Grade 9 Maths Student Textbook PDF

MATHEMATICS MATHEMATICS Mathematics STUDENT TEXTBOOK GRADE 9 STUDENT TEXTBOOK...

MATHEMATICS MATHEMATICS Mathematics STUDENT TEXTBOOK GRADE 9 STUDENT TEXTBOOK GRADE 9 Student Textbook Grade 9 FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA MINISTRY OF EDUCATION MINISTRY OF EDUCATION Price ETB 167.00  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 GRADE 9 STUDENT TEXTBOOK 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) FEDERAL DEMOCRATIC REPUBLIC OF ETHIOPIA HAWASSA UNIVERSITY MINISTRY OF EDUCATION First Published xxxxx 2022 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. © 2022 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: xxxxxxxx PRINTING P.O.Box xxxxxx xxxxxxx, ETHIOPIA Under Ministry of Education Contract no. xxxxxxxxxxx ISBN: 978-999944-2-046-9 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 8 units namely: 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 Exercises 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 Description of the Concept Set _ _ _ _ _ _ _ _ _ _ _ 2 1’.2 Notion of Set _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 5 1’.3 Operations on Sets _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 9 1’.4 Application of Operations on Sets _ _ _ _ _ _ _ _ _ 14 Summary _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 14 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 Unit 1 THE NUMBER SYSTEM……………………… 17 1.1 Revision on Natural Numbers and Integers _ _ _ _ 18 1.2 Rational Numbers_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 32 1.3 Irrational Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 39 1.4 Real Numbers _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 48 1.5 Applications _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 80 Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 82 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 84 Unit 2 SOLVING EQUATIONS ……………………… 87 2.1 Revision of Linear Equation in One Variable _ _ _ 88 2.2 Systems of Linear Equations in Two Variables _ _ 91 2.3 Solving Non-linear Equations _ _ _ _ _ _ _ _ _ _ _ _ 100 2.4 Some Applications of Solving Equations _ _ _ _ _ _ _ 120 Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 126 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 127 Unit 3 SOLVING INEQUALITIES …………………… 130 3.1 Revision on Linear Inequalities in One Variable _ 131 3.2 Systems of Linear Inequalities in Two Variables _ 135 3.3 Inequalities Involving Absolute Value _ _ _ _ _ _ _ 144 3.4 Quadratic Inequalities _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 149 3.5 Applications on Equations_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 153 Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 158 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 159 Unit 4 INTRODUCTION TO TRIGONOMETRY …….. 162 4.1 Revision on Right-angled Triangles_ _ _ _ _ _ _ _ _ 163 4.2 The Trigonometric Ratios _ _ _ _ _ _ _ _ _ _ _ _ _ _ 167 Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 177 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 178 Unit 5 REGULAR POLYGONS ……………………… 181 5.1 Sum of Interior Angles of a Convex Polygons _ _ _ 182 5.2 Sum of Exterior Angles of a Convex Polygons _ _ _ 189 5.3 Measures of Each Interior Angle and Exterior Angle of a Regular Polygon _ _ _ _ _ _ _ _ _ _ _ _ _ 195 5.4 Properties of Regular Polygons: pentagon, _ _ _ _ _ _ _ _ _ _ _ 197 hexagon, octagon and decagon Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 205 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 206 __ Unit 6 CONGRUENCY AND SIMILARITY ………..… 209 6.1 Revision on Congruency of Triangles _ _ _ _ _ _ _ _ 210 6.2 Definition of Similar Figures _ _ _ _ _ _ _ _ _ _ _ _ _ 214 6.3 Theorems on Similar Plane Figures_ _ _ _ _ _ _ _ _ 219 6.4 Ratio of Perimeters of Similar Plane Figures _ _ _ 229 6.5 Ratio of Areas of Similar Plane Figures. _ _ _ _ _ _ 232 6.6 Construction of Similar Plane Figures _ _ _ _ _ _ _ 236 6.7 Applications on Similarities _ _ _ _ _ _ _ _ _ _ _ _ _ 238 Summary _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 243 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 244 Unit 7 VECTORS IN TWO DIMENTIONS …………… 247 7.1 Scalar and Vector Quantities _ _ _ _ _ _ _ _ _ _ _ _ _ 248 7.2 Representation of a Vector _ _ _ _ _ _ _ _ _ _ _ _ _ 251 7.3 Vectors Operations _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 257 7.4 Position Vector _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _267 7.5 Applications of Vectors in Two Dimensions _ _ _ _ 272 Summary_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 275 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 276 Unit 8 Statistics and Probability …………………….. 263 Unit 8 Statistics and Probability …………………….. 279 8.1 Statistical Data _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 281 8.2 Probability _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 313 Summary _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 323 Review Exercise _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 325 Mathematics Grade 9 Unit 1’ UNIT FURTHER ON SETS LEARNING OUTCOMES At the end of this unit, you will be able to: Explain facts about sets Describing sets in different ways Define operations on sets Demonstrate set operations using Venn diagram Apply rules and principles of set theory for practical situations. Main Contents 1.1 Description of the Concept Set 1.2 Notion of Set 1.3 Operations on Sets 1.4 Application of Operations on Sets Key Terms Summary Review Exercise 1 Mathematics Grade 9 Unit 1’ Introduction In Grade 7 you have learnt basic definition and operations involving sets. The concept of 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. 1’.1 Description of the concept set Activity 1’.1 1. Define sets 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 all the colors in the rainbow. d. Collection of consonants of the English alphabet. e. Collection of hardworking teachers in a school. A set is a collection of well- defined objects or elements. When we say that 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. 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 a set 𝐴, we say that “ 𝑎 belongs to 𝐴” the Greek symbol ∈(epsilon) is used to denote the phrase ‘belongs to’. Thus, we write 𝑎 ∈ 𝐴. If 𝑏 is not an element of a set 𝐴, we write 𝑏 ∉ 𝐴 and read ′′𝑏 does not belong to 𝐴’’. Example 1 1. The set of students in your class is a well-defined set, since the elements of the set are clearly known. 2 Mathematics Grade 9 Unit 1’ 2. The collection of handsome boys in your school. This is not a well-defined set because it is difficult to list members of the set. 3. Consider A as a set of vowels in English alphabet, then 𝑂 ∈ 𝐴, I, but 𝑏 ∉ 𝐴. 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 a 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’.1.1 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 English statement form( in words). Example 1 a. The set of all even natural numbers less than 10. b. The set of whole numbers greater than 1 and less than 20. ii) Complete listing method (Roster Method) In this method all the elements of the sets are completely listed, where the elements are separated by commas and are enclosed within braces { }. Example 2 a. The set of all even positive integers less than 7 is described in listing method as {2, 4, 6}. b. The set of all vowels in the English alphabet is described in listing method as {𝑎, 𝑒, 𝑖, 𝑜, 𝑢 }. 3 Mathematics Grade 9 Unit 1’ Note iii) 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 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 the element 3 indicate that the elements in the set continue in that manner up to and including the last element 99. b. Naming the set of whole numbers by 𝕎 , we can describe it as 𝕎 = {0, 1, 2, 3, … }. 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 prime factors of 36. b. The set of natural numbers less than 100 and divisible by 5. c. The set of non-negative integers. d. The set of even natural numbers. e. The set of integers divisible by 3. f. B is the set of positive even number below or equal to 30 g. The set of rational numbers between 2 and 8. h. Up to now you have leant three methods of describing a set. But there are sets which cannot be described by these three methods. Here is another method of describing the set. 4 Mathematics Grade 9 Unit 1’ iv) 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 satisfy a set or property of a set. In 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 or (:). Example 4 Describe the following sets using set builder method. i) 𝐴 = { 1,2,3 … 10} can be described as 𝐴 = {𝑥: 𝑥 ∈ ℕ and 𝑥 < 11}. You read this as “A is the set of all elements of natural numbers less than 11.’’ ii) Let 𝐴 = {0, 2, 4, …. }. This can be described as 𝐴 = {𝑥: 𝑥 is an elemet of non negative even integrs}. Exercise 1’.3 1. Write the following sets using set builder method. a. 𝐷 = {1, 3, 5 …. } b. 𝐴 = {2, 4, 6, 8} c. C = {1, 4, 9, 16, 25} d. 𝐸 = {4, 6, 8, 10, 12, 14, 15, 16, 18, 20 , … , 52 } e. F = {-10, …., -3, -2, -1, 0, 1, 2, … , 5} f. {1, 4, 9, … , 81} 1’.2 The Notion of Sets Empty Set A set which does not contain any element is called an empty set or void set or null set. The empty set is denoted by the symbol { } or Ø. Example 1 Let 𝐴 = {𝑥 ∶ 1 < 𝑥 < 2, 𝑥 is a natural number}. Then 𝐴 is the empty set, because there is no natural number between 1 and 2. 5  Mathematics Grade 9 Unit 1’ Finite and Infinite Sets 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 1. A set of natural numbers up to 10, is a finite set because it has definite (limited) number of elements. 2. The set of African countries is a finite set while the set of whole numbers is an infinite set. 3. If 𝐴 = {1, 2, 3, 4, 5, 6, 7, 8, 9,10}, then set 𝐴 has finite number of elements and the number of elements of 𝐴 is denoted by 𝑛(𝐴), that is,𝑛(𝐴) = 10. Exercise 1’.4 1. Identify empty set from the list below. a. {𝑥 ∶ 𝑥 ∈ 𝑁 and 5 < 𝑥 < 6} b. 𝐵 = {𝑂} c. The set of odd natural numbers divisible by 2. d. 𝐶 = { } 2. Identify the following sets as finite or infinite sets. a. The set of days in a week b. The set of all real numbers c. 𝐵 = {𝑥 ∶ 𝑥 is an even prime number} d. 𝐶 = {𝑥 ∶ 𝑥 is a multiple of 5} e. 𝐷 = {𝑥 ∶ 𝑥 is a factor of 30} f. 𝑃 = {𝑥 ∶ 𝑥 ∈ 𝑍, 𝑥 < −1} Equivalent Sets Definition 1’.2 Two sets 𝐴 and 𝐵 are said to be equivalent, written as 𝐴 ↔ 𝐵 (or 𝐴~𝐵), if there is a one-to- one correspondence between them. Observe that two finite sets A and B are equivalent, if and only if they have equal number of elements. Mathematically we write this as 𝑛 (𝐴) = 𝑛 (𝐵). 6 Mathematics Grade 9 Unit 1’ Example 1 If 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. Equal Sets Definition 1’.3 Two sets 𝐴 and 𝐵 are said to be equal if and only if they have exactly the same or identical elements. Example 2 𝐴 = {1, 2, 3, 4} and 𝐵 = {4, 3, 2, 1}, then 𝐴 = 𝐵. Subsets (⊆) Definition 1’.4 A set 𝐴 is said to be a subset of set 𝐵 if every element of 𝐴 is also an element of 𝐵. Mathematically it is denoted as 𝐴 ⊆ 𝐵. If set 𝐴 is not a subset of set 𝐵, then it is denoted as 𝐴 ⊈ 𝐵. Example 3 Let 𝐴 = {1, 2, 3}. Then {1, 2} ⊆ 𝐴. Similarly, other subsets of set A are: {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }. Note i) 𝐴ny set is a subset of itself. ii) Empty set is a sub set of every set. iii) If a set 𝐴 is finite with n elements, then the number of subsets of 𝐴 is 2𝑛. 7 Mathematics Grade 9 Unit 1’ Proper Subset (⊂) Definition 1’. 5 If 𝐴 ⊆ 𝐵 and 𝐴 ≠ 𝐵, then 𝐴 is called the proper subset of 𝐵 and it can be written as 𝐴 ⊂ 𝐵. Example 4 Given, 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. Note also that 𝐵 ⊄ A. Note i) For any set 𝐴, 𝐴 is not a proper subset of it self ii) The number of proper subsets of set 𝐴 is 2𝑛 − 1. iii) Empty set is the proper subset of any other sets. Exercise 1’.5 1. Identify equal and equivalent sets or neither. a. A = {1, 2, 3} and B = {4, 5} b. P = {q, s, m} and Q = {6, 9, 12} c. A = {2} and B = {x : x ∈ N, x is an even prime number}. d. A = {x : x ∈ W, x < 5} and B = {x : x ∈ N, x ≤ 5} e. D = {x : x is a multiple of 30} and E = {x : x is a factor of 10} 2. 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 } 3. Express the relationship of the following, 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 below 10} and 𝐷 = {1, 2, 4, 8}, c. 𝐸 = {1,2} and 𝐹 = {𝑥 | 0 < 𝑥 < 3, 𝑥 ∈ 𝑍} 8 Mathematics Grade 9 Unit 1’ 4. List all the sub sets of set 𝐴 = { 1, 3, 5 }. How many sub sets does it have? 5. List all the proper sub sets of set 𝐷 = { 𝑎, 𝑏}. How many proper subsets does it have? 1.3 Operations on Sets Definition 1’. 6 A universal set (usually denoted by U) is a set which has elements of all the related sets, without any repetition of elements. 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. Activity 1’.2 Let the universal set is the set of natural numbers 𝑁 and 𝐴 = {1,2,3,4,5,6} and 𝐵 = {1,3,5,7,9}. Can you write a set consisting of all natural numbers that are in 𝐴 and are in 𝐵? Can you write a set consisting of all natural numbers that are in 𝐴 or are in 𝐵? Can you write a set consisting of all natural numbers that are in 𝐴 and are not in 𝐵? The three most important set operations namely union(∪), intersection(∩) and complement(') are discussed below 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}. Definition 1’. 8 The intersection of two sets A and 𝐵, 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}. 9 Mathematics Grade 9 Unit 1’ Note Two sets A and 𝐵 are disjoint sets if 𝐴 ∩ 𝐵 = ∅. Example 1 Let 𝐴 = {0, 1, 3, 5, 7} and 𝐵 = {1, 2, 3, 4, 6, 7}. Then, find 𝐴 ∪ 𝐵 and 𝐴 ∩ 𝐵. Solution: 𝐴 ∪ 𝐵 = {0, 1, 2, 3, 4, 5, 6, 7}, and 𝐴 ∩ 𝐵 = {1, 3, 7 }. Example 2 Let 𝐴 be the set of positive even integers and 𝐵 be the set of positive multiples of 3. 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, 4, 6, 8, 9, 10, 12, 14, 15, 16,….} A ∩ B = {𝑥:𝑥 is a positive integer that is both even and multiple of 3}. = {𝑥: 𝑥 is a positive multiple of 6} = {6,12,18,24, …. } Note i) Law of ∅ and 𝑈: ∅ ∩ 𝐴 = ∅, 𝑈 ∩ 𝐴 = 𝐴. ii) Commutative law: 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴. iii) Associative Law: (𝐴 ∩ 𝐵) ∩ 𝐶 = 𝐴 ∩ (𝐵 ∩ 𝐶). Definition 1’. 9 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}. 10 Mathematics Grade 9 Unit 1’ Note 𝐴 − 𝐵 can be also written as 𝐴\𝐵. Definition 1’. 10 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}. Example 3 a. If 𝑈 = {0,1,2,3,4}, and 𝐴 = {3,4}, then 𝐴′ = {3,4}. b. Let 𝑈 = {1, 2, 3, … , 10}, 𝐴 = {𝑥 | 𝑥 is a positive factors of 10}, and 𝐵 = {𝑥 | 𝑥 is an odd integer in U}. Then, 𝐴′ = { 1,3,4,6,7, 8,9}, 𝐵 ′ = {2,4,6,8,10}. c. Find (𝐴 ∪ 𝐵)′ and (𝐴′ ∩ 𝐵′) using question (𝑏) above. First, we find 𝐴 ∪ 𝐵. Hence, 𝐴 ∪ 𝐵 = {1, 2, 3, 5, 7, 9,10} and (𝐴 ∪ 𝐵)′ = {4,6,8}. From question (b) above we also conclude that (𝐴′ ∩ 𝐵′) = {4,6,8}. In general, for any two sets 𝐴 and 𝐵, we have (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′. Exercise 1’.6 1. Let 𝐴 = {0, 2, 4, 6, 8} and 𝐵 = {0, 1, 2, 3, 5, 7, 9}. Then, find A ∪ B and A ∩ B. 2. Let 𝐴 be the set of positive odd integers and B be the set of positive multiples of 5. Then, find A ∪ B and A ∩ B. a. If 𝑈 = {0,1,2,3,4,5}, and 𝐴 = {4,5}, then find 𝐴′. b. Let 𝑈 = {1, 2, 3, … , 20}, 𝐴 = {𝑥 | 𝑥 is a positive factors of 20} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈}. Find 𝐴′ , 𝐵 ′ , (𝐴 ∪ 𝐵)′ and (𝐴′ ∩ 𝐵′). 11 Mathematics Grade 9 Unit 1’ Theorem 1.1 For any two sets 𝐴 and 𝐵, each of the following holds. (𝐴′)′ = 𝐴, 𝐴′ = 𝑈 − 𝐴, 𝐴 − 𝐵 = 𝐴 ∩ 𝐵 ′ , (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵 ′ , (𝐴 ∩ 𝐵)′ = 𝐴′ ∪ 𝐵 ′ , 𝐴 ⊆ 𝐵 ⟺ 𝐵 ′ ⊆ 𝐴′. Venn diagrams: A Venn diagram is a schematic or pictorial representative 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. Example 4 Let 𝑈 be the set of one digits numbers, 𝐴 be the set of one digits even numbers, 𝐵 be the set of positive prime numbers less than 10. We illustrate the sets using a Venn diagram as follows. From the Venn diagram we observe 𝐴 ∩ 𝐵 = {2}, 𝐴\𝐵 = {0,4,6,8}, 𝐵\𝐴 = {3,5,7}, and 𝑈\(𝐴 ∪ 𝐵) = {1,9}. Exercise 1’.7 1. Let 𝑈 = {1, 2, 3, … , 60}, 𝐴 = {𝑥 | 𝑥 is a positive factors of 40} and 𝐵 = {𝑥 | 𝑥 is an odd integer in 𝑈}. Find 𝐴′ , 𝐵 ′ , (𝐴 ∪ 𝐵)′ and (𝐴′ ∩ 𝐵′). 2. From the given Venn diagram find each of the following: a. A∩ 𝐵 b. 𝐴 ∪ 𝐵 c. (𝐴 ∪ 𝐵)′ d. 𝑈\(𝐴 ∪ 𝐵) e. (𝐵\𝐴) 12 Mathematics Grade 9 Unit 1’ Symmetric Difference of Two Sets Definition 1’. 11 Symmetric Difference For two sets A and B the symmetric difference b/n these two sets is denoted by 𝐴∆𝐵 and is defined as 𝐴∆𝐵 = (𝐴\𝐵) ∪ (𝐵\𝐴) or = 𝐴 ∪ 𝐵)\(𝐴 ∩ 𝐵). Using Venn diagram , the shaded part represents 𝐴∆𝐵. Example 5 Consider 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 = (𝐴\𝐵) ∪ (𝐵\𝐴)= {1, 4, 8} ∪ {3, 7} = {1, 3, 4, 7, 8}. Example 6 Given sets 𝐴 = {𝑑, 𝑒, 𝑓} and 𝐵 = {4, 5, 6}. Then find 𝐴∆𝐵. Solution: First, let us find 𝐴\𝐵 = {𝑑, 𝑒, 𝑓} and 𝐵\𝐴 = {4,5,6}. Hence, 𝐴∆𝐵 = (𝐴\𝐵) ∪ (𝐵\𝐴)= {𝑑, 𝑒, 𝑓} ∪ {4, 5, 6} = {𝑑, 𝑒, 𝑓, 4, 5, 6}. Cartesian Product of Two Sets Definition 1’. 12 Cartesian Product of two Sets The Cartesian product of two sets 𝐴 and 𝐵, is denoted by 𝐴 × 𝐵, is the set of all ordered pairs (𝑎, 𝑏) where 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵. This also can be expressed as 𝐴 × 𝐵 = {(𝑎, 𝑏): 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵}. Example 7 Let 𝐴 = {2, 4, 6} 𝑎𝑛𝑑 𝐵 = {1, 3}. Then find i) 𝐴 × 𝐵 ii) 𝐵 × 𝐴 Solution: 13 Mathematics Grade 9 Unit 1’ i) 𝐴 × 𝐵 = {(2, 1), (2, 3), (4, 1), (4, 3), (6, 1), (6, 3)} ii) 𝐵 × 𝐴 = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6)}. Example 8 If 𝐴 × 𝐵 = {(4, 1), (4, 4), (2, 1), (2, 4), (3, 1), (3, 4)}, then find set 𝐴 and 𝐵. Solution: 𝐴 is the set of all first component of 𝐴 × 𝐵 = {4,2,3}, and 𝐵 is the set of all second component of 𝐴 × 𝐵 = {1,4}. Exercise 1’.8 1. Given 𝐶 = {0, 2, 3, 4, 5} and 𝐷 = {0, 1, 2, 3, 5, 7, 9}. Then find 𝐶∆𝐷. 2. Let 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑑, 𝑒, 𝑓}. Then find i) 𝐴 × 𝐵 ii) 𝐵 × 𝐴 3. If 𝐴 × 𝐵 = {(7, 6), (7, 4), (5, 4), (5, 6), (1, 4), (1, 6)} 4. If 𝐴∆𝐵 = ∅, then what can be said about the two sets? 5. For any two sets 𝐴 and 𝐵, can we generalize 𝐴∆𝐵 = 𝐵∆𝐴 ? Justify your answer. Exercise 1’.9 1. Let 𝐴 = {𝑎, 𝑏, 𝑐} and 𝐵 = {𝑑, 𝑒, 𝑓}. Then find i) 𝐴 × 𝐵 ii) 𝐵 × 𝐴 2. If 𝐴 × 𝐵 = {(7,6), (7,4), (5,4), (5,6), (1,4), (1,6)}. 3. If 𝐴 = {𝑎, 𝑏, 𝑐} , 𝐵 = {𝑏, 𝑐, 𝑑} and 𝐶 = {𝑑, 𝑒}, the find 𝐴 × (𝐵 ∪ 𝐶) 4. If 𝐴 = {6, 9, 11}. Then find 𝐴 × 𝐴. 5. 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 _____________________. 1.4 Application Example 8 Let 𝐴 and 𝐵 be two finite sets such that 𝑛(𝐴) = 20, 𝑛(𝐵) = 28, and 𝑛(𝐴 ∪ 𝐵) = 36, then find 𝑛(𝐴 ∩ 𝐵). 14 Mathematics Grade 9 Unit 1’ Solution: Using the formula 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵) we have 𝑛(𝐴 ∩ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∪ 𝐵) = 20 + 28 − 36 = 48 − 36 = 12 Exercise 1’.10 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 clubs. b. How many students are not members of both clubs. c. How many students are only in Nature club. Review Exercise on unit one’ 1. If 𝐵 ⊆ 𝐴, 𝐴 ∩ 𝐵 ′ = {1, 4, 5}, and 𝐴 ∪ 𝐵 = {1, 2, 3, 4, 5, 6}, then find set 𝐵. 2. Let 𝐴 = {2, 4, 6, 7, 8, 9}, 𝐵 = {1, 3, 5, 6, 10} and 𝐶 ={ 𝑥: 3𝑥 + 6 = 0 or 2𝑥 + 6 = 0}. Find a. 𝐴 ∪ 𝐵. b. Is (𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶)? 3. Suppose 𝑈 be the set of one digit numbers and 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′, and g) 𝑈′ 4. Let 𝑈 = {1, 2, 3, 4, … …. ,10}, 𝐴 = {1, 3, 5, 7}, 𝐵 = {1, 2, 3, 4}, {3, 4, 6, 7}. 5. In a group of people, 42 drink tea, 27 drink coffee, and 60 people are in all. If every person in the group drinks at least one of the two drinks, find the number of people who drink both tea and coffee. Hint use a Venn diagram. 6. Select 𝐴′ from the choices below. A. {2, 3, 4, 6, 8,10} 15 Mathematics Grade 9 Unit 1’ B. {2, 5, 6, 7, 8, 9} C. {2, 4, 6, 8, 9,10} D. {2, 3, 5, 8, 9,10} E. {1, 2, 3, 4, 7, 10} F. ∅ 7. 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 not 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} 16 Mathematics Grade 9 Unit 1 UNIT THE NUMBER SYSTEM LEARNING OUTCOMES At the end of this unit, you will be able to: 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 Main Contents 1.1 Revision on Natural Numbers and Integers 1.2 Rational Numbers 1.3 Irrational Numbers 1.4 Real Numbers 1.5 Applications Key Terms Summary Review Exercise 17 Mathematics Grade 9 Unit 1 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. 1.1 Revision on Natural Numbers and Integers Activity 1.1 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? 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 is 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, … }. 18  Mathematics Grade 9 Unit 1 Exercise 1.1 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? 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? 1.1.1 Euclid’s division lemma Activity 1.2 1. In a book store there are 115 different books to be distributed for 8 students. If the book store shares these books equally, how many books each student will 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, from the process of dividing one positive integer by another, you will get remainder and quotient as described in the following theorem. Theorem 1.1 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 = ሺ𝑏 × 𝑞ሻ + 𝑟 with 0 ≤ 𝑟 < 𝑏. 19 Mathematics Grade 9 Unit 1 In theorem 1.1, 𝑎 is called the dividend, 𝑞 is called the quotient, 𝑏 is called the divisor, and 𝑟 is called the remainder. Example 1 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 𝑏 = 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 also 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. Divide 2,574 by 8 and determine the quotient and remainder. 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. 20 Mathematics Grade 9 Unit 1 Note For two positive integers 𝑎 and 𝑏 in the division algorithm, 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 1.2 1. For each of the following pairs of numbers, let 𝑎 be the first number of the pair and 𝑏 is the second number. Find 𝑞 and 𝑟 for each pair such that 𝑎 = ሺ𝑏 × 𝑞ሻ + 𝑟, where 0 ≤ 𝑟 < 𝑏. a. 14, 3 b. 116, 7 c. 25, 36 d. 987, 16 e. 570,6 2. Find all positive integers, when divided by 4 leaves remainder 3. 3. A man has 68 Birr. He plans to buy items such that each costs 7 Birr. If he needs 5 Birr to remain in his pocket, what is the maximum number of items he can buy? 1.1.2 Prime numbers and composite numbers In this subsection, you will confirm important facts about prime and composite numbers. The following activity (activity 1.3) will help you to refresh your memory. Activity 1.3 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 _____________ 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. 21 Mathematics Grade 9 Unit 1 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. Example 1 Is 249 is divisible by 3? Why? Solution: 249 = 3 × 83 , so that 249 is divisible by 3. Definition 1.1 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. Example 2 Is 18 a composite number? Why? 22 Mathematics Grade 9 Unit 1 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 3 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 1.3 1. List prime numbers less than 30. 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? 5. Which one of the following is true about composite numbers? a. They have 3 pairs of factors c. They are always prime numbers b. They do not have factors d. They have more than two factors 23 Mathematics Grade 9 Unit 1 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. 1.1.3 Divisibility test In the previous lesson, you practiced how to get the quotient and remainder while we 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 1.4 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. 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. 24 Mathematics Grade 9 Unit 1 9, if the sum of its digits is divisible by 9. 10, if its unit digit is 0. Example 1 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 ሺ2 + 3 + 3 + 4ሻ is 12 and it is divisible by 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 10 since its unit digit is not zero. 2,334 is divisible by 6 since it is divisible by 2 and 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 not divisible by 9 since the sum of the digits ( 2 + 3 + 3 + 4 = 12 ) is not divisible by 9. Exercise 1.4 1. Using divisibility test, check whether the following numbers are divisible by 2,3,4,5,6,8,9 and 10: a. 384 b. 3,186 c. 42,435 2. Given that 74,3𝑥2 be a number where 𝑥 is its tens place. If this number is divisible by 4, what is (are) the possible value(s) of 𝑥? 3. Find the least possible value of the blank so that the number 3457__40 is divisible by 4. 4. Fill the blank with the smallest possible digit that makes the given number 81231_37 is divisible by 9. 25 Mathematics Grade 9 Unit 1 Definition 1.2 Prime factorization The expression of a 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 2 6 = 2 × 3, since both 2 and 3 are prime we stop the process. Example 3 30 = 2 × 15 and 15 = 3 × 5. 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. Example 4 Express 72 as prime factors. Solution: By divisibility test, 72 is divisible by 2. So that 2 is one of its factor and it will be written as 2 × 36 as shown at 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. Finally, the prime factorization of 72 is given as 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32. 26 Mathematics Grade 9 Unit 1 Exercise 1.5 1. Express each of the following numbers as prime factorization form a. 21 b. 70 c. 105 d. 252 e. 360 2. 180 can be written as 2𝑎 × 3𝑏 × 5𝑐. Then find the value of 𝑎, 𝑏 and 𝑐. Example 5 Find the prime factorization of 456. 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. 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. For instance, 18 can be written as 18 = 2 × 32 , 2 and 3 are factors of 18 and this is a unique factorization. The following theorem 1.2 is stated without proof to generalize to generalize factors of composite numbers. Theorem 1.2 Fundamental theorem of arithmetic Every composite number can be expressed (factored) as a product of primes and this factorization is unique. Exercise 1.6 1. Express each of the following as prime factorization form. a. 1,848 b. 1,890 c. 2,070. d. 134,750 2. Find the four prime numbers when they multiply gives 462. 27 Mathematics Grade 9 Unit 1 1.1.4 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 1.5 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. Consider 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. Definition 1.3 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. GCFሺ𝑎, 𝑏ሻ 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 are 1,2,4,7,8,14,28,56. Here 1,2 and 4 are the common factors for both 36 and 56. The greatest one from these common factors is 4. Hence GCFሺ36,56ሻ = 4. You can also observe this from the following Venn diagram. 28 Mathematics Grade 9 Unit 1 Figure 1.1 Factors of 36 and 56 Activity 1.6 Let 𝑎 = 36, 𝑏 = 56. Then, write 1. the prime factorization of 𝑎 and 𝑏. 2. the common prime factors of 𝑎 and 𝑏 with the least power. 3. take the product of the common prime factors you found in (2) if they are two or more. 4. compare your result in (3) with GCFሺ36,56ሻ you got in example 1 above. The above activity leads you to use another alternating 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, each power to the smallest number of times it appears in the prime factorization of the given numbers. Example 2 Use prime factorization to find GCFሺ12,18,36ሻ. 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 ሺ12,18 and 36ሻ. 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, GCFሺ12,18,36ሻ = 6. 29 Mathematics Grade 9 Unit 1 Exercise 1.7 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 and 1,080 Least common multiple Activity 1.7 For this activity, you need to use pen and pencil. Write the natural numbers from1 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 and 8? Definition 1.4 For any two natural numbers 𝑎 and 𝑏 the Least Common Multiple of 𝑎 and 𝑏 denoted by LCMሺ𝑎, 𝑏ሻ , is the smallest multiple of both 𝑎 and 𝑏. 30 Mathematics Grade 9 Unit 1 Example 3 Find Least Common Multiple of 6 and 9, that is, LCMሺ6,9ሻ. Solution: 6,12,18,24,30,... 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, LCMሺ6,9ሻ is 18. Remark We can also use the prime factorization method to determine LCM of two or more natural numbers. The common multiple contains all the prime factors of each given number. The LCM is the product of each of these prime factors to the greatest number of times it appears in the prime factorization of the numbers. Example 4 Find the Least Common Multiple of 6,10 and 16, that is, LCMሺ6,10,16ሻ 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 LCMሺ6,10,16ሻ = 24 × 3 × 5 = 240. Exercise 1.8 Find the least common multiples (LCM) of the following list of numbers a. 6,15 b. 14,21 c. 4,15,21 d. 6,10,15,18 31 Mathematics Grade 9 Unit 1 Activity 1.8 Consider two natural numbers 15 and 42. Then, find a. LCMሺ15,42ሻ and GCFሺ15,42ሻ. b. 15 × 42 c. What is the product of GCFሺ𝑎, 𝑏ሻ and LCMሺ𝑎, 𝑏ሻ? 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, GCFሺ𝑎, 𝑏ሻ × LCMሺ𝑎, 𝑏ሻ = 𝑎𝑏. Exercise 1.9 1. 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 2. Find GCFሺ16,24ሻ × LCM ሺ16,24ሻ. 1.2 Rational Numbers In section 1.1, 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 1.9 Given integers 7, −2, 6,0 and − 3. a. Divide one number by another. 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 32 Mathematics Grade 9 Unit 1 Definition 1.5 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 −6 0 The numbers 4 , −6, 0, 10 are rational numbers. Here, -6 and 0 can be written as 1 and 1, respectively. So we can conclude every natural number and integer is a rational number. You can see this relation using figure. 1.2. Figure 1.2 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. iv) 𝑥 is said to be in simplest (lowest form) if 𝑎 and 𝑏 are relatively prime or GCFሺ𝑎, 𝑏ሻ = 1. 33 Mathematics Grade 9 Unit 1 Example 2 Categorize each of the following as proper, improper or mixed fraction 7 2 1 7 , , 5 4 , 3 and 6. 8 9 Solution: 7 2 and are proper fractions, 8 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. Solution: 𝑚 𝑚 ሺ𝑙×𝑛ሻ+𝑚 1 ሺ3×4ሻ+1 13 For three integers 𝑙, 𝑚, 𝑛 where 𝑛 ≠ 0: 𝑙 =𝑙+ =. So that 3 4 = =. 𝑛 𝑛 𝑛 4 4 Exercise 1.10 1. Write true if the statement is correct and false otherwise. Give justification if the answer is false. 𝑎 𝑐 a. Any integer is a rational number. c. For 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℤ, 𝑏 + 𝑑 is a rational number. 35 9 2 11 b. The simplest form of 45 is 7. d. The simplest form of 5 4 is. 2 𝑎 𝑐 𝑎 𝑐 2. If and are two rational numbers, show that ቀ𝑏ቁ × ቀ𝑑ቁ is also a rational number. 𝑏 𝑑 3. Zebiba measures the length of a table and she reads 54 cm and 4 mm. Express this 𝑎 measurement in terms of cm in lowest form of. 𝑏 1 4. A rope of 5 3 meters is to be cut into 4 pieces of equal length. What will be length of each piece? 34 Mathematics Grade 9 Unit 1 1.2.1 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 1.10 1. Perform each of the following divisions. 3 5 a. b. 5 9 14 −2 c.− 15 d. 7 2. Write the numbers 0.2 and 3.31 as a fraction form. From the above activity 1.10, you may observe the following: 𝑎 Any rational number can be written as decimal form by dividing the numerator 𝑎 by the 𝑏 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. 3 In the above activity 1.10 ሺ1ሻ, when you perform the division 5 you obtain a decimal 0.6 which 5 −14 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 1 23 = 0.333 … = 0. 3̅ and ̅̅̅̅. = 0.232323 … = 0. 23 3 99 35 Mathematics Grade 9 Unit 1 Exercise 1.11 1. Write the following fractions using the notation of repeating decimals 8 1 a. c. 9 11 5 14 b. 6 d. 9 1 2. Using long division we get that 7 = 0. ̅̅̅̅̅̅̅̅̅̅ 142857. Can you predict the decimal expression 2 3 4 5 6 of , , , and without actually doing long division? If so how? 7 7 7 7 7 3. What can you conclude from question no. 2 above? 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 1 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 so we write 7.3×10 73 this as a number of tenths. So that, 7.3 = = 10 (since it has 1 digit after a decimal 10 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 −0.18×100 −18 we write this as a number of hundredths. Hence, −0.18 = 100 = 100 and after −9 simplification we obtain. 50 36 Mathematics Grade 9 Unit 1 Exercise 1.12 1. Write down each of the following as a fraction form: i) a. 0.3 b. 3.7 c. 0.77 d. −12.369 ii) a. 0.6 b. 9.5 c. −0.48 d. −32.125 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 fraction, 4 −5 change the fraction into decimal. That is, = 1. 3̅ and 2 = −2.5. Now locate each of these by 3 bold mark on the number line as shown in figure 1.3. Figure1.3 Exercise 1.13 4 1 Locate the rational numbers 3, 4, 5 , 0. 4̅ and 2 3 on the number line. 1.2.2 Conversion of repeating decimals into fractions In section 1.2.1, 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 1 Write each of the following decimals as a fraction (ratio of two integers) a. 0. 5̅ ̅̅̅̅ b. 2. 12 37 Mathematics Grade 9 Unit 1 Solution: 𝑎 a. We need to write 0. 5̅ as 𝑏 form. Now let 𝑑 = 0. 5̅. Then, 10𝑑 = 5. 5̅ (multiplying both side of 𝑑 = 0. 5̅ by 10 since there is one repeating digit) − 𝑑 = 0. 5̅ (we use subtraction to eliminate the repeating part) 5 9𝑑 = 5. (now dividing both sides by 9 we have 𝑑 = 9 ). 5 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 ̅̅̅̅ (multiplying both side of 𝑑 = 2. 12 100𝑑 = 212. 12 ̅̅̅̅ by 100 since there are two repeating digits) − 𝑑 = 2. ̅12 ̅̅̅ ( we use subtraction to eliminate the repeating part) 210 70 99𝑑 = 210. Now, dividing both sides by 99 results 𝑑 = = 33 99 70 Hence, the fraction form of 2. ̅̅ 12̅̅ is. 33 Example 2 Represent each of the following in fraction form. 𝐚. 0.12̅ ̅̅̅̅ b. 1.216 Solution: 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 solving for 𝑑, that is dividing both sides by 99 11 11 we obtain 𝑑 =. Hence, the fraction form of 0.12̅ is 99. 99 ̅̅̅̅. Now b. Using similar procedure of (a), let 𝑑 = 1.216 ̅̅̅̅ (multiplying both sides of 𝑑 = 1216. 16 1000𝑑 = 1216. 16 ̅̅̅̅ by 1000) −10𝑑 = ̅̅̅̅ (multiplying both sides of 𝑑 = 12. 16 12. 16 ̅̅̅̅ by 10) 990𝑑 = 1204 (dividing both sides by 990) 38 Mathematics Grade 9 Unit 1 1204 602 we obtain 𝑑 = ̅̅̅̅ is 602. = 495. Hence, the fraction form of 1.216 990 495 Exercise 1.14 1. Represent each of the following decimals as simplest fraction form. a. 2. 6̅ b. 0. ̅̅ 14̅̅ ̅̅̅̅ c. 0.716 ̅̅̅̅ d. 1.3212 ̅̅̅̅ e. −0.53213 1.3 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 1.3.1 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. 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 because52 = ሺ−5ሻ2 = 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 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. 39 Mathematics Grade 9 Unit 1 Example 1 a. √4 = 2 since 22 = 4 3 b. √9 = √32 = 3 = 1 c. −√9 = −√32 = −3 4 1 d. √0.0016 = √ሺ0.04ሻ2 = 0.04 = 100 ሺits simplest form is 25ሻ. Exercise 1.15 Find the numbers without radical sign a. √25 b. −√36 c. √0.04 d. −√0.0081 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 side is √7 cm is 7 cm2. From the figure, when the side length becomes greater, the area becomes greater, too. Hence , √5 < √7 since 5 < 7. This leads to the following conclusion When 𝑎 > 0, 𝑏 > 0, if 𝑎 < 𝑏, then √𝑎 < √𝑏. Example 1 Compare √5 and √6 Solution: Since 5 < 6 then √5 < √6. 40 Mathematics Grade 9 Unit 1 Exercise 1.16 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 provide 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 repeating decimal. Hence √2 is not rational number. Similarly, check √3 and √7 are not terminating and not repeating. Such types of numbers are called irrational numbers. Figure 1.4 Scientific calculator Definition 1.6 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 1 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 41 Mathematics Grade 9 Unit 1 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 = 3.1415926 …which is neither terminating nor repeating, so that it is irrational. This example (𝑎, 𝑏& 𝑑) leads you to the fact that, there are decimals which are neither repeating nor terminating. Exercise 1.17 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 1 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) 42 Mathematics Grade 9 Unit 1 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 1.5 Location of √2 on the number lne 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 1.6 could indicate how other irrational numbers are constructed on the number line using √2. Figure 1.6 43 Mathematics Grade 9 Unit 1 Exercise 1.18 1. Between which natural numbers are the following numbers? a. √3 b. √5 c. √6 2. Locate the following on the number line a. √3 b. √5 c.−√3 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. 1.3.2 Operations on irrational numbers In section 1.3.1, 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 1.11 1. Identify whether 1 + √2 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 1 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 44 Mathematics Grade 9 Unit 1 c. Using distribution of addition over multiplication (2 + √2) × (1 + √5) = ሺ2 × 1ሻ + (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 Exercise 1.19 1. Calculate each of the following a. √3 × √5 e. (2 + √3) × (−2 + √3) 2 b. 2√5 × √7 f. (√3 + √2) c. −√2 × √6 g. (√7 + √3)(√7 − √3) 1 10 2 d. × h. (√6 − √10) √5 √5 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 1.12 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. √𝑏 45 Mathematics Grade 9 Unit 1 Example 1 Calculate √2 −√18 a. b. √3 √2 Solutions: √2 2 −√18 18 a. = ට3 b. = −ට 2 = −√9 = −3 √3 √2 When 𝑎 > 0, 𝑏 > 0, then 𝑎√𝑏 = √𝑎2 𝑏, √𝑎2 𝑏 = 𝑎√𝑏 Example 2 Convert each of the following in √𝑎 form a. 3√2 b. 5√5 Solutions: a. 3√2 = √32 × 2 = √18 b. 5√5 = √52 × 5 = √125 Exercise 1.20 1. Calculate √27 √12 √14 −√15 a. b. c. d. √3 √3 √7 √3 2. Convert each of the following in √𝑎 form a. 4√2 b. 5√3 c. −3√7 d. 7√6 Activity 1.13 Give an example which satisfies: a. two neither repeating nor terminating decimals whose sum is rational b. two neither repeating nor terminating decimals whose sum is irrational c. any two irrational numbers whose difference is rational d. any two irrational numbers whose difference is irrational 46 Mathematics Grade 9 Unit 1 Example 1 Simplify each of the following a. 0.131331333 … + 0.535335333 … b. 0.4747747774 … − 0.252552555 … Solutions: a. 0.131331333 … + 0.535335333 … = 0.666 … b. 0.4747747774 … − 0.252552555 …. = 0.222 … Example 2 Simplify each of the following. a. 2√3 + 4√3 b. 3√5 − 2√5 Solutions: a. 2√3 + 4√3 = ሺ2 + 4ሻ√3 = 6√3 b. 3√5 − 2√5 = ሺ3 − 2ሻ√5 = √5 Example 3 Simplify each of the following. a. √8 + √2 c. √18 + √50 b. √12 − 5√3 d. √72 − √8 Solutions: First, convert √𝑎2 𝑏 into 𝑎√𝑏 form. a. √8 + √2 = √4 × 2 + √2 = √22 × 2 + √2 = 2√2 + √2 = ሺ2 + 1ሻ√2 = 3√2 b. √12 − 5√3 = √22 × 3 − 5√3 = 2√3 − 5√3 = ሺ2 − 5ሻ√3 = −3√3 c. √18 + √50 = √32 × 2 + √52 × 2 = 3√2 + 5√2 = 8√2 d. √72 − √8 = √22 × 32 × 2 − √22 × 2 = √ሺ2 × 3ሻ2 × 2 − √22 × 2 = 6√2 − 2√2 = 8√2 47 Mathematics Grade 9 Unit 1 Exercise 1.21 Simplify each of the following. a. √3 + 4√3 f. √80 − √20 b. √5 − √45 g. 5√8 + 6√32 c. 2√5 − 4√5 h. √8 + √72 3 d. √18 + 2√2 i. √12 − √48 + ට4 e. 0.12345 … − 0.111 … j. 2.1010010001 … + 1.0101101110 … 1.4 Real Numbers In the previous two sections you have learnt about rational numbers and irrational numbers. Rational numbers are either terminating or repeating. You can locate these numbers on the number line. You have also discussed, it is possible to locate irrational numbers which are neither repeating nor terminating on the number line. Activity 1.14 1. Can you think of a set which consists both rational numbers and irrational numbers? 2. What can you say about the correspondence between the points on the number line and decimal numbers? Based on the reply for the above questions and recalling the previous lessons, you can observe that every decimal number (rational or irrational) corresponds to a point on the number line. So that there should be a set which consists both rational and irrational numbers. This leads to the following definition. Definition 1.7 Real Numbers A number is called a real number, if and only if it is either a rational number or an irrational number. The set of real numbers is denoted by ℝ, and is described as the union of the sets of rational and irrational numbers. We write this mathematically as ℝ ={𝑥: 𝑥 is a rational number or irrational number}. 48 Mathematics Grade 9 Unit 1 The following diagram indicates the set of real number,ℝ , is the union of rational and irrational numbers. Figure 1.6 Comparing real numbers You have seen that there is a one to one correspondence between a point on the number line and a real number. ▪ Suppose two real numbers 𝑎 and 𝑏 are given. Then one of the following is true 𝑎 < 𝑏 or 𝑎 = 𝑏 or 𝑎 > 𝑏 (This is called trichotomy property). ▪ For any three real numbers 𝑎, 𝑏 and 𝑐, if 𝑎 < 𝑏 and 𝑏 < 𝑐 , then 𝑎 < 𝑐 (called transitive property order) Applying the above properties we have ▪ For any two non-negative real numbers 𝑎 and 𝑏 if 𝑎2 < 𝑏 2 , then 𝑎 < 𝑏. Example 1 Compare each pair (you can use Scientific calculator whenever it is necessary). 𝟓 a. −2, −8 b. , 0.8 𝟖 √2 √3 2 c. 3 , 0.34 d. 6 , 3 49 Mathematics Grade 9 Unit 1 Solution: a. The location of these two numbers on the number line help us to compare the given numbers. Here, −8 is located at the left of −2 so that −8 < −2. 5 5 b. Find the decimal representation of , that is 8 = 0.625. 8 5 Hence, 8 < 0.8. √2 √2 c. Using a scientific calculator we approximate = 0.4714 …. Then, 3 > 0.34. 3 2 √3 3 2 2 d. Here use the third property given above. Square the two numbers ቀ ቁ = and ቀ ቁ = 6 36 3 4 16 = 36. 9 √3 2 Hence, it follows < 3. 6 Exercise 1.22 Compare each of the following pairs (using > 𝑜𝑟 0, 𝑛 𝑛 √𝑏 is defined as √𝑏 = ቐ the negative 𝑛𝑡ℎ root of 𝑏 , 𝑖𝑓 𝑏 < 0 and 𝑛 is odd, 0, if 𝑏 = 0. 57 Mathematics Grade 9 Unit 1 Notations 𝑛 √𝑏 called radical expression where √ is radical sign, 𝑛 is index and 𝑏 is radicand. When the index is not written, the radical sign indicates square root. 𝒕𝒉 Definition 1.12 The (𝟏ൗ𝒏) power 1 𝑛 If 𝑏 ∈ ℝ and n is a positive integer greater than 1, then 𝑏 𝑛 = √𝑏. Example 1 Find : a. √9 b. √0.01 4 c. √81 d. 5√−100,000 Solution: a. √9 = 3 because 32 = 9. b. √0.01 = 0.1 because ሺ0.1ሻ2 = 0.01. 4 c. √81 = 3 because 34 = 81. d. 5√−100,000 = −10 because ሺ−10ሻ5 = −100,000. Example 2 Write each of the following in exponential form. 3 1 a. √3 b. √6 c. 4 √10 Solution: 1 a. √3 = 32 1 3 b. √6 = 63 1 1 c. 4 = 10−4 √10 58 Mathematics Grade 9 Unit 1 Exercise 1.26 1. Express each of the following in exponential form 4 2 a. √5 d. (√81) 7 3 2 b. √7 e. ට 5 3 c. √34 2. Simplify each of the following 1 3 √125 a. ሺ−27ሻ3 c. √625 1 b. 325 d. √0.09 Laws of exponent Activity 1.17 For the given table below, i) Determine the simplified form of each term ii) Compare the value you obtained for (I) and (II) for each row No I II 1 23 × 33 ሺ2 × 3ሻ3 2 25 25−3 23 3 ሺ32 ሻ3 ሺ33 ሻ2 4 ሺ3 × 4ሻ2 32 × 42 5 22 × 23 22+3 The above activity 1.17 leads you to have the following laws of exponents 59 Mathematics Grade 9 Unit 1 For any 𝑎, 𝑏 ∈ ℝ and 𝑛, 𝑚 ∈ ℕ , the following holds: ✓ 𝑎𝑛 × 𝑎𝑚 = 𝑎𝑛+𝑚 𝑎𝑛 ✓ = 𝑎ሺ𝑛−𝑚ሻ , where 𝑎 ≠ 0 𝑎𝑚 ✓ ሺ𝑎𝑛 ሻ𝑚 = ሺ𝑎𝑚 ሻ𝑛 = 𝑎𝑛𝑚 ✓ ሺ𝑎 × 𝑏ሻ𝑛 = 𝑎𝑛 × 𝑏 𝑛 𝑎𝑛 𝑎 𝑛 ✓ = ቀ𝑏ቁ , for 𝑏 ≠ 0. 𝑏𝑛 1 𝑡ℎ 𝑚 1 Note that these rules also applied for ቀ𝑛ቁ power and rat