TO TRIGONOMETRY …….. 185 5.1 Revision on Right-angled Triangles _ _ _ _ 186 5.2 Trigonometric Ratios _ _ _ _ _ _ _ _ _ _ _ _ _ 190 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 202 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ 204 Unit 6 REGULAR POLYGONS ……………………… 207 6.1 Sum of Interior Angles of a Convex Polygon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ 208 6.2 Sum of Exterior Angles of a Convex Polygon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 216 6.3 Measures of Each Interior Angle and Exterior Angle of a Regular Polygon_ _ _ 221 6.4 Properties of Regular Polygons_ _ _ _ _ _ 224 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 233 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ 235 Unit 7 CONGRUENCY AND SIMILARITY ……….… 239 7.1 Revision on Congruency of Triangles _ _ 240 7.2 Definition of Similar Figures _ _ _ _ _ _ _ _ 245 7.3 Theorems on Similar Plane Figures _ _ _ 249 7.4 Ratio of Perimeters of Similar Plane Figures _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ __ _ _ 259 7.5 Ratio of Areas of Similar Plane Figures_ 262 7.6 Construction of Similar Plane Figure_ _ _ 265 7.7 Applications of Similarities_ _ _ _ _ _ _ _ _ 267 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 273 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ 274 Unit 8 VECTORS IN TWO DIMENTIONS …………… 277 8.1 Vector and Scalar Quantities _ _ _ _ _ _ _ 279 8.2 Representation of a Vector_ _ _ _ _ _ _ _ 282 8.3 Vectors Operations_ _ _ _ _ _ _ _ _ _ _ _ _ 288 8.4 Position Vector _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 299 8.5 Applications of Vectors in Two Dimensions__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 303 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 307 Review Exercise_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 308 Unit 9 Statistics and Probability …………………….. 311 9.1 Statistical Data_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 313 9.2 Probability _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 346 Summary_ _ _ _ __ _ _ _ __ _ _ _ _ _ _ _ _ _ 358 Review Exercise_ _ _ _ _ _ _ _ _ __ _ _ _ _ _ 360  Unit 1: Further on Sets UNIT FURTHER ON SETS 1 Unit Outcomes Explain facts about sets. Describe sets in different ways. Define operations on sets. Demonstrate set operations using Venn diagram. Apply rules and principles of set theory for practical situations. Unit Contents 1.1 Sets and Elements 1.2 Set Description 1.3 The Notion of Sets 1.4 Operations on Sets 1.5 Application Summary Review Exercise 1  Unit 1: Further on Sets subset symmetric difference set description Venn diagram empty set intersection absolute complement union proper subset complement set Introduction In Grade 7 you have learnt basic definition and operations involving sets. The concept of a set serves as a fundamental part of the present day mathematics. Today this concept is being used in almost every branch of mathematics. We use sets to define the concepts of relations and functions. In this unit, you will discuss some further definitions, operations and applications involving sets. Activity 1.1 1. Define a set in your own words. 2. Which of the following are well defined sets and which are not? Justify your answer. a. Collection of students in your class. b. Collection of beautiful girls in your class. c. Collection of consonants of the English alphabet. d. Collection of hardworking teachers in a school. 1.1 Sets and Elements A set is a collection of well-defined objects or elements. When we say a set is well- defined, we mean that if an object is given, we are able to determine whether the object is in the set or not. 2  Unit 1: Further on Sets Note i) Sets are usually denoted by capital letters like 𝐴, 𝐵, 𝐶, 𝑋, 𝑌, 𝑍, etc. ii) The elements of a set are represented by small letters like 𝑎, 𝑏, 𝑐, 𝑥, 𝑦, 𝑧, etc. If 𝑎 is an element of set 𝐴, we say “𝑎 belongs to 𝐴”. The Greek symbol ∈ (epsilon) is used to denote the phrase “belongs to”. Thus, we write 𝒂 ∈ 𝑨 if 𝑎 is a member of set 𝐴. If 𝑏 is not an element of set 𝐴, we write 𝒃 ∉ 𝑨 and read as “𝑏 does not belong to set 𝐴” or “𝑏 is not a member of set A”. Figure 1.1 Example 1 a. The set of students in your class is a well-defined set since the elements of the set are clearly known. b. The collection of kind students in your school. This is not a well-defined set because it is difficult to list members of the set. c. Consider 𝐺 as a set of vowel letters in English alphabet. Then 𝑎 ∈ 𝐺, 𝑜 ∈ 𝐺, 𝑖 ∈ 𝐺, but 𝑏 ∉ 𝐺. Example 2 Suppose that 𝐴 is the set of positive even numbers. Write the symbol ∈ or ∉ in the blank spaces. a. 4 _____ 𝐴 b. 5_____ 𝐴 c. −2_____ 𝐴 d. 0____ 𝐴 Solution: The positive even numbers include 2, 4, 6, 8, …. Therefore, a. 4 ∈ 𝐴 b. 5 ∉ 𝐴, c. −2 ∉ 𝐴, d. 0 ∉ 𝐴. 3  Unit 1: Further on Sets Exercise 1.1 1. Which of the following is a well-defined set? Justify your answer. a. A collection of all boys in your class. b. A collection of efficient doctors in Black Lion Hospital. c. A collection of all natural numbers less than 100. d. The collection of songs by Artist Tilahun Gessese. 2. Let 𝐴 = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: a. 5____𝐴 b. 8 ____ 𝐴 c. 0____𝐴 d. 4____ 𝐴 e. 7____𝐴 1.2 Set Description Sets can be described in the following ways. i) Verbal method (Statement form) In this method, the well-defined description of the elements of the set is written in an ordinary English language statement form (in words). Example 1 a. The set of whole numbers greater than 1 and less than 20. b. The set of students in this mathematics class. ii) Listing Methods a) Complete listing method (Roster Method) In this method, all elements of the set are completely listed. The elements are separated by commas and are enclosed within set braces, { }. Example 2 a. The set of all even positive integers less than 7 is described in complete listing method as {2, 4, 6}. b. The set of all vowel letters in the English alphabet is described in complete listing method as {𝑎, 𝑒, 𝑖, 𝑜, 𝑢 }. 4  Unit 1: Further on Sets b) Partial listing method We use this method, if listing of all elements of a set is difficult or impossible but the elements can be indicated clearly by listing a few of them that fully describe the set. Example 3 Use partial listing method to describe the following sets. a. The set of natural numbers less than 100. b. The set of whole numbers. Solution: a. The set of natural numbers less than 100 are 1, 2, 3 , … , 99. So, naming the set as 𝐴, we can express 𝐴 by partial listing method as 𝐴 = {1, 2, 3, … , 99}. The three dots after element 3 and the comma above indicate that the elements in the set continue in that manner up to 99. b. Naming the set of whole numbers by 𝕎 , we can describe it as 𝕎 = {0, 1, 2, 3, … }. So far, you have learnt three methods of describing a set. However, there are sets which cannot be described by these three methods. Here, below is another method of describing a set. iii) Set builder method (Method of defining property) The set-builder method is described by a property that its member must satisfy the common property. This is the method of writing the condition to be satisfied by a set or property of a set. In set brace, write the representative of the elements of a set, for example 𝑥, and then write the condition that 𝑥 should satisfy after the vertical line (|) or colon (:) Figure 1.2 5  Unit 1: Further on Sets Note The set of natural numbers, whole numbers, and integers are denoted by ℕ, 𝕎, and ℤ, respectively. They are defined as ℕ = {1, 2, 3,... } 𝕎 = {0, 1, 2, 3,... }, ℤ = {.... −3, −2, −1, 0 , 1, 2, 3,... }. Example 4 Describe the following sets using set builder method. i) Set 𝐴 = {1, 2, 3 … 10} can be described in set builder method as: 𝐴 = {𝑥 | 𝑥 ∈ ℕ and 𝑥 < 11}. We read this as “𝐴 is the set of all elements of natural numbers less than 11.” ii) Let set 𝐵 = {0, 2, 4, …. }. This can be described in set builder method as: 𝐵 = {𝑥 | 𝑥 ∈ ℤ and 𝑥 is a non-negative even integer} or 𝐵 = {2𝑥 | 𝑥 = 0, 1, 2, 3, … } 𝑜𝑟 𝐵 = {2𝑥 | 𝑥 ∈ 𝕎}. Exercise 1.2 1. Describe each of the following sets using a verbal method. a. 𝐴 = { 5, 6, 7, 8, 9} b. 𝑀 = {2, 3, 5, 7, 11, 13} c. 𝐺 = {8, 9, 10, …. } d. 𝐸 = {1, 3, 5, … , 99} 2. Describe each of the following sets using complete and partial listing method (if possible): a. The set of positive even natural numbers below or equal to 10. b. The set of positive even natural numbers below or equal to 30. c. The set of non-negative integers. d. The set of even natural numbers. e. The set of natural numbers less than 100 and divisible by 5. f. The set of integers divisible by 3. 3. List the elements of the following sets: 6  Unit 1: Further on Sets a. 𝐴 = {3𝑥 | 𝑥 ∈ 𝕎} b. 𝐵 = {𝑥 | 𝑥 ∈ ℕ and 5 < 𝑥 < 10} 4. Write the following sets using set builder method. a. 𝐴 = {1, 3, 5 …. } b. 𝐵 = {2, 4, 6, 8} c. 𝐶 = {1, 4, 9, 16, 25} d. 𝐷 = {4, 6, 8, 10, … , 52 } e. 𝐸 = {−10,... , −3, −2, −1, 0, 1, 2, … , 5} f. 𝐹 = {1, 4, 9, …. } 1.3 The Notion of Sets Empty set, Finite set and Infinite set Empty Set A set which does not contain any element is called an empty set, void set or null set. The empty set is denoted mathematically by the symbol { } or Ø. Example 1 Let set 𝐴 = {𝑥 | 1 < 𝑥 < 2, 𝑥 ∈ ℕ}. Then, 𝐴 is an empty set, because there is no natural number between numbers 1 and 2. Finite set and Infinite set Definition 1.1 A set which consists of a definite number of elements is called a finite set. A set which is not finite is called an infinite set. Example 2 Identify the following sets as finite set or infinite set. a. The set of natural numbers up to 10 b. The set of African countries c. The set of whole numbers Solution: a. Let 𝐴 be a set and 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. 7  Unit 1: Further on Sets Thus, it is a finite set because it has definite (limited) number of elements. b. The set of African countries is a finite set. c. The set of whole numbers is an infinite set. Note The number of elements of set 𝐴 is denoted by 𝑛ሺ𝐴ሻ. For instance, in the above example 2𝑎, 𝑛ሺ𝐴ሻ = 10. Read 𝑛ሺ𝐴ሻ as number of elements of set 𝐴 Exercise 1.3 1. Identify empty set from the list below. a. 𝐴 = {𝑥 | 𝑥 ∈ ℕ and 5 < 𝑥 < 6} b. 𝐵 = {0} c. C is the set of odd natural numbers divisible by 2. d. 𝐷 = { } 2. Sort the following sets as finite or infinite sets. a. The set of all integers b. The set of days in a week c. 𝐴 = {𝑥 ∶ 𝑥 is a multiple of 5} d. 𝐵 = {𝑥 ∶ 𝑥 ∈ 𝑍, 𝑥 < −1} e. 𝐷 = {𝑥 ∶ 𝑥 is a prime number} Equal Sets, Equivalent Sets, Universal Set, Subset and Proper Subset Equal Sets Definition 1.2 Two sets 𝐴 and 𝐵 are said to be equal if and only if they have exactly the same or identical elements. Mathematically, it is denoted as 𝐴 = 𝐵. 8  Unit 1: Further on Sets Example 1 Let 𝐴 = {1, 2, 3, 4} and 𝐵 = {4, 3, 2, 1}. Then, 𝐴 = 𝐵. Set 𝐴 and set 𝐵 are equal. Equivalent Sets Definition 1.3 Two sets 𝐴 and 𝐵 are said to be equivalent if there is a one-to-one correspondence between the two sets. This is written mathematically as 𝐴 ↔ 𝐵 (or 𝐴~𝐵). Note Observe that two finite sets 𝐴 and 𝐵 are equivalent, if and only if they have equal number of elements and we write mathematically this as 𝑛 ሺ𝐴ሻ = 𝑛 ሺ𝐵ሻ. Example 2 Consider two sets 𝐴 = {1, 2, 3, 4} and 𝐵 = {Red, Blue, Green, Black}. In set 𝐴 there are four elements and in set 𝐵 also there are four elements. Therefore, set 𝐴 and set 𝐵 are equivalent. Universal Set (∪) Definition 1.4 A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. Example 3 Let set 𝐴 = {2, 4, 6,... } and 𝐵 = {1, 3, 5,... }. The universal set U consists of all natural numbers, such that 𝑈 = {1, 2, 3, 4,... }. Therefore, as we know all even and odd numbers are part of natural numbers. Hence, set U has all the elements of set A and set B. 9  Unit 1: Further on Sets Subset (⊆) Definition 1.5 Set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵. Figure 1.3 shows this relationship. Mathematically, we write this as 𝑨 ⊆ 𝑩. If set 𝐴 is not a subset of set 𝐵, then it is written as 𝐴 ⊈ 𝐵. Figure 1.3 Example 4 Let 𝐴 = {1, 2, 3} and 𝐵 = {1, 2, 3, 4} be sets. Here, set 𝐴 is a subset of set 𝐵, or 𝐴 ⊆ 𝐵, since all members of set 𝐴 are found in set 𝐵. In the above set 𝐴, find all subsets of the set. How many subsets does set 𝐴 have? Solution: The subsets of set 𝐴 are {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }. The number of subsets of set 𝐴 is 8. Note i) 𝐴ny set is a subset of itself. ii) Empty set is a subset of every set. iii) If set 𝐴 is finite with 𝑛 elements, then the number of subsets of set 𝐴 is 2𝑛. In the above Example 3.b, 𝑛ሺ𝐴ሻ = 3. Then, the number of subsets is 23 = 8. Proper Subset (⊂) Definition 1.6 If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called the proper subset of set 𝐵 and it can be written as 𝐴 ⊂ 𝐵. 10  Unit 1: Further on Sets Example 5 Given that sets 𝐴 = {2, 5, 7} and 𝐵 = {2, 5, 7, 8}. Set 𝐴 is a proper subset of set 𝐵, that is, 𝐴 ⊂ 𝐵 since 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵. Observe also that 𝐵 ⊄ A. In the above set 𝐴, find all the proper subsets. How many proper subsets does set 𝐴 have? Solution: The proper subsets of set 𝐴 are {2}, {5}, {7}, {2, 5}, {5, 7}, {2, 7}, { }. There are seven subsets. Note i) For any set 𝐴, 𝐴 is not a proper subset of itself. ii) The number of proper subsets of set 𝐴 is 2𝑛 − 1. iii) Empty set is the proper subset of any other sets. iv) If set 𝐴 is subset of set 𝐵 ሺ𝐴 ⊆ 𝐵ሻ, conversely 𝐵 is super set of 𝐴 written as 𝐵 ⊃ 𝐴. Exercise 1.4 1. Identify equal sets, equivalent sets or which are neither equal nor equivalent. a. 𝐴 = {1, 2, 3} and 𝐵 = {4, 5} b. 𝐶 = {𝑞, 𝑠, 𝑚} and 𝐷 = {6, 9, 12} c. 𝐸 = {3, 7, 9, 11} and 𝐹 = {3, 9, 7, 11} d. 𝐺 = {𝐼, 𝐽, 𝐾, 𝐿} and 𝐻 = {𝐽, 𝐾, 𝐼, 𝐿} e. 𝐼 = {𝑥 | 𝑥 ∈ 𝕎, 𝑥 < 5} and 𝐽 = {𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 5} f. 𝐾 = {𝑥 | 𝑥 is a multiple of 30} and 𝐿 = {𝑥 | 𝑥 is a factor of 10} 2. List all the subsets of set 𝐻 = {1, 3, 5}. How many subsets and how many proper subsets does it have? 3. Determine whether the following statements are true or false. a. {𝑎, 𝑏} ⊄ {𝑏, 𝑐, 𝑎} b. {𝑎, 𝑒} ⊆ {𝑥 | 𝑥 is a vowel in the English alphabet} c. {𝑎} ⊂ { 𝑎, 𝑏, 𝑐 } 11  Unit 1: Further on Sets 4. Express the relationship of the following sets, using the symbols ⊂, ⊃, or = a. 𝐴 = {1, 2, 5, 10} and 𝐵 = {1, 2, 4, 5, 10, 20} b. 𝐶 = {𝑥 |𝑥 is natural number less than 10} and 𝐷 = {1, 2, 4, 8} c. 𝐸 = {1, 2} and 𝐹 = {𝑥 | 0 < 𝑥 < 3, 𝑥 ∈ ℤ} 5. Consider sets 𝐴 = {2, 4, 6}, 𝐵 = {1, 3 7, 9, 11} and 𝐶 = {4, 8, 11}, then a. Find the universal set b. Relate sets 𝐴, 𝐵, 𝐶 and 𝑈 using subset. 1.4 Operations on Sets There are several ways to create new sets from sets that have already been defined. Such process of forming new set is called set operation. The three most important set operations namely Unionሺ∪ሻ, Intersectionሺ∩ሻ, Complementሺ′ሻ and Difference ሺ−ሻ are discussed below. Union and Intersection Activity 1.2 Let the universal set is the set of natural numbers ℕ which is less than 12, and sets 𝐴 = {1, 2, 3, 4, 5, 6} and 𝐵 = {1, 3, 5, 7, 9}. a. Can you write a set consisting of all natural numbers that are in 𝐴 or in 𝐵? b. Can you write a set consisting of all natural numbers that are in 𝐴 and in 𝐵? c. Can you write a set consisting of all natural numbers that are in 𝐴 and not in 𝐵? Figure 1.4 Venn diagrams A Venn diagram is a schematic or pictorial representation of the sets involved in the discussion. Usually sets are represented as interlocking circles, each of which is 12  Unit 1: Further on Sets enclosed in a rectangle, which represents the universal set. Figure 1.4 above is an example of Venn diagram. Definition 1.7 The union of two sets 𝐴 and 𝐵, which is denoted by 𝑨 ∪ 𝑩, is the set of all elements that are either in set 𝐴 or in set 𝐵 (or in both sets). We write this mathematically as 𝐴 ∪ 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵}. Figure 1.5 Definition 1.8 The intersection of two sets 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵, is the set of all elements that are both in set 𝐴 and in set 𝐵. We write this mathematically as 𝐴 ∩ 𝐵 ={𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵} Figure 1.6 Note Two sets 𝐴 and 𝐵 are disjoint if 𝐴 ∩ 𝐵 = ∅ Figure 1.7 Example 1 Let 𝐴 = {0, 1, 3, 5, 7} and 𝐵 = {1, 2, 3, 4, 6, 7} be sets. Draw the Venn diagram and find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. 13  Unit 1: Further on Sets Solution: Figure 1.8 shows Venn diagram of set 𝐴 and 𝐵. Thus, 𝐴 ∪ 𝐵 = {0, 1, 2, 3, 4, 5, 6, 7} and 𝐴 ∩ 𝐵 = {1, 3, 7}. Figure 1.8 Example 2 Let 𝐴 = {2, 4, 6, 8, 10, … } and 𝐵 = {3, 6, 9, 12, 15, … } be sets. Then, find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. Solution: 𝐴 ∪ 𝐵 = {𝑥 | 𝑥 is a positive integer that is either even or a multiple of 3} = {2, 3, 6, 9, 12, 15,.... } 𝐴 ∩ 𝐵 = {𝑥 | 𝑥 is a positive integer that is both even and a multiple of 3} = {6, 12, 18, 24,.... } Note i) Law of ∅ and 𝑈: ∅ ∩ 𝐴 = ∅, 𝑈 ∩ 𝐴 = 𝐴. ii) Commutative law: 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴. iii) Associative Law: ሺ𝐴 ∩ 𝐵ሻ ∩ 𝐶 = 𝐴 ∩ ሺ𝐵 ∩ 𝐶ሻ. Exercise 1.5 1. Let 𝐴 = {0, 2, 4, 6, 8} and 𝐵 = {0, 1, 2, 3, 5, 7, 9}. Draw the Venn Diagram and find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. 2. Let 𝐴 be the set of positive odd integers less than 10 and 𝐵 is the set of positive multiples of 5 less than or equal to 20. Find a) 𝐴 ∪ 𝐵, b) 𝐴 ∩ 𝐵. 3. Let 𝐶 = {𝑥 | 𝑥 is a factors of 20}, 𝐷 = {𝑦 | 𝑦 is a factor of 12}. Find a) 𝐶 ∪ 𝐷, b) 𝐶 ∩ 𝐷. 14  Unit 1: Further on Sets Complement of sets Definition 1.9 Let 𝐴 be a subset of a universal set 𝑈. The absolute complement (or simply complement) of 𝐴, which is denoted by 𝑨′ , is defined as the set of all elements of 𝑈 that are not in 𝐴. We write this mathematically as 𝐴′ = {𝑥: 𝑥 ∈ 𝑈 and 𝑥 ∉ 𝐴}. Figure 1.9 Example 1 a. Let 𝑈 = {0, 1, 2, 3, 4} and 𝐴 = {3, 4}. Then, 𝐴′ = {0, 1, 2}. Example 2 Let 𝑈 = {1, 2, 3, … , 10} be a universal set, 𝐴 = {𝑥 | 𝑥 is a positive factor of 10 in 𝑈} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈} be sets. a. Find 𝐴′ and 𝐵′. b. Find ሺ𝐴 ∪ 𝐵ሻ′ and 𝐴′ ∩ 𝐵′. What do you observe from the answers? Solution: a. 𝐴 = {1, 2, 5, 10}, 𝐵 = {1, 3, 5, 7, 9}, Thus, 𝐴′ = {3, 4, 6, 7, 8, 9}, 𝐵 ′ = {2, 4, 6, 8, 10}. b. First, we find 𝐴 ∪ 𝐵. Hence, 𝐴 ∪ 𝐵 = {1, 2, 3, 5, 7, 9,10} and ሺ𝐴 ∪ 𝐵ሻ′ = {4, 6, 8}. On the other hand, from 𝐴′ and 𝐵 ′ , we obtain 𝐴′ ∩ 𝐵′ = {4, 6, 8}. Hence, we immediately observe ሺ𝐴 ∪ 𝐵ሻ′ = 𝐴′ ∩ 𝐵′. In general, for any two sets 𝐴 and 𝐵 , ሺ𝐴 ∪ 𝐵ሻ′ = 𝐴′ ∩ 𝐵′. It is called the first statement of De Morgan’s law. 15  Unit 1: Further on Sets De Morgan’s Law For the complement set of 𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵, 1st statement: (𝐴 ⋃ 𝐵ሻ′ = 𝐴′ ⋂ 𝐵 ′ , 2nd statement: (𝐴 ⋂ 𝐵ሻ′ = 𝐴′ ⋃ 𝐵 ′. Figure 1.10 Figure 1.11 Exercise 1.6 1. If the universal set 𝑈 = {0, 1, 2, 3, 4, 5}, and 𝐴 = {4, 5}, then find 𝐴′. 2. Let the universal set 𝑈 = {1, 2, 3, … , 20}, 𝐴 = {𝑥 |𝑥 is a positive factor of 20} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈}. Find 𝐴′ , 𝐵 ′ , ሺ𝐴 ∪ 𝐵ሻ′ and 𝐴′ ∩ 𝐵 ′. 3. Let the universal set be 𝑈 = {𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 10}. When 𝐴 = {2, 5, 9}, and 𝐵 = {1, 5, 6, 8}, find a) 𝐴′ ⋂ 𝐵′ and b) 𝐴′ ⋃ 𝐵′. Difference of sets Definition 1.10 The difference between two sets 𝐴 and 𝐵, which is denoted by 𝐴 − 𝐵 , is the of all elements in 𝐴 and not in 𝐵; this set is also called the relative complement of 𝐴 with respect to 𝐵. We write this mathematically as 𝐴 − 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐵}. Figure 1.12 Note The notation 𝐴 − 𝐵 can be also written as 𝐴\𝐵. 16  Unit 1: Further on Sets Example 1 If sets 𝐴 = {0, 1, 2, 3, 4} and 𝐵 = {3, 4}, then 𝐴 − 𝐵 or 𝐴\𝐵 = {0, 1, 2}. Example 2 Let 𝑈 be a universal set of the set of one-digit numbers, 𝐴 be the set of even numbers, 𝐵 be the set of prime numbers less than 10. Find the following: a. 𝐴 − 𝐵 or 𝐴\𝐵 b. 𝐵 − 𝐴 or 𝐵\𝐴 c. 𝐴∪𝐵 d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ or 𝑈\ሺ𝐴 ∪ 𝐵ሻ Solution: Here, 𝐴 = {0, 2, 4, 6, 8}, 𝐵 = {2, 3, 5, 7}. Then, we illustrate the sets using a Venn diagram as follows. From the Venn diagram we observe: a. 𝐴 − 𝐵 = {0, 4, 6, 8} b. 𝐵 − 𝐴 = {3 ,5, 7} c. 𝐴 ∪ 𝐵 = {0, 2, 3, 4, 5, 6, 7, 8} d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ = {1, 9} Figure 1.13 Example 3 For the same sets in Example 2, find the following. What can you say from Example 2 a. and b.? What about d. and Example 2, a.? a. 𝐴′ b. 𝑈 − 𝐴 c. 𝐵′ d. 𝐴 ∩ 𝐵′ Solution: a. 𝐴′ = {1, 3, 5, 7, 9} b. 𝑈 − 𝐴 = {1, 3, 5, 7, 9} c. 𝐵′ = {0, 1, 4, 6, 8, 9} d. 𝐴 ∩ 𝐵′ = {0, 4, 6, 8} From a. and b., we can say, 𝐴′ = 𝑈 − 𝐴. From d. and Example 2, a., we can say, 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′. 17  Unit 1: Further on Sets Theorem 1.1 For any two sets 𝐴 and 𝐵, each of the following holds true. ሺ𝐴′ ሻ′ = 𝐴 𝐴′ = 𝑈 − 𝐴 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′ 𝐴 ⊆ 𝐵 ⟺ 𝐵 ′ ⊆ 𝐴′ Exercise 1.7 From the given Venn diagram, find each of the following: a. 𝐴 − 𝐵 or 𝐴\𝐵 b. 𝐵 − 𝐴 or 𝐵\𝐴 c. 𝐴 ∪ 𝐵 d. 𝑈 − ሺ𝐴 ∪ 𝐵ሻ or 𝑈\ሺ𝐴 ∪ 𝐵ሻ Symmetric Difference of Two Sets Definition 1.11 Symmetric Difference For two sets A and B, the symmetric difference between these two sets is denoted by 𝐴∆𝐵 and is defined as: 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ , which is ሺ𝐴 − 𝐵ሻ ∪ ሺ𝐵 − 𝐴ሻ or = ሺ𝐴 ∪ 𝐵ሻ\ሺ𝐴 ∩ 𝐵ሻ In the Venn diagram, the shaded part represents 𝐴∆𝐵 Figure 1.14 Example 1 Consider sets 𝐴 = {1, 2, 4, 5, 8} and 𝐵 = {2, 3, 5, 7}. Then, find 𝐴∆𝐵. Solution: First, let us find 𝐴\𝐵 = {1, 4, 8} and 𝐵\𝐴 = {3, 7}. Hence, 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ Figure 1.15 18  Unit 1: Further on Sets = {1, 4, 8} ∪ {3, 7} = {1, 3, 4, 7, 8}. Or 𝐴∆𝐵 = ሺ𝐴 ∪ 𝐵ሻ\ሺ𝐴 ∩ 𝐵ሻ = {1, 3, 4, 7, 8}. Example 2 Given sets 𝐴 = {𝑑, 𝑒, 𝑓} and 𝐵 = {4, 5, 6}. Then, find 𝐴∆𝐵. Solution: First, we find 𝐴\𝐵 = {𝑑, 𝑒, 𝑓} and 𝐵\𝐴 = {4,5,6}. Hence, 𝐴∆𝐵 = ሺ𝐴\𝐵ሻ ∪ ሺ𝐵\𝐴ሻ = {𝑑, 𝑒, 𝑓} ∪ {4, 5, 6} = {𝑑, 𝑒, 𝑓, 4, 5, 6}. Figure 1.16 Exercise 1.8 1. Given 𝐴 = {0, 2, 3, 4, 5} and 𝐵 = {0, 1, 2, 3, 5, 7, 9}. Then, find 𝐴∆𝐵. 2. If 𝐴∆𝐵 = ∅, then what can be said about the two sets? 3. For any two sets 𝐴 and 𝐵, can we generalize 𝐴∆𝐵 = 𝐵∆𝐴 ? Justify your answer. Cartesian Product of Two Sets Definition 1.12 Cartesian Product of Two Sets The Cartesian product of two sets 𝐴 and 𝐵, denoted by 𝐴 × 𝐵, is the set of all ordered pairs ሺ𝑎, 𝑏ሻ where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. This also can be expressed as 𝐴 × 𝐵 = {ሺ𝑎, 𝑏ሻ: 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵}. Example 1 Let 𝐴 = {1, 2} and 𝐵 = {𝑎, 𝑏}. Then, find a) 𝐴 × 𝐵 b) 𝐵 × 𝐴 Solution: a. 𝐴 × 𝐵 = {ሺ1, 𝑎ሻ, ሺ1, 𝑏ሻ, ሺ2, 𝑎ሻ, ሺ2, 𝑏ሻ} b. 𝐵 × 𝐴 = {ሺ𝑎, 1ሻ, ሺ𝑎, 2ሻ, ሺ𝑏, 1ሻ, ሺ𝑏, 2ሻ} 19  Unit 1: Further on Sets Example 2 If 𝐴 × 𝐵 = {ሺ1, 𝑎ሻ, ሺ1, 𝑏ሻ, ሺ2, 𝑎ሻ, ሺ2, 𝑏ሻ, ሺ3, 𝑎ሻ, ሺ3, 𝑏ሻ}, then find sets 𝐴 and 𝐵. Solution: 𝐴 is the set of all first components of 𝐴 × 𝐵, that is, 𝐴 = {1, 2, 3}, and 𝐵 is the set of all second components of 𝐴 × 𝐵, that is, 𝐵 = {𝑎, 𝑏}. Exercise 1.9 1. Let 𝐴 = {1, 2, 3} and 𝐵 = {𝑒, 𝑓}. Then, find a) 𝐴 × 𝐵 b) 𝐵 × 𝐴. 2. If 𝐴 × 𝐵 = {ሺ7,6ሻ, ሺ7,4ሻ, ሺ5,4ሻ, ሺ5,6ሻ, ሺ1,4ሻ, ሺ1,6ሻ}, then find sets 𝐴 and 𝐵. 3. If 𝐴 = {𝑎, 𝑏, 𝑐} , 𝐵 = {1, 2, 3} and 𝐶 = {3, 4}, then find 𝐴 × ሺ𝐵 ∪ 𝐶ሻ. 4. If 𝐴 = {6, 9, 11}, then find 𝐴 × 𝐴. 5. If the number of elements of set 𝐴 is 6 and the number of elements of set 𝐵 is 4, then the number of elements of 𝐴 × 𝐵 is _____________________. 1.5 Application Number of Elements of union of two sets For the two subsets 𝐴 and 𝐵 of a universal set 𝑈, the following formula on the number of elements holds. That is 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ. Especially in the case 𝐴 ∩ 𝐵 = ∅, thus, Figure 1.17 𝑛ሺ𝐴 ∩ 𝐵ሻ = 0, and the following holds: 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ. If 𝐴 ∩ 𝐵 ≠ ∅, then 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ. Figure 1.18 20  Unit 1: Further on Sets Example Let 𝐴 and 𝐵 be two finite sets such that 𝑛ሺ𝐴ሻ = 20, 𝑛ሺ𝐵ሻ = 28, and 𝑛ሺ𝐴 ∪ 𝐵ሻ = 36, then find 𝑛ሺ𝐴 ∩ 𝐵ሻ. Solution: Using the formula 𝑛ሺ𝐴 ∪ 𝐵ሻ = 𝑛ሺ𝐴ሻ + 𝑛ሺ𝐵ሻ − 𝑛ሺ𝐴 ∩ 𝐵ሻ we have 36 = 20 + 28 − 𝑛ሺ𝐴 ∩ 𝐵ሻ. This gives 𝑛ሺ𝐴 ∩ 𝐵ሻ = ሺ20 + 28ሻ − 36 = 48 − 36 = 12. Exercise 1.10 1. Let 𝐴 and 𝐵 be two finite sets such that 𝑛ሺ𝐴ሻ = 34, 𝑛ሺ𝐵ሻ = 46 and 𝑛ሺ𝐴 ∪ 𝐵ሻ = 70. Then, find 𝑛ሺ𝐴 ∩ 𝐵ሻ. 2. There are 60 people attending a meeting. 42 of them drink tea and 27 drink coffee. If every person in the meeting drinks at least one of the two drinks, find the number of people who drink both tea and coffee. (Hint: Use a Venn diagram). 21  Summary and Review Exercise Summary 1. A set is a collection of well-defined objects or elements. When we say a set is well-defined, we mean that given an object we are able to determine whether the object is in the set or not. 2. Sets can be described in the following ways: Verbal method (Statement form) In this method, the well-defined description of the elements of the set is written in an ordinary English language statement form (in words). Complete listing method (Roster Method) In this method all the elements of the sets are completely listed. The elements are separated by commas and are enclosed within set brace, { }. Partial listing method We use this method, if listing of all elements of a set is difficult or impossible but the elements can be indicated clearly by listing a few of them that fully describe the set. Set builder method (Method of defining property) The set-builder method is described by a property that its member must satisfy. This is the method of writing the condition to be satisfied by a set or property of a set. 3. A set which does not contain any element is called an empty set, void set or null set. The empty set is denoted mathematically by the symbol { } or Ø. 4. Two sets A and B are said to be equal if and only if they have exactly same or identical elements. Mathematically, we write this as 𝐴 = 𝐵. 5. Set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵. Mathematically, we write this as 𝐴 ⊆ 𝐵. Any set is a subset to itself. Empty set is a sub set of every set. 22  Summary and Review Exercise If set 𝐴 is finite with 𝑛 elements, then the number of subsets of set 𝐴 is 2𝑛. 6. If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called the proper subset of 𝐵 and it can be written as 𝐴 ⊂ 𝐵. For any set 𝐴, 𝐴 is not a proper subset to itself. If set 𝐴 is finite with 𝑛 elements the number of proper subsets of set 𝐴 is 2𝑛 − 1. Empty set is a proper subset of any other sets. 7. A universal set (usually denoted by 𝑈) is a set which has elements of all the related sets, without any repetition of elements. 8. The union of two sets 𝐴 and 𝐵, which is denoted by 𝐴 ∪ 𝐵, is the set of all elements that are either in set 𝐴 or in set B (or in both sets). We write this mathematically as 𝐴 ∪ 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵}. 9. The intersection of two sets 𝐴 and 𝐵, which is denoted by 𝐴 ∩ 𝐵, is the set of all elements that are in set 𝐴 and in set 𝐵. We write this mathematically as 𝐴 ∩ 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵}. 10. The difference between two sets 𝐴 and 𝐵, which is denoted by 𝐴 − 𝐵, is the set of all elements in set 𝐴 and not in set 𝐵; this set is also called the relative complement of set 𝐴 with respect to set 𝐵. We write this mathematically as 𝐴 − 𝐵 = 𝐴\𝐵 = {𝑥 | 𝑥 ∈ 𝐴 and 𝑥 ∉ 𝐵}. 11. Let 𝐴 be subset of a universal set 𝑈. The absolute complement (or simply complement) of set 𝐴, which is denoted by 𝐴′, is defined as the set of all elements of 𝑈 that are not in 𝐴. We write this mathematically as 𝐴′ = {𝑥 | 𝑥 ∈ 𝑈 and 𝑥 ∉ 𝐴}. 12. A Venn diagram is a schematic or pictorial representation of the sets involved in the discussion. Usually sets are represented as interlocking circles, each of which is enclosed in a rectangle, which represents the universal set. 23  Summary and Review Exercise 13. For two sets 𝐴 and 𝐵 the symmetric difference between these two sets is denoted by 𝐴∆𝐵 and is defined as 𝐴∆𝐵 = (𝐴\𝐵) ∪ (𝐵\𝐴) = 𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵). 14. For any two finite sets 𝐴 and 𝐵, 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵). Review Exercise 1. Express following sets using the listing method. a. 𝐴 is the set of positive factors of 18 b. 𝐵 is the set of positive even numbers below or equal to 30 c. 𝐶 = {2𝑛 | 𝑛 = 0, 1, 2, 3, … } d. 𝐷 = {𝑥 | 𝑥 2 = 9} 2. Express following sets using the set-builder method. a. {2, 4, 6, ….. } b. {1, 3, 5, … , 99} c. {1, 4, 9, … , 81} 3. Find all the subsets of the following sets. a. {3, 4, 5} b. {𝑎, 𝑏} 4. Find 𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵 of the following. a. 𝐴 = {2, 3, 5, 7, 11} and 𝐵 = {1, 3, 5, 8, 11} b. 𝐴 = {𝑥 | 𝑥 is the factor of 12} and 𝐵 = {𝑥 | 𝑥 is the factor of 18} c. 𝐴 = {3𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 20 } and 𝐵 = {4𝑥 | 𝑥 ∈ ℕ, 𝑥 ≤ 15} 5. If 𝐵 ⊆ 𝐴, 𝐴 ∩ 𝐵 ′ = {1, 4, 5}, and 𝐴 ∪ 𝐵 = {1, 2, 3, 4, 5, 6}, then find set 𝐵. 6. Let 𝐴 = {2, 4, 6, 7, 8, 9}, 𝐵 = {1, 3, 5, 6, 10} and 𝐶 = {𝑥 | ∈ ℤ, 3𝑥 + 6 = 0 or 2𝑥 + 6 = 0}. Find a) 𝐴 ∪ 𝐵 b) Is (𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶)? 24  Summary and Review Exercise 7. Suppose the universal set 𝑈 be the set of one-digit numbers, and set 𝐴 = {𝑥 | 𝑥 is an even natural number less than or equal to 9 }. Describe each set by complete listing method: a. 𝐴′ b. 𝐴 ∩ 𝐴′ c. 𝐴 ∪ 𝐴′ d. (𝐴′)′ e. 𝜙 \ 𝑈 f. 𝜙′ g. 𝑈′ 8. Let 𝑈 = {1, 2, 3, 4, … …. ,10}, 𝐴 = {1, 3, 5, 7}, 𝐵 = {1, 2, 3, 4}. Evaluate 𝑈\(𝐴∆𝐵). 9. Consider a universal set 𝑈 = {1, 2, 3,... , 14}, 𝐴 = {2, 3, 5, 7, 11}, 𝐵 = {2, 4, 8, 9, 10, 11}. Then, which one of the following is true? A. (𝐴 ∪ 𝐵)′ = {1, 4, 6, 12, 13, 14} B. 𝐴 ∩ 𝐵 = 𝐴′ ∪ 𝐵 ′ C. 𝐴∆𝐵 = (𝐴 ∩ 𝐵)′ D. 𝐴\𝐵 = {3, 5, 7} E. None 10. Let 𝐴 = {3, 7, 𝑎2 } and 𝐵 = {2, 4, 𝑎 + 1, 𝑎 + 𝑏} be two sets and all the elements of the two sets are integers. If 𝐴 ∩ 𝐵 = {4, 7}, then find a and b. In addition, find 𝐴 ⋃ 𝐵. 11. In a survey of 200 students in Motta Secondary School, 90 students are members of Nature club, 31 students are members of Mini-media club, 21 students are members of both clubs. Answer the following questions. a. How many students are members of either of the clubs? b. How many students are not members of either of the clubs? 25  Summary and Review Exercise c. How many students are only in Nature club? 12. A survey was conducted in a class of 100 children and it was found out that 45 of them like Mathematics whereas only 35 like Science and 10 students like both subjects. How many like neither of the subjects? A) 70 B) 30 C) 100 D) 40 26  Unit 2: The Number System UNIT THE NUMBER SYSTEM 2 Unit Outcomes Describe rational numbers. Locate rational numbers on number line. Describe irrational numbers. Locate some irrational numbers on a number line. Define real numbers. Classify real numbers as rational and irrational. Solve mathematical problems involving real numbers. Unit Contents 2.1 Revision on Natural Numbers and Integers 2.2 Rational Numbers 2.3 Irrational Numbers 2.4 Real Numbers 2.5 Applications Summary Review Exercise 27  Unit 2: The Number System perfect square significant digits greatest common significant figures factor division algorithm principal nth root terminating decimal fundamental theorem of prime factorization least common arithmetic multiple (LCM) rationalizing factor repeating decimal real number irrational number scientific notation radicand rationalization bar notation composite number Introduction In the previous grades, you learned number systems about natural numbers, integers and rational numbers. You have discussed meaning of natural numbers, integers and rational numbers, the basic properties and operations on the above number systems. In this unit, after revising those properties of natural numbers, integers and rational numbers, you will continue to learn about irrational and real numbers. 2.1 Revision on Natural Numbers and Integers Activity 2.1 1. List five members of :- a. Natural numbers b. Integers 2. Select natural numbers and integers from the following. a. 6 b. 0 c. −25 3. What is the relationship between natural numbers and integers? 28  Unit 2: The Number System 4. Decide if the following statements are always true, sometimes true or never true and provide your justification. a. Natural numbers are integers b. Integers are natural numbers c. -7 is a natural number 5. Draw diagram which shows the relationship of Natural numbers and Integers. The collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word ‘distinct’ means that the objects of a set must be all different. From your grade 7 mathematics lessons, you recall that The set of natural numbers, which is denoted by ℕ expressed as ℕ = {1, 2, 3, … }. The set of integers, which is denoted by ℤ is expressed as ℤ = {… , −2, −1, 0, 1, 2, 3, … }. Example Categorize each of the following as natural numbers and integers. 3, −2, 11, 0, −18, 15, 7 Solution: All the given numbers are integers and 3, 11, 15 and 7 are natural numbers. Exercise 2.1 1. Categorize each of the following as natural numbers and integers. 8, −11 , 23, 534, 0, −46, −19, 100 2. What is the last integer before one thousand? 29  Unit 2: The Number System 3. Consider any two natural numbers 𝑛1 and 𝑛2. a. Is 𝑛1 + 𝑛2 a natural number? Explain using example. b. Is 𝑛1 − 𝑛2 a natural number? Explain using example. c. What can you conclude from (a) and (b)? 4. If the perimeter of a triangle is 10 and lengths of the sides are natural numbers, find all the possible lengths of sides of the triangle. 5. Assume 𝑚 and 𝑛 are two positive integers and 𝑚 + 𝑛 < 10. How many different values can the product 𝑚𝑛 (𝑚 multiplied by 𝑛) have? 2.1.1 Euclid’s Division lemma Activity 2.2 1. In a book store there are 115 different books to be distributed to 8 students. If the book store shares these books equally, how many books will each student receive and how many books will be left? 2. Divide a natural number 128 by 6. What is the quotient and remainder of this process? Can you guess a remainder before performing the division process? From your activity, using the process of dividing one positive integer by another, you will get remainder and quotient as described in the following theorem. Theorem 2.1 Euclid’s Division lemma Given a non-negative integer 𝑎 and a positive integer 𝑏, there exist unique non- negative integers 𝑞 and 𝑟 satisfying 𝑎 = 𝑏 × 𝑞 + 𝑟 with 0 ≤ 𝑟 < 𝑏. In theorem 2.1, 𝑎 is called the dividend, 𝑞 is called the quotient, 𝑏 is called the divisor, and 𝑟 is called the remainder. 30  Unit 2: The Number System Example Find the unique quotient and remainder when a positive integer. a. 38 is divided by 4 b. 5 is divided by 14 c. 12 is divided by 3 d. 2,574 is divided by 8 Solution: a. Here, it is given that the dividend is 𝑎 = 38 and the divisor is = 4. So that we need to determine the unique numbers 𝑞 and 𝑟.When we divide 38 by 4, we get a quotient 𝑞 = 9 and a remainder 𝑟 = 2. Hence, we can write this as 38 = 4 × 9 + 2. b. The number 5 is less than the divisor 14. So, the quotient is 0 and the remainder is 5. That is, 5 = 14 × 0 + 5. c. When we divide 12 by 3, we obtain 4 as a quotient and the remainder is 0. That is 12 = 3 × 4 + 0. d. Using long division if 2,574 is divided by 8, we get 321 as a quotient and 6 as a remainder. We can write 2,574 = 8 × 321 + 6. 31  Unit 2: The Number System Note For two positive integers 𝑎 and 𝑏 in the division lemma, we say 𝑎 is divisible by 𝑏 if the remainder 𝑟 is zero. In the above example (c), 12 is divisible by 3 since the remainder is 0. Exercise 2.2 1. For each of the following pairs of numbers, let 𝑎 be the first number of the pair and 𝑏 be the second number. Find 𝑞 and 𝑟 for each pair such that 𝑎 = 𝑏 × 𝑞 + 𝑟, where 0 ≤ 𝑟 < 𝑏. a. 14, 3 b. 116, 7 c. 570, 6 d. 25, 36 e. 987, 16 2. Find the unique quotient and remainder when 31 is divided by 6. 3. Find four positive integers when divided by 4 leaves remainder 3. 4. A man has Birr 68. He plans to buy items such that each costs Birr 7. If he needs Birr 5 to remain in his pocket, what is the maximum number of items he can buy? (5m+1)(5m+3)(5m+6) 5. Find the remainder of for m is non-negative integer. 5 2.1.2 Prime numbers and composite numbers In this subsection, you will confirm important facts about prime and composite numbers. The following activity (activity 2.3) will help you to refresh your memory. Activity 2.3 1. Fill in the blanks to make the statements correct using the numbers 3 and 12. a. ____________is a factor of _____________. b. ____________is divisible by ____________. c. ____________is a multiple of _____________. 32  Unit 2: The Number System 2. For each of the following statements write ‘true’ if the statement is correct and ‘false’ otherwise. If your answer is false give justification why it is false. a. 1 is a factor of all natural numbers. b. There is no even prime number. c. 23 is a prime number. d. If a number is natural number, it is either prime or composite. e. 351 is divisible by 3. f. 22 × 3 × 7 is the prime factorization of 84. g. 63 is a multiple of 21. 3. Write factors of : a. 7 b. 15 Observations Given two natural numbers ℎ and 𝑝, ℎ is called a multiple of 𝑝 if there is a natural number 𝑞 such that ℎ = 𝑝 × 𝑞. In this way we can say: ▪ 𝑝 is called a factor or a divisor of ℎ. ▪ ℎ is divisible by 𝑝. ▪ 𝑞 is also a factor or divisor of ℎ. ▪ ℎ is divisible by 𝑞. Hence, for any two natural numbers ℎ and 𝑝, ℎ is divisible by 𝑝 if there exists a natural number 𝑞 such that ℎ = 𝑝 × 𝑞. Definition 2.1 Prime and composite numbers A natural number that has exactly two distinct factors, namely 1 (one) and itself is called a prime number whereas a natural number that has more than two factors is called a composite number. 33  Unit 2: The Number System Example 1 Is 18 a prime number or a composite number? Why? Solution: Observe that, 18 = 1 × 18 , 18 = 2 × 9 or 18 = 3 × 6.This indicates 1, 2, 3, 6, 9 and 18 are factors of 18. Hence, 18 is a composite number. Example 2 Find a prime number(s) greater than 50 and less than 55. Solution: The natural numbers greater than 50 and less than 55 are 51, 52, 53 and 54. 51 = 3 × 17, so that 3 and 17 are factors of 51. 52 = 2 × 26, so that 2 and 26 are factors of 52, and 54 = 2 × 27, hence 2 and 27 are factors of 54. Therefore, these three numbers are composite numbers. But 53 = 1 × 53. Hence, 1 and 53 are the only factors of 53 so that 53 is a prime number. Therefore, 53 is a prime number greater than 50 and less than 55. Exercise 2.3 1. Is 21 a prime number or a composite number? Why? 2. Write true if the statement is correct and false otherwise a. There are 7 prime numbers between 1 and 20. b. 4, 6, 15 and 21 are composite numbers. c. The smallest composite number is 2. d. 101 is the prime number nearest to 100. e. No prime number greater than 5 ends with 5. 3. Is 28 a composite number? If so, list all of its factors. 4. Can we generalize that ‘if a number is odd, then it is prime’? Why? 34  Unit 2: The Number System 5. Which one of the following is true about composite numbers? A. They have 3 pairs of factors B. They are always prime numbers C. They do not have factors D. They have more than two factors Note ❖ 1 is neither prime nor composite. ❖ 2 is the only even prime number. ❖ Factors of a number are always less than or equal to the number. 2.1.3 Divisibility test In the previous lesson, you practiced how to get the quotient and remainder while you divide a positive integer by another positive integer. Now you will revise the divisibility test to check whether 𝑎 is divisible by 𝑏 or not without performing division algorithm. Activity 2.4 Check whether the first integer is divisible by the second or not without using division algorithm. a. 2584, 2 b. 765, 9 c. 63885, 6 d. 7964, 4 e. 65475, 5 From your activity, you may check this either by using division algorithm or by applying rules without division. Every number is divisible by 1. You need to perform the division procedure to check divisibility of one natural number by another. The following rules can help you to determine whether a number is divisible by 2, 3, 4, 5, 6, 8, 9 and 10. Divisibility test by 7 will not be discussed now because it is beyond the scope of this level. Divisibility test:- It is an easy way to check whether a given number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10 without actually performing the division process. 35  Unit 2: The Number System A number is divisible by 2, if its unit digit is divisible by 2. 3, if the sum of its digits is divisible by 3. 4, if the number formed by its last two digits is divisible by 4. 5, if its unit digit is either 0 or 5. 6, if it is divisible by 2 and 3. 8, if the number formed by its last three digits is divisible by 8. 9, if the sum of its digits is divisible by 9. 10, if its unit digit is 0. Example Using divisibility test check whether 2,334 is divisible by 2, 3, 4, 5, 6, 8, 9 and 10. Solution: 2,334 is divisible by 2 since its unit digit 4 is divisible by 2. 2,334 is divisible by 3 since the sum of the digits (2 + 3 + 3 + 4) is 12 and it is divisible by 3. 2,334 is not divisible by 4 since its last two digits, that is 34 is not divisible by 4. 2,334 is not divisible by 5 since its unit digit is not either 0 or 5. 2,334 is divisible by 6 since it is divisible by 2 and 3. 2,334 is not divisible by 8 since its last three digits 334 is not divisible by 8. 2,334 is not divisible by 9 since the sum of the digits ( 2 + 3 + 3 + 4 = 12 ) is not divisible by 9. 2,334 is not divisible by 10 since its unit digit is not zero. Exercise 2.4 1. Using divisibility test, check whether the following numbers are divisible by 36  Unit 2: The Number System 2, 3, 4, 5, 6, 8, 9 and 10: a. 384 b. 3,186 c. 42,435 2. Given that 74,3𝑥2 is a number where 𝑥 is its tens place. If this number is divisible by 8, what is (are) the possible value(s) of 𝑥? 3. Find the least possible value of the blank space so that the number 3457__40 is divisible by 4. 4. Fill the blank space with the smallest possible digit that makes the given number 81231_37 is divisible by 9. Definition 2.2 Prime factorization The expression of a composite number as a product of prime numbers is called prime factorization. Consider a composite number which we need to write it in prime factorized form. Recall that a composite number has more than two factors. The factors could be prime or still composite. If both of the factors are prime, we stop the process by writing the given number as a product of these prime numbers. If one of the factors is a composite number, we will continue to get factors of this composite number till all the factors become primes. Finally, express the number as a product of all primes which are part of the process. Example 1 Express each of the following numbers as prime factorization form. a. 6 b. 30 c. 72 Solution: a. 6 = 2 × 3, since both 2 and 3 are prime, we stop the process. b. 30 = 2 × 15 and 15 = 3 × 5. 37  Unit 2: The Number System So that, 30 = 2 × 3 × 5. The right hand side of this equation is the prime factorization of 30. The process of prime factorization can be easily visualized by using a factor tree as shown below. c. By divisibility test, 72 is divisible by 2. So that 2 is one of its factors and it will be written as 2 × 36 as shown in the right side. Again 36 is divisible by 2. Hence, 36 = 2 × 18 and this process will continue till we get the factors as primes. Hence, the prime factorization of 72 is given as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. Also we can observe 72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = 23 × 32 72 = 4 × 18 = 2 × 2 × 2 × 3 × 3 = 23 × 32 72 = 3 × 24 = 3 × 3 × 2 × 2 × 2 = 23 × 32. The following theorem is stated without proof to generalize factors of composite numbers. Theorem 2.2 Fundamental theorem of arithmetic Every composite number can be expressed (factored) as a product of primes and this factorization is unique. Example 2 Express 456 as prime factorization form. Solution: To determine the prime factors, we will divide 456 by 2. If the quotient is also a composite number, again using a divisibility test we will search the prime number which divides the given composite number. Repeat this process till the quotient is prime. 38  Unit 2: The Number System That is, 456  2 = 228 228  2 = 114 114  2 = 57 57  3 = 19, we stop the process since 19 is prime. Hence, 456 = 23 × 3 × 19. Exercise 2.5 1. Express each of the following numbers as prime factorization form. a. 21 b. 70 c. 105 d. 252 e. 360 f. 1, 848 2. 180 can be written as 2𝑎 × 3𝑏 × 5𝑐. Then, find the value of 𝑎, 𝑏 and 𝑐. 3. Find the four prime numbers whose product is 462. 2.1.4 Greatest common factor and least common multiple In this subsection, you will revise the basic concepts about greatest common factors and least common multiples of two or more natural numbers. Greatest common factor Activity 2.5 1. Given the numbers 12 and 16: a. Find the common factors of the two numbers. b. Find the greatest common factor of the two numbers. 2. Given the following three numbers 24, 42 and 56: a. Find the common factors of the three numbers. b. Find the greatest common factor of the three numbers. 39  Unit 2: The Number System Definition 2.3 i) Given two or more natural numbers, a number which is a factor of these natural numbers is called a common factor. ii) The Greatest Common Factor (GCF) or Highest Common Factor (HCF) of two or more natural numbers is the greatest natural number of the common factors. GCF(𝑎, 𝑏) is to mean the Greatest Common Factor of 𝑎 and 𝑏. Example 1 Find the greatest common factors of 36 and 56. Solution: Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 and factors of 56 are1, 2, 4, 7, 8, 14, 28, 56. Here 1,2 and 4 are the common factors of both 36 and 56. The greatest one from these common factors is 4. Hence GCF(36, 56) = 4. You can also observe this from the following Venn Figure 2.1 Factors of 36 and 56 diagram. Activity 2.6 Let 𝑝 = 36, 𝑞 = 56. Then, write a. The prime factorization of 𝑝 and 𝑞. b. Write common prime factors of 𝑝 and 𝑞 with the least power. c. Take the product of the common prime factors you found in the above (𝑏) if they are two or more. d. Compare your result in (𝑐) with GCF(36, 56) you got in example 1 above. 40  Unit 2: The Number System The above activity leads you to use another alternate approach of determining GCF of two or more natural numbers using prime factorization. Using this method, the GCF of two or more natural numbers is the product of their common prime factors and the smallest number of times each power appears in the prime factorization of the given numbers. Example 2 Use prime factorization to find GCF(12, 18, 36). Solution: Let us write each of the given three numbers as prime factorization form. 12 = 22 × 31 18 = 21 × 32 36 = 22 × 32 In the above factorizations, 2 and 3 are common prime factors of the numbers (12,18 and 36). Further the least power of 2 is 1 and least power of 3 is also 1. So that the product of these two common prime numbers with least power of each is 2 × 3 = 6. Hence, GCF(12, 18, 36) = 6. Exercise 2.6 Find the greatest common factors (GCF) of the following numbers. 1. Using Venn diagram method a. 12, 18 b. 24, 64 c. 45, 63, 99 2. Using prime factorization method a. 24, 54 b. 108, 104 c. 180, 270, 1,080 41  Unit 2: The Number System Least common multiple Activity 2.7 For this activity, you need to use pen and pencil. Write the natural numbers from 1 to 60 using your pen. Encircle the number from the list which is a multiple of 6 using pencil. Underline the number on the list which is a multiple of 8 using pencil. Using the above task: a. Collect the numbers from the list which are both encircled and underlined in a set. b. What is the least common number from the set you found in (a)? c. What do you call the number you get in (b) above for the two numbers 6 Definition 2.4 For any two natural numbers 𝑎 and 𝑏 , the Least Common Multiple of 𝑎 and 𝑏 denoted by LCM(𝑎, 𝑏) is the smallest multiple of both 𝑎 and 𝑏. Intersection Method Example 1 Find Least Common Multiple of 2 and 3, that is, LCM(2, 3). Solution: 2, 4, 𝟔, 8, 10, 𝟏𝟐, 14, 16,... are multiples of 2 and 3, 𝟔, 9, 𝟏𝟐, 15, 𝟏𝟖, … are multiples of 3. Hence, 𝟔, 𝟏𝟐, 𝟏𝟖, … are common multiples of 2 and 3. The least number from the common multiples is 6. Therefore, LCM(2, 3) is 6. Example 2 Find Least Common Multiple of 6 and 9, that is, LCM(6, 9). 42  Unit 2: The Number System Solution: 6, 12, 𝟏𝟖, 24, 30, 𝟑𝟔... are multiples of 6 and 9, 𝟏𝟖, 27, 𝟑𝟔, 45, … are multiples of 9. Hence, 𝟏𝟖, 𝟑𝟔, 𝟓𝟒, … are common multiples of 6 and 9. The least number from the common multiples is 18. Therefore, LCM(6, 9) is 18. Factorization Method Example 3 Find the Least Common Multiple of 2, 3 and 5, that is, LCM(2, 3, 5) using factorization method. Solution: 2,3 and 5 are prime numbers. Taking the product of these prime numbers gives us LCM(2,3,5) = 2 × 3 × 5 = 30. Example 4 Find the Least Common Multiple of 6,10 and 16, that is, LCM(6, 10, 16) using factorization method. Solution: Writing each by prime factorization, we have 6=2×3 10 = 2 × 5} The prime factors that appear in these factorization are 2, 3 and 5. 16 = 24 Taking the product of the highest powers gives us LCM(6, 10, 16) = 24 × 3 × 5 = 240. Activity 2.8 Consider two natural numbers 15 and 42. Then, find a. LCM(15,42) and GCF(15, 42). b. 15 × 42 43  Unit 2: The Number System c. What is the product of GCF(15, 42) and LCM(15, 42)? d. Compare your results of (b) and (c). e. What do you generalize from (d)? From the above activity you can deduce that For any two natural numbers 𝑎 and 𝑏, GCF(𝑎, 𝑏) × LCM(𝑎, 𝑏) = 𝑎𝑏. Example 5 Find GCF(12, 18) × LCM(12, 18). Solution: The product of GCF(𝑎, 𝑏) and LCM(𝑎, 𝑏) is the product of the two numbers 𝑎 and 𝑏 for any two natural numbers 𝑎 and 𝑏. Hence GCF(12, 18) × LCM(12, 18) = 12 × 18 = 216. Exercise 2.7 1. Find the least common multiples (LCM) of the following list of numbers using both intersection method and prime factorization method. a. 6, 15 b. 14, 21 c. 4, 15, 21 d. 6, 10, 15, 18 2. Find GCF and LCM of each of the following numbers a. 4 and 9 b. 7 and 48 c. 12 and 32 d. 16 and 39 e. 12, 16 and 24 f. 4, 18 and 30 3. Find GCF(16, 24) × LCM (16, 24). 4. In a school, the number of participants in Sport, Mini media and Anti Aids club are 60, 84 and 108 respectively. Find the minimum number of rooms required. In each room the same number of participants are to be seated and all of them being in the same club. 44  Unit 2: The Number System 2.2 Rational Numbers In section 2.1, you have learnt about natural numbers and integers. In this section you will extend the set of integers to the set of rational numbers. You will also discuss how to represent rational numbers as decimals and locate them on the number line. Activity 2.9 Given integers 7, −2, 6, 0 and − 3. a. Divide one number by another (except division by 0). b. From the result obtained in (a) which of them are integers? c. What can you conclude from the above (a) and (b)? Now let us discuss about rational numbers Definition 2.5 Rational Numbers 𝑎 Any number that can be written in the form 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 ≠ 0 is called a rational number. The set of rational numbers is denoted by ℚ and is described as 𝑎 ℚ = ቄ𝑏 : 𝑎, 𝑏 ∈ ℤ 𝑎𝑛𝑑 𝑏 ≠ 0ቅ. Example 1 3 −13 The numbers , −6, 0, are rational numbers. 4 10 −6 0 Here, −6 and 0 can be written as and , 1 1 respectively. So we can conclude every natural number and integer is a rational number. You can see this relation using figure 2.2. Figure 2.2 45  Unit 2: The Number System Note 𝑎 Suppose 𝑥 = 𝑏 𝜖 ℚ, 𝑥 is a fraction with numerator 𝑎 and denominator 𝑏, i) If 𝑎 < 𝑏, then 𝑥 is called proper fraction. ii) If 𝑎 ≥ 𝑏, then 𝑥 is called improper fraction. 𝑎 𝑎 iii) If 𝑦 = 𝑐 𝑏 , where 𝑐 𝜖 ℤ and 𝑏 is a proper fraction, then 𝑦 is called a mixed fraction (mixed number). iv) 𝑥 is said to be in simplest (lowest form) if 𝑎 and 𝑏 are relatively prime or GCF(𝑎, 𝑏) = 1. Example 2 Categorize each of the following as proper, improper or mixed fraction 7 2 1 7 , ,54, and 6. 8 9 3 7 2 Solution: 8 and are proper fractions, 9 7 6 and 6 are improper fraction (since 6 = 1 ) and 3 1 5 4 is a mixed fraction. Example 3 1 Express 3 4 as improper fraction. 𝑚 𝑚 (𝑙×𝑛)+𝑚 Solution: For three integers 𝑙, 𝑚, 𝑛 where 𝑛 ≠ 0: 𝑙 =𝑙+ =. 𝑛 𝑛 𝑛 1 (3×4)+1 13 So that 3 4 = =. 4 4 Exercise 2.8 1. Express each of the following integers as fraction. a. 5 b. −3 c. 13 2. Write true if the statement is correct and false otherwise. Give justification if the answer is false. 46  Unit 2: The Number System a. Any integer is a rational number. 35 9 b. The simplest form of 45 is 7. 𝑎 2 11 c. For 𝑎, 𝑏 ∈ ℤ, is a rational number. d. The simplest form of 5 4 is. 𝑏 2 𝑎 𝑐 𝑎 𝑐 3. If and are two rational numbers, show that 𝑏 × 𝑑 is also a rational number. 𝑏 𝑑 4. Zebiba measures the length of a table and she reads 54 cm and 4 mm. Express 𝑎 this measurement in terms of cm in lowest form of 𝑏. 1 5. A rope of 5 3 meters is to be cut into 4 pieces of equal length. What will be the length of each piece? 2.2.1 Representation of rational numbers by decimals In this subsection, you will learn how to represent rational numbers by decimals and locate the rational number on the number line. Activity 2.10 1. Perform each of the following divisions. 3 5 14 −2 a. b. c.− 15 d. 5 9 7 2. Write the numbers 0.2 and 3.31 as a fraction form. From the above activity 2.10, you may observe the following: 𝑎 Any rational number can be written as decimal form by dividing the 𝑏 numerator 𝑎 by the denominator 𝑏. 𝑎 When we change a rational number 𝑏 into decimal form, one of the following cases will occur The division process ends when a remainder of zero is obtained. Here, the decimal is called a terminating decimal. The division process does not terminate but repeats as the remainder never become zero. In this case the decimal is called repeating decimal. 47  Unit 2: The Number System 3 In the above activity 2.10 (1), when you perform the division 5 , you obtain a 5 −14 decimal 0.6 which terminates, = 0.555 … and = −0.933 … are repeating. 9 15 Notation To represent repetition of digit/digits we put a bar notation above the repeating digit/digits. Example 1 Write the following fractions using the notation of repeating decimals. 1 23 a. b. 3 99 Solution: 1 23 a. = 0.333 … = 0. 3̅ ̅̅̅̅. b. 99 = 0.232323 … = 0. 23 3 Converting terminating decimals to fractions Every terminating decimal can be written as a fraction form with a denominator of a power of 10 which could be 10, 100, 1000 and so on depending on the number of digits after a decimal point. Example 2 Convert each of the following decimals to fraction form. a. 7.3 b. −0.18 Solution: a. The smallest place value of the digits in the number 7.3 is in the tenths column 7.3×10 73 so we write this as a number of tenths. So that, 7.3 = = 10 (since it has 1 10 digit after a decimal point, multiply both the numerator and the denominator by 10). b. The smallest place value of the digits in the number −0.18 is in the hundredths column so we write this as a number of hundredths. 48  Unit 2: The Number System −0.18×100 −18 −9 Hence, −0.18 = = and after simplification, we obtain. 100 100 50 Exercise 2.9 1. Write the following fractions using the notation of repeating decimals. 8 1 5 14 a. b. c. 6 d. 9 11 9 2. Convert each of the following as a fraction form. a. 0.3 b. 3.7 c. 0.77 d. −12.369 e. 0.6 f. 9.5 g. −0.48 h. −32.125 1 3. Using long division, we get that 7 = 0. ̅142857 ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅. Can you predict the decimal 2 3 4 5 6 expression of , , , and without doing long division? If so, how? 7 7 7 7 7 4. What can you conclude from question no. 3 above? Representing rational numbers on the number line Example 1 4 −5 Locate the rational numbers −5, 3, 3 and on the number line. 2 Solution: You can easily locate the given integers −5 and 3 on a number line. But to locate a 4 −5 fraction, change the fraction into decimal. That is, 3 = 1. 3̅ and 2 = −2.5. Now locate each of these by bold mark on the number line as shown in figure 2.3. Figure 2.3 49  Unit 2: The Number System 2.2.2 Conversion of repeating decimals into fractions In section 2.2.1, you have discussed how to convert the terminating decimals into fractions. In this subsection, you will learn how repeating decimals can be converted to fractions. Example 2 Represent each of the following decimals as a simplest fraction form (ratio of two integers). a. 0. 5̅ b. 2. ̅12 ̅̅̅ Solution: 𝑎 a. We need to write 0. 5̅ as 𝑏 form. Now let 𝑑 = 0. 5̅. Then, 10𝑑 = 5. 5̅ (multiplying both sides of 𝑑 = 0. 5̅ by 10) − 𝑑 = 0. 5̅ (we use subtraction to eliminate the repeating part) 9𝑑 = 5. 5 5 Now dividing both sides by 9, we have 𝑑 = 9. Hence, the fraction form of 0. 5̅ is 9. 𝑎 ̅̅̅̅ as form. Now let 𝑑 = 2. 12 b. We need to write 2. 12 ̅̅̅̅. Then, 𝑏 100𝑑 = 212. ̅̅ 12̅̅ (multiplying both sides of 𝑑 = 2. ̅12 ̅̅̅ by 10 ) −𝑑 = 2. ̅̅ 12̅̅ (we use subtraction to eliminate the repeating part) 99𝑑 = 210. 210 70 Now, dividing both sides by 99 results 𝑑 = = 33. Hence, the fraction form of 99 70 ̅̅̅̅ is. 2. 12 33 Example 3 Represent each of the following in fraction form. 𝐚. 0.12̅ ̅̅̅̅ b. 1.216 Solution: 50  Unit 2: The Number System a. Let 𝑑 = 0.12̅, now 100𝑑 = 12. 2̅ ( multiplying both sides of 𝑑 = 0.12̅ by 100) −10𝑑 = 1. 2̅ (multiplying both sides of 𝑑 = 0.12̅ by 10) 90𝑑 = 11. 11 Solving for 𝑑, that is dividing both sides by 90 we obtain 𝑑 = 90. Hence, the 11 fraction form of 0.12̅ is 90. ̅ ̅ ̅ ̅. Now b. Using similar procedure of (a), let 𝑑 = 1.216 ̅ ̅ ̅ ̅ (multiplying both sides of 𝑑 = 1216. 16 1000𝑑 = 1216. 16 ̅ ̅ ̅ ̅ by 1000) −10𝑑 = 12. ̅16 ̅ ̅ ̅ (multiplying both sides of 𝑑 = 12. ̅16 ̅ ̅ ̅ by 10) 990𝑑 = 1204. 1204 602 Dividing both sides by 990, we obtain 𝑑 = = 495. Hence, the fraction form 990 602 ̅ ̅ ̅ ̅ is of 1.216. 495 Exercise 2.10 4 1 1. Locate the rational numbers 3, 4, , 0. 4̅ and 2 on the number line. 5 3 2. Represent each of the following decimals as simplest fraction form. a. 2. 6̅ ̅̅̅̅ b. 0. 14 ̅̅̅̅ c. 0.716 ̅̅̅̅ d. 1.3212 ̅̅̅̅ e. −0.53213 2.3 Irrational Numbers Recall that terminating and repeating decimals are rational numbers. In this subsection, you will learn about decimals which are neither terminating nor repeating 2.3.1 Neither repeating nor terminating numbers Do you recall a perfect square? A perfect square is a number that can be expressed as a product of two equal integers. For instance, 1, 4, 9 are perfect squares since 1 = 12 , 4 = 22 , 9 = 32. 51  Unit 2: The Number System A square root of a number 𝑥 is a number 𝑦 such that 𝑦 2 = 𝑥. For instance, 5 and − 5 are square roots of 25 because 52 = (−5)2 = 25. Consider the following situation. There is a square whose area is 2 cm2. What is the length of a side of this square? The length of a side is supposed to be 𝑠 cm. Since 12 = 1, area of the square which side is 1 cm is 1 cm2. Likewise, 22 = 4, area of the square whose side is 2 cm is 4 cm2. From the above, 1 < 𝑠 < 2. In a similar way, 1.42 < 2 < 1.52 ⇒ 1.4 < 𝑠 < 1.5 1.412 < 2 < 1.422 ⇒ 1.41 < 𝑠 < 1.42 1.4142 < 2 < 1.4152 ⇒ 1.414 < 𝑠 < 1.415 Repeating the above method, we find 𝑠 = 1.414 …. This number is expressed as 𝑠 = √2, and √2 = 1.414 …. The sign √ is called a radical sign. Example 1 Find the number without radical sign. a. √4 b. √9 c. −√9 d. √0.0016 Solution: 3 a. √4 = 2 since 22 = 4 b. √9 = √32 = 3 = 1 4 1 c. −√9 = −√32 = −3 d. √0.0016 = √(0.04)2 = 0.04 = 100 = 25 Consider the magnitude of √5 and √7. Which one is greater? The area of square with side length √5 cm is 5 cm2. The area of square with side length √7 cm is 7 cm2. From the figure, when the side length becomes greater, the area becomes greater, too. 52  Unit 2: The Number System Hence, √5 < √7 since 5 < 7. This leads to the following conclusion. When 𝑎 > 0, 𝑏 > 0, if 𝑎 < 𝑏, then √𝑎 < √𝑏. Example 2 Compare √5 and √6. Solution: Since 5 < 6 then √5 < √6. Exercise 2.11 1. Find the numbers without radical sign. a. √25 b. −√36 c. √0.04 d. −√0.0081 2. Compare the following pairs of numbers. a. √7, √8 ̅ b. √3, √9 c. √0.04, √0.01 d. −√3 , −√4 What can you say about the square root of a number which is not a perfect square? To understand the nature of such numbers, you can use a scientific calculator and practice the following. Let us try to determine √2 using a calculator. Press 2 and then the square root button. A value for √2 will be displayed on the screen of the calculator as √2 ≈ 1.4142135624. The calculator provides a terminated result due to its capacity (the number of digit you get may be different for different calculators). But this result is not a terminating or Figure 2.4 Scientific calculator 53  Unit 2: The Number System repeating decimal. Hence √2 is not rational number. Similarly, check √3 and √7 are not terminating and repeating. Such types of numbers are called irrational numbers. Definition 2.6 A decimal number that is neither terminating nor repeating is an irrational number. Remark In general, if 𝑎 is natural number that is not perfect square, then √𝑎 is an irrational number. Example Determine whether each of the following numbers is rational or irrational. 22 a. √25 b. √0.09 c. 0.12345 … d. 0.010110111 … e. f. 𝜋 7 Solution: a. √25 = √52 = 5 which is rational. b. √0.09 = √0.32 = 0.3 which is rational number. c. 0.12345… is neither terminating nor repeating, so it is irrational. d. 0.010110111… is also neither repeating nor terminating, so it is irrational number. 22 e. is a fraction form so that it is rational. 7 f. 𝜋 which is neither terminating nor repeating, so that it is irrational. This example (c & d) leads you to the fact that, there are decimals which are neither repeating nor terminating. 54  Unit 2: The Number System Exercise 2.12 Determine whether each of the following numbers is rational or irrational. 6 a. √36 b. √7 c. d. √0.01 5 e. −√13 f. √11 g. √18 Locating irrational number on the number line Given an irrational number of the form √𝑎 where 𝑎 is not perfect square. Can you locate such a number on the number line? Example Locate √2 on the number line. Solution: We know that 2 is a number between perfect squares 1 and 4. That is 1 < 2 < 4. Take a square root of all these three numbers, that is, √1 < √2 < √4. Therefore 1 < √2 < 2. Hence, √2 is a number between 1 and 2 on the number line. (You can also show √2 ≈ 1.4142 … using a calculator). To locate √2 on the number line, you need a compass and straightedge ruler to perform the following. Draw a number line. Label an initial point 0 and points 1 unit long to the right and left of 0. Construct a perpendicular line segment 1 unit long at 1. Figure 2.5 Location of √2 on the number line 55  Unit 2: The Number System Draw a line segment from the point corresponding to 0 to the top of the 1unit segment and label its length as 𝑐. Using Pythagorean Theorem, 12 + 12 = 𝑐 2 so that 𝑐 = √2 unit long. Open the compass to the length of 𝑐. With the tip of the compass at the point corresponding to 0, draw an arc that intersects the number line at 𝐵. The distance from the point corresponding to 0 to 𝐵 is √2 unit. The following figure 2.6 could indicate how other irrational numbers are constructed on the number line using √2. Figure 2.6 Exercise 2.13 1. Locate the following on the number line. a. √3 b. √5 c.−√3 2. Between which natural numbers are the following numbers? a. √3 b. √5 c. √6 3. Write ‘True’ if the statement is correct and ‘False’ otherwise. a. Every point on the number line is of the form √𝑛 , where 𝑛 is a natural number. b. The square root of all positive integers is irrational. 56  Unit 2: The Number System 2.3.2 Operations on irrational numbers In section 2.3.1, you have seen what an irrational number is and how you represent it on the number line. In this section, we will discuss addition, subtraction, multiplication and division of irrational numbers. Activity 2.11 1. Identify whether 1 + √2 is rational or irrational. 2. Is the product of two irrational numbers irrational? Justify your answer with examples. When 𝑎 > 0, 𝑏 > 0, then √𝑎 × √𝑏 = √𝑎𝑏. Example Calculate each of the following. 2 1 2 a. √2 × √3 b. × c. (2 + √2) × (1 + √5) d. (√5 + √3) √3 √3 Solutions: a. √2 × √3 = √2 × 3 = √6 1 2 1×2 2 2 b. × = = =3 √3 √3 √3×√3 √3×3 c. Using distribution of addition over multiplication (2 + √2) × (1 + √5) = (2 × 1) + (2 × √5) + (√2 × 1) + (√2 × √5) = 2 + 2√5 + √2 + √10 2 2 2 d. (√5 + √3) = (√5 + √3)(√5 + √3) = (√5) + 2 ∙ √5√3 + (√3) = 8 + 2√15 57  Unit 2: The Number System Exercise 2.14 1. Calculate each of the following. a. √3 × √5 b. 2√5 × √7 1 10 c. −√2 × √6 d. × √5 √5 2 e. (2 + √3) × (−2 + √3) f. (√3 + √2) 2 g. (√7 + √3)(√7 − √3) h. (√6 − √10) 2. Decide whether the following statements are always true, sometimes true or never true and give your justification. a. The product of two irrational numbers is rational. b. The product of two irrational numbers is irrational. c. Any irrational number can be written as a product of two irrational numbers. d. Irrational numbers are closed with respect to multiplication. Activity 2.12 1. Consider irrational numbers √2, √3, √5 and √8. Divide each of these numbers by √2. 2. What do you conclude from the result you obtained in (1) above. √𝑎 𝑎 When 𝑎 > 0, 𝑏 > 0, then = ට𝑏