Mathematics Grade 8 Student Textbook PDF
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2015
Misganaw Mintesnot, Getenet Zerihun
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This textbook is aimed at 8th-grade math students, covering topics such as rational numbers, squares, square roots, and linear equations for Ethiopian students. It seeks to engage students in the formulation and construction of mathematics knowledge and skills.
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MATHEMATICS GRADE 8 Student Textbook Writers Misganaw Mintesnot (MSc) Getenet Zerihun (MSc) Editors Adamu Ayele (MSc) Birilew Belayneh (PhD) Tilahun Abebaw (PhD) Illustrator: Bek...
MATHEMATICS GRADE 8 Student Textbook Writers Misganaw Mintesnot (MSc) Getenet Zerihun (MSc) Editors Adamu Ayele (MSc) Birilew Belayneh (PhD) Tilahun Abebaw (PhD) Illustrator: Bekalu Tarekegn (PhD) Team Leader: Berie Getie (MSc) በአማራ ብሔራዊ ክልላዊ መንግሥት ትምህርት ቢሮ The textbook is prepared by Amhara National Regional State Education Bureau and the Scholars Council with budget allocated by the Amhara National Regional State Government. ©2015 Amhara National Regional State Education Bureau. All rights reserved. The Scholars Council Rationales for the Curriculum Reform Curriculum relevance and its appropriateness to develop higher order thinking skills have been a subject of discussion in Ethiopia for many years. Various studies have been conducted to identify and propose reform ideas to the general education quality and efficiency problems. The main and most comprehensive ones are the Education Roadmap study and the Cambridge Education study. These studies indicated that the general education curriculum is staffed with many subjects; some textbooks are overloaded with factual content; school contents are not adequately related with students’ lives; and indigenous knowledge and real world problems are not integrated in the school curriculum. The studies also showed that the curriculum does not integrate ICT and has gaps to meet the needs of children with special educational needs. In addition, the studies demonstrated that curriculum materials do not thoroughly cultivate 21st Century skills and competencies such as lifelong learning, critical thinking, problem solving, creative and innovative thinking, communication and cooperative skills, leadership and decision making skills, technological skills, cultural identity and international citizenship. Consequently, these studies recommended the need to reform the curriculum. Based on the recommendations, extended discussions and consultative meetings were conducted at national and regional levels with relevant stakeholders, teachers, parents, and educational leaders. Following these consultations, a new national curriculum framework, content flow chart, learning competencies, and syllabus have been prepared national level. Based on such documents, student textbooks and teacher guides of different subjects are prepared. The mathematics textbooks intend to engage students in the formulation and construction of mathematics knowledge and skill based on their day to day experiences and previous knowledge. Learners are expected to actively take part in drawing their prior knowledge and experience in the learning process. How to Use the Book The book is prepared to enable student learn in active and participatory manner using their prior learning and experiences from the immediate environment. Students and teachers carry out activities and solve problems using their experiences and knowledge. In this process, students not only learn mathematical concepts and ideas but also develop the necessary learning to learn skills. Such practice also helps students to deeply understand the contents. To this end, teachers are expected to teach using the proposed learning and teaching strategies and learning processes. It is essential that students and teachers appreciate the processes involved in learning the contents and not merely focus on memorization of concepts and mathematical procedures. Hence, teachers are expected to employ the proposed techniques and implement all activities as they are designed by considering the objectives and contents of the textbook. In addition, teacher can select and use other methods and approaches based on students’ capacity and needs. I Dear Students! You have to use the textbook with care. Learning largely determines the future of a generation. Learning is a base for any social, human, and economic development. The textbook contents and activities are designed to promote your active participation in class. By carrying out and studying all the activities, contents and questions provided in the textbook, you are required to develop deep understandings and skills. Effort, exercise, and perseverance are important to succeed in your academic career. You will enjoy and find learning mathematics to be fun! Make sure that you bring your textbook to class and use it during the teaching and learning process. Dear Parents! Textbooks have significant roles to facilitate student learning. Thus, you are required to help and advice students to handle and use textbooks with care. Moreover, you are expected to motivate students to take textbooks to school and direct and support them to work the activities given by teachers. You should also visit your child’s school to discuss with teachers about learning and behavioral change, identify gaps, and correct them through follow up and advising. II Content Unit 1 RATIONAL NUMBERS 1 1.1 The Concept of Rational Numbers 2 1.2 Decimal and Fraction forms of Rational Numbers 6 1.3 Representation of Rational Numbers on the Number Line 10 1.4 Comparing and Ordering Rational Numbers 14 1.5 Absolute Values of Rational Numbers 19 1.6 Operations and their Properties on Rational Numbers 22 1.7 Applications of Rational Numbers in Calculating Interest and Loans 28 Unit 2 SQUARES, SQUARE ROOTS, CUBES AND CUBE ROOTS 37 2.1 Squares and Square Roots 38 2.2 Approximation of Square Roots of Rational Number 48 2.3 Cubes and Cube Roots 52 Unit 3 LINEAR EQUATIONS AND INEQUALITIES 61 3.1 Solving Linear Equations 62 3.2 Solving Linear Inequalities 71 3.3 Graph of Linear Equations and Linear Inequalities 83 Unit 4 SIMILARITY OF FIGURES 91 4.1 Similar Plane Figures 92 4.2 Tests for Similarity of Triangles 100 4.3 Perimeter and Area of similar Triangles 106 III Grade 8 MATHEMATICS Unit 5 THEOREMS ON TRIANGLES 113 5.4 Theorems of Triangles 114 5.5 Exterior Angles of a Triangle 123 5.6 Theorems on Right-Angled Triangles 126 Unit 6 LINES AND ANGLES IN A CIRCLE 135 6.1 Circles and Lines 136 6.2 Central Angles and Inscribed Angles of circles 142 6.3 Angles Formed by Two Intersecting Chords in a Circle 147 Unit 7 SOLID FIGURES AND MEASUREMENTS 157 7.1 Solid Figures 158 7.2 Surface Area of Solid Figures 165 7.3 Volume of Solid Figures 175 Unit 8 INTRODUCTION TO PROBABILITY 183 8.1 The Concept of Probability 184 8.2 Probability of Simple Event 193 IV Unit 1 RATIONAL NUMBERS Learning outcomes: After completing this unit, you will be able to: ª identify the relationships among the numbers systems: natural numbers (ℕ), whole numbers (𝕎), intergers (ℤ) and rational numbers (ℚ); ª define rational numbers; ª represent rational numbers on a number line; ª compare and order rational numbers; ª define absolute value of rational numbers; ª compute basic operations on rational numbers; ª appreciate application of rational numbers in real life problems. Key terms rational number compound interest comparing rational numbers opposite rational number operations on rational numbers ordering rational numbers terminating decimals absolute value simple interest repeating decimals Grade 8 MATHEMATICS Introduction As human experience, needs and knowledge developed, the set of integers were not sufficient to describe the day-to-day activities and thus a new set of numbers, the set of rational numbers were developed. In your previous grades, you have learned about the sets of natural numbers, the set of whole numbers, the set of integers and fractions. In this unit, you will learn about the set of rational numbers, the four fundamental operations on rational numbers and some applications of rational numbers. 1.1 The Concept of Rational Numbers In your previous grades you have learned about different number systems; the set of natural numbers, the set of whole numbers and the set of integers. Natural Numbers Natural numbers, which are also called “counting numbers”, are numbers that start with the number 1 and they do not involve negatives, fractions or decimals. You have started counting in your early ages with this set of numbers. The set of natural numbers is denoted by and it is given by: ℕ={ 1, 2, 3, 4, 5, }. Whole Numbers As the set of natural numbers was not enough to express the whole idea of counting, the idea of “zero” was developed. The set of natural numbers and zero “0” form a new class of numbers called the set of whole numbers, denoted by . That is, 𝕎 = { 0, 1, 2, 3, 4, } and observe that Í . Integers Whole numbers were not sufficient in order to represent situations like ‘profit and loss’, ‘temperatures below 0oC, 'altitudes below sea level' etc. Thus, it was obvious, to extend the set of whole numbers by including negative numbers like , 4, 3, 2, 1 and the set of integers, denoted by , was formed by combining the whole numbers and negative numbers. That is, = , 3, 2, 1, 0,1, 2, 3, Observe that, you have the following relationships between the three given sets of numbers: Í Í . Fractions In our daily life there are quantities that cannot be expressed by the set of numbers that are mentioned above. For example, if you divide a bread into three equal parts, then each part cannot be expressed using integers. Thus, it was needed to introduce fractions. Fractions are numbers which can be written in the form ba , where a and b 2 Unit 1፡Rational numbers a are whole numbers and b is not equal to zero ( b ¹ 0). In the fraction , a is called b the numerator and b is called the denominator. Operations on Fractions c Given two fractions a and , where a, b, c and d are natural numbers and b ¹ 0 b d and d ¹ 0 : a c (a × d ) + (b × c) 1 + = (Sum of fractions) b d b×d a × d b × c = a c 2 (Difference of fractions) b d b×d a c a×c 3 × = (Product of fractions) b d b×d a c a d a×d 4 ÷ = × = (Division of fractions and c ¹ 0 ) b d b c b×c Rational Numbers Definition 1.1 a A number that can be written in the form , where a and b are integers and b ¹ 0 is b called a rational number. The set of rational numbers, denoted by ℚ, is defined by: a a, b and b 0. b Example 1.1 Show that each one of the following numbers is a rational number. 1 3 5 , , , 0, 1, 5 , 0.3 2 2 6 Solution 1 a i. is a rational number as it can writen in the form where a = 1 and b = 2. 2 b 3 a ii. is a rational number as it can be written in the form , where a = 3 and b = 2. 2 b a iii. 5 is a rational number as it can be written in the form , where a = −5 and b = 6. 6 b 0 a iv. 0= , it is written in the form of , where a = 0 and b = 1 are integers; so it is 1 b a rational number; 3 Grade 8 MATHEMATICS -15 a v. -1.5 = , it is written in the form of , where a = ˗15 and b = 10; so it is 10 b a rational number; 3 a where a = 3 and b = 10; so it is a vi. 0.3 = , it is written in the form of , 10 b rational number; Note: a The rational number when b ¹ 0, is called a fraction, a is called the numerator and b b is called the denominator. Relationships between Integers and Rational Numbers In your previous grades, you have learned the following relationships: Í Í , where is the set of natural numbers, is the set of whole numbers and is the set of integers. a If a is an integer, then a = a and is a rational number. 1 1 This implies, every integer is a rational and hence the set of rational numbers includes the set of integers, that is, Í . Thus, we have the following relationships between the set of natural numbers, the set of whole numbers, the set of integers and the set of rational numbers: NÍÍÍ.These relationships are given in the following Venn diagram. ℚ ℤ ℕ Figure 1.1. Relationships of ℕ, 𝕎, ℤ and ℚ 4 Unit 1፡Rational numbers Equality of Rational Numbers Activity 1.1 Find the values of the variables that make the following fractions equal. 1 2 b c 3 e 8 12 g 36 a = = = = b = = = = 4 a 16 64 d 7 16 f 56 h a c From your responses in Activity 1.1, observe that two fractions = , where a, b, c b d a c and d natural numbers, are equal written as = , if a × d = b × c. b d The same method can be used to define the equality of two rational numbers. Definition 1.2 a c Two rational numbers and , where a, b, c and d are integers and b ≠ 0, d ≠ 0 are b d a c said to be equal, written as = , if a d = b c. b d Example 1.2 3 9 Show that the rational numbers and are equal. 4 12 Solution 3 × 12 = 36 and 4 × 9 = 36. 3 9 This implies 3 × 12 = 4 × 9 and hence =. 4 12 Example 1.3 a 3 Find the value of a so that rational numbers and are equal. 3 18 Solution a 6 If = , then a × 9 = 3 × 6. 3 9 9a = 18 18 Therefore, a = = 2. 9 5 Grade 8 MATHEMATICS Note: a a a×c Given a rational number , if c is a nonzero integer, then =. b b b×c Example 1.4 3 Find another three different forms of the rational number. 4 Solution 3 3×2 6 3 3×3 9 3 3 × 4 12 = = = = = = 4 4×2 8 4 4 × 3 12 4 4 × 4 16 6 9 Therefore, , and 12 are three different forms of 3. 8 12 16 4 Exercise 1.1 x 1 For any two integers x and y, is always a rational number? Why? y 3 x 2 If = , then what is the value of x? 8 24 5 3 Give at least three different rational numbers which are equal to 6. 4 Give five rational numbers which are not integers. 5 In a class, 47 students sat for mathematics final examination, 15 of them failed in the examination and the rest passed the examination. Write the ratio of the number students who failed in the examination to the number of students who passed in the examination. 6 Express the number of female students out of total number of students in your class as a rational number. 1.2 Decimal and Fraction forms of Rational Numbers Activity 1.2 1 Convert each of the following fractions to decimal form. 2 13 c 2 a b 8 6 6 Unit 1፡Rational numbers 2 Convert each of the following decimals to fraction form. a 4.53 b 0.26 c 0.3 From your responses in Activity 1.2, observe that you can convert a number in fraction form into a decimal form and any number in decimal form into fraction. When a given number in fraction form is changed to decimal form, there are two possibilities: 5 a the digits after the decimal point terminate; for example, = 1.25 4 b the digits after the decimal point do not terminate, but some of the digits 2 repeat themselves, for example, = 0.6666 3 Definition 1.3 A number in decimal form with only finite number of digits after the decimal point is called a terminating decimal. A number in decimal form with some digits are repeating themselves after the decimal point is called a repeating decimal. Example 1.5 a 0.375, 2.25 and 0.8 are examples of terminating decimal numbers. b 0.666… , 1.252525… and 3.2444… are examples of repeating decimals. Notation Repeating decimals can be denoted by putting: i. a dot at the top of the repeating digit, if there is only one repeating digit, for example, 0.666… = 0.6 and 3.2444…=3.24 ; etc. ii. a bar at the top of the repeating digits, if there are two or more repeating digits, for example, 1.252525… = 1.25 and 35.132132132 = 35.132 ; etc. Note To convert fractions to decimals, divide the numerator by the denominator. Converting Decimal to Fraction I. To convert a terminating decimal to fraction form, if there are k digits after the decimal 10k point, then multiply the number by and you will get a number in fraction form. 10k 7 Grade 8 MATHEMATICS Example 1.6 Convert each of the following decimals to fraction from. a 0.03 b 1.2 c 13.205 Solution 0.03 102 a 0.03 = (there are two digits after the decimal point) 102 3 = 100 1.2 10 b 1.2 = (there is one digit after the decimal point) 10 12 = 10 13.205 103 c 13.205 = (there are three digits after the decimal point) 103 = 13205 1000 II. Steps to convert repeating decimals to fractions. a Represent the repeating decimal number by a variable, say x; b If there are only k repeating digits but no nonrepeating digits after the decimal point, then multiply both sides by 10k. c Subtract the first equation from the second and then solve for the variable, the result is a fraction form of the number; d If there are n non-repeating decimals and k repeating decimals after the decimal point, first multiply x by 10n and then by 10n+k, subtract the second expression from the first expression and solve for x. Example 1.7 Convert the number 0.7 to fraction form. Solution Let x = 0.777 (1) Since it has only one repeating digit after the decimal point, multiply both sides by 10 10 x = 7.777 (2) Subtract the first equation from the second equation: 8 Unit 1፡Rational numbers 10 x x = (7.777 ) 0.777 7.777 9x = 7 0.777 7 7.0 x= 9 7 Therefore, 0.77777…. = 0.7= 9 Example 1.8 Convert the number 4.325 to fraction form. Solution Let x = 4.325 = 4.3252525… (1) There are one non-repeating and two repeating digits after the decimal point. Multiply both sides of equation (1) first by 10 and then by 103 = 1000. 10 x = 43.252525… (2) 1000x = 4325.252525…. (3) Subtract the second equation from the third equation 1000x − 10x = (4325.252525….) − (43.252525…) 990 x = 4282 4325.252525 4282 43.252525 x= 990 4282 4282 2141 Therefore, 4.3252525… = 4.325 = = 990 495 Exercise 1.2 1 Convert each of the following fraction to decimal form. 1 7 c 119 a b 3 6 20 2 Convert each of the following decimals to fraction form. a 0. 5 c 6. 345 e 1.35 b 1.5 d 2.3 12 f 35.729 9 Grade 8 MATHEMATICS 3 Show that 2.9 = 3 3 4 Are 0. 3 and equal? Why? 10 1.3 Representation of Rational Numbers on the Number Line Activity 1.3 Represent each of the following integers on a number line. a −5 b 0 c 7 To locate an integer m on a number line, first determine the sign of the number (negative or positive). a if m is a positive integer, then m is located at m units to the right of the point represented by 0, the origin, on the number line. b if m is a negative integer, then m is located at − m units to the left of point represented by 0, the origin, on the number line. Example 1.9 Locate -6 and 3 on a number line. Solution First draw a number line with an appropriate units and locate point represent by zero. Then, move 6 units to the left of 0 to locate − 6 and move 3 units to right of 0 to locate 3 as shown in Figure 1.2 Figure 1.2. A number line Locating a Rational number a Consider a rational number , where a and b are integers and b ¹ 0. b a Case 1: is a proper fraction, that is, a is closer to 0 than b on the number line. b a a There are two subcases to consider: > 0 and < 0. b b 10 Unit 1፡Rational numbers a Subcase 1: Suppose > 0. b Divide the line segment with end points represented by 0 and 1 on the number line a into b equal parts. The ath point from 0 to 1 is represented by. b Example 1.10 Locate 4 on the number line. 5 Solution 4 4 is a proper fraction and > 0. 5 5 Step 1: Draw a number line and locate the integers 0 and 1 on the number line. Step 2: Divide the line segment with end points 0 and 1 into 5 equal parts. 4 Step 3: The 4th point from 0 to 1 is the point represented by as in the Figure 1.3. 5 Figure 1.3. Locating rational numbers on the number line a Subcase 2: Suppose < 0 , where a < 0 and b > 0. b Divide the line segment with end points represented by −1and 0 on the number line a into b equal parts. The (− a)th point from 0 to −1 to the left is represented by. b Example 1.11 3 Locate 4 on the number line. Solution 3 3 4 is a proper fraction with 4 0 and b < 0. a a r =p+ Subcase 1: Suppose b > 0 and b b. Step 1: Draw a number line and locate the points by the numbers p and p+1. Step 2: Divide the line segment on the number line with end points represented by p a and p +1 into b equal parts. Then the r th point from p to p + 1 is represented by. b Example 1.12 4 Locate on the number line. 3 Solution: 4 1 First write =1+. 3 3 Then divide the line segment with end points 1 and 2 on the number line into 3 equal parts. 4 Then the first point from 1 to 2 is the point represented by. as shown below. 3 0 1 4 3 2 a a r Subcase 2: Suppose < 0 and = p + with p < 0, r < 0 and b > 0. b b b 12 Unit 1፡Rational numbers Step 1: Draw a number line and locate the points represented by p ˗ 1 and p on the number line. Step 2: Divide the line segment on the number line with end points represented by a p -1 and p into b equal parts. Then the (−r)th point from p to p 1 is represented by. b Example 1.13 7 Locate 4 on the number line. Solution 7 3 First write = 1 . 4 4 Then divide the line segment with end points −2 and −1 on the number line into 4 equal parts. 7 Then the third point from −1 and −2 is the point represented by. 4 -2 -74 -1 0 Definition 1.4 Opposite rational numbers are located on the number line in opposite directions from 0 at equal distance. That is, opposite rational numbers are numbers that have the same magnitude, but they are different in signs. Example 1.14 Determine the opposites of each of the following rational numbers. a 5 b 0.333… - 2 Solution 5 5 5 a Since - is located to the left of 0 at a distance of from 0 and is 2 2 5 2 located to the right of 0 at equal distance, we conclude that is the opposite 5 2 of - or vice versa. 2 Figure 1.5. Opposite Rational Numbers on the Number Line 13 Grade 8 MATHEMATICS 1 1 b = 0.333… is located to the right of 0 and = 0.333… is located to the left 3 3 of 0 at equal distance. Thus, the opposite of 0. 333… is 0.333…. Exercise 1.3 1 Insert at least three rational numbers between the following pairs of numbers. a 1 and 1 d 3 and 2 2 1 4 1 b and e and 0 3 5 2 1 1 c and f 0.2 and 0.3 3 5 2 Locate the following rational numbers on the number line. 14 5 a 3.9 c 3 e 5 6 b 2.5 7 d 17 f 3 8 3 How many integers are located between 0. 3 and 1.5 on the number line? Why? 4 How many rational numbers are located between 0. 3 and 1.5 on the number line? Why? 1.4 Comparing and Ordering Rational Numbers 1.4.1 Comparing Rational Numbers Activity 1.4 1 Compare the following pairs of fractions. 1 2 b 2 2 a and and 4 4 5 7 2 Compare the following pairs of decimals. a 0.122 and 0.112 b 1.321 and 2.321 If the two rational numbers are in fraction form you can compare them in two ways: i. transform the two fractions to decimal forms and compare the decimal forms or ii. transform the two fractions to fraction forms with the same denominators and compare the numerators. 14 Unit 1፡Rational numbers Comparison Methods for Rational Numbers a c Given two rational numbers and b d i. if the rational numbers have the same denominator (that is if b = d) and b > 0, then a c < if a < c; b b ii. if the rational numbers have different denominators, then convert them to rational numbers having the same denominator and compare the numerators. That is, if b > 0 a c ad cb <. and d > 0, then b < d whenever b d d b Note Any negative rational number is less than any positive rational numbers and any negative rational numbers is less than zero. Example 1.15 Compare each of the following pairs of rational numbers. a 3 and 5 7 5 5 1 c and 3 e 3 and 2 2 2 9 3 1 d 1 and 0 b and 4 2 3 Solution 3 5 a The rational numbers and have the same denominator. Since 3 2 , we have >. Therefore, 4 > 2 4 4 c Since every negative rational number is less than every positive rational 5 7 number < 3 9 1 d Since any negative rational number is less than 0, we have 3 14 > 13 > 15 So, 18 > 14 > 13 > 15 12 12 12 12 3 7 13 5 Therefore, > > > (in decreasing order) 2 6 12 4 Exercise 1.5 1 Arrange each of the following rational sets of numbers in increasing/or ascending order. 1 1 6 4 7 2 a , , , , , 2 3 13 5 18 19 1 3 1 b , , 0.5, 2, 5, 3 4 3 2 Arrange the following rational numbers in decreasing/or descending order. 3 , 0.7 , 4 , 2 , − 22 , 3.14 8 5 7 7 18 Unit 1፡Rational numbers 1.5 Absolute Values of Rational Numbers From your knowledge in the previous grades, recall that the distance between two points on the number line is the length of the straight path connecting them. Activity 1.6 1 a How far is the point represented by 12 from 0 on a number line? b How far is the point represented by ˗12 from 0 on a number line? 2 What do you observe from the questions 1(a ) and 1(b)? From your responses in Activity 1.6 observe that the distances from the origin to each of the points represented by 12 and ˗12 are equal and this distance is called the absolute of the numbers. Definition 1.5 The absolute value of a rational number x, denoted by x , is defined by: x, if x 0 x 0, if x 0 x, if x 0 Example 1.19 Find the absolute value each of the following numbers. a −28 d 0 b 7 3 c 5 Solution: a 28 < 0 implies 28 = 28 = 28 b 7 > 0 implies 7 = 7 3 3 3 3 c < 0 implies = =. 5 5 5 5 d 0 = 0, by definition of absolute value. 19 Grade 8 MATHEMATICS Note: For any rational number a, a is the distance from the origion to the point represented by a. Thus, a 0. Exercise 1.6 1 Evaluate each of the following. a 13 + 13 c 0 e 9 4 5 3 b 10 6 + 4 d 12 12 f − 4 2 Find the distance between points represented by the following rational numbers. 2 2 a 2 and 5 b −2 and 5 3 3 Equations involving absolute values Activity 1.7 1. How many points are located on the number line that are 4 units away from the origin? 2. What are the corresponding number(s) represented these point(s) in (1)? Geometrically the expression x = 3 means that the point with coordinate x is 3 units from 0 on the number line. Obviously, the number line contains two points that are 3 units from the origin: one on the right of the origin and the other on the left of the origin. Thus x = 3 has two solutions x = 3 and x = −3 Definition 1.6 For any non-negative rational number a, x = a implies x = a or x = −a Example 1.20 Solve each of the following equations. a x =8 b 2 x 1 = 11 c x = 9 20 Unit 1፡Rational numbers Solution a x = 8 implies x = 8 or x = 8 b 2 x 1 = 11 implies 2 x 1 = 11 or 2 x 1= 11 2x = 12 or 2x = 10 x = 6 or x = 5 Therefore, the solutions of the equation 2 x 1 = 11 is x = −5 or x = 6. c Since the absolute value of any number is non-negative, the equation x = 9 has no solution. That is, there is no rational number whose absolute value is −9. Properties of Absolute values Activity 1.8 Compare each of the following pairs of numbers. 3 3 a −3 and |−3| e and 4 4 b |3| and |−3| f |3 + 4| and |3| + |4| c |5| and |−5| g |3 + (−4)| and |3| + |4| d |3 (-4)| and |3| |−4| Note For any two rational numbers a and b the following relations are true. a a≤ a a a d = , b¹0 b b b |a| = |−a| c |ab| = |a||b| e |a + b| ≤ |a| + |b| Example 1.21 i 3 £ 3 because 3 = 3 and 3 £ 3 ii 3 = 3 because 3 = 3 and 3 = 3 iii 2 × 5 = 2 × 5 because 2 × 5 = 10 = 10 and 2 × 5 = 2 × 5 = 10 iv 2 × 5 = 2 × 5 because 2 5 10 10 and 2 5 2 5 10 3 3 v = because 3 = 3 4 4 4 4 21 Grade 8 MATHEMATICS vi 3 + 5 ≤ 3 + 5 because 3 + 5 = 2 and 3 + 5 = 3 + 5 = 8 Exercise 1.7 1 Evaluate each of the following expressions. a 6 x + 2 x – 3 when x = 3 b m m + 3 when m = 1 2 x 7 c when x = 3 d x 4 5 y when x = 2 and y = 6 e x + y when x = 3 and y = 1 f y x when y = −7 and x = 3 2 Find the possible value (s) of x that make the given equations true. a 2x = 5 4x f x = 0 d = 10 5 b x 1 = 3 g 7 x 2 = 14 e 3x = 0 c 3x 2 = 4 1.6 Operations and their Properties on Rational Numbers In Grade 7, you have learnt about the four basic operations; addition, subtraction, multiplication and division on integers. In this section you will learn the four basic operations; addition, subtraction, multiplication and division on rational numbers and their properties. 1.6.1 Addition and Subtraction of Rational Numbers Activity 1.9 1 Find each of the following sums. 1 3 2 5 a + b + 2 2 5 3 2 Find each of the following differences. 2 3 4 2 a − b − 5 5 7 5 22 Unit 1፡Rational numbers Adding and subtracting of rational numbers is defined in the same way as addition and subtraction of fractions are defined. Definition 1.7 a c For any two rational numbers and , with b ¹ 0 and d ¹ 0 : b d a × d + b × c = a c a c a c i the sum of and ; denoted by + is defined by + b d b d b d b×d a c a c ii the difference of and ; denoted by b d is defined by a c a × d b × c b d = b d b×d Example 1.22 Compute each of the following. 3 5 3 5 13 12 a + b 8 2 c + 8 2 17 23 Solution 5 3 × 2 + 8 × 5 6 + 40 + = 3 46 a = = 8 2 8×2 16 16 5 3 × 2 8 × 5 6 40 = 3 34 b = = 8 2 8×2 16 16 c 13 12 + = 13 × 23 + 17 × 12 = 299 + 204 = 503 17 23 17 × 23 391 391 Example 1.23 3 1 A husband is 32 years old and his wife is 27 years old. What is the age 4 8 difference between the husband and his wife? Solution First change the mixed fractions to improper fractions. 3 131 1 217 32 = and 27 =. 4 4 8 8 3 1 131 217 Then 32 27 = 2 8 4 8 = 131 × 8 4 × 217 4×8 23 Grade 8 MATHEMATICS 1048 868 = 4×8 45 5 = =5 8 8 5 Therefore, the age difference between the husband and his wife is 5 years. 8 Properties of Addition of Rational Numbers Activity 1.10 Compute each of the following and compare your results. 1 3 3 1 c 1 1 a + and + + 0 and 0 + 2 2 2 2 2 2 2 5 1 2 5 1 1 1 1 1 b d + + and + + and 5 3 2 5 3 2 2 2 2 2 From your responses in Activity 1.2, observe that the two sums in each of (a), (b), (c) and (d) are equal and these results are true for any group of rational numbers. a c e Let , and , where b ¹ 0 , d ¹ 0 and f ¹ 0, be rational numbers. Then: b d f a c c a i + = + (Addition on the set of rational numbers is commutative) b d d b a c e a c e ii + + = + d + f (addition on the set of rational numbers is associative) b d f b a a a iii +0= =0 + (0 is the additive identity in addition of rational numbers) b b b a a a a is the additive inverse of a a iv + =0= + ( b b b b b b Exercise 1.8 1 Compute each of the following. a 17 + 22 d 5 26 1 g 22 + 3 3 1 19 9 13 7 5 h 2.3 − 0.2 5 1 7 e 2 + b 16 6 + 24 6 13 6 17 8 c + f 0.2 + + 21 15 8 9 24 Unit 1፡Rational numbers 3 1 2 A woman bought 3kg of rice for 57 Birr, 2kg of wheat for 45 Birr and 10kg of 4 4 1 Teff for 580 Birr. How much money did she spend in buying all the listed items? 2 1 3 Part of a 100m2 farming land is covered by lemon and orange plants. If of the 1 4 land is covered by orange, of it is covered by lemon and the rest not covered by 5 plants, then a how much of the farm land is covered by lemon and orange? b how much of the farm land is not covered by any plant? a c e 4 For any three rational numbers , and , justify whether each of the following are true or not. b d f a c e a c e = d f i b d f b e a c e a e c ii + = + + + + f b d f b f d a c e a c e + = + iii b d f b d f 1.6.2 Multiplication and Division of Rational Numbers Activity 1.11 1 Compute each of the following products. 1 5 2 5 × a 2 4 b 3 × 6 2 Compute each of the following divisions. 1 5 2 5 ÷ a 2 ÷ 4 b 3 6 Multiplication and division of rational numbers are defined in the same way as multiplications and divisions of fractions. Definition 1.8 a c For any two rational numbers and , where b ¹ 0, d ¹ 0; b d a c a c a c a×c ac × = =. i the product of and , denoted × , is defined by: b d b×d bd b d b d 25 Grade 8 MATHEMATICS a c a c ii the quotient of and , (for c ¹ 0), denoted by b ÷ d , is defined by: b d a c a d a×d ad ÷ = × = =. b d b c b×c bc Example 1.24 Compute each of the following openations. 23 1 23 −33 a − 15 × 62 c − × 15 62 34 17 28 ÷ b 5 6 d −2 ÷ 37 Solution a − 23 × 1 = (−23) × 31 −23 = 15 62 15 × 62 930 34 17 34 6 34 × 6 204 b ÷ = × = = 5 6 5 17 5 × 17 85 23 −3 (−23) × (−3) 69 c − × = = 15 62 15 × 62 930 d −2 ÷ 28 −2 37 (−2) × 37 −74 = × = = 37 1 28 1 × 28 28 Properties of Multiplication on Rational Numbers: Activity 1.12 Compute each of the following and compare your results. 1 2 2 1 1 2 1 1 2 1 1 a ´ and ´ c and 3 5 5 3 3 5 2 3 5 3 2 1 2 1 1 2 1 2 1 1 2 1 1 1 b and d and 5 3 2 5 3 2 5 2 3 5 3 2 3 From your responses in Activity 1.14, observe that the two results in each case are equal. a c e Let , and , where b0, d ¹ 0 and f ¹ 0, be rational numbers. Then b d f a c c a i × = × (multiplication of rational numbers is commutative) b d d b 26 Unit 1፡Rational numbers a c e a c e × × = × × ii b d f b d f (multiplication of rational numbers is associative) a c e a c a e iii iii. × + = × + × (multiplication is distributive over addition b d f b d b f in the set of rational numbers) a c e a e c e iv + × = × + × (multiplication is distributive over b d f b f d f addition in the set of rational numbers) a a a v ×1= =1× (1 is the multiplicative identity in the set of real numbers) b b b Note a a b ab ab b If is a rational number with a ≠ 0 and b ≠ 0, × = = = 1. Thus b b a ba ab a 1 a a is called the multiplicative inverse of and denoted by 1 b b a That is, b. b a Example 1.25 Find the multiplicative inverse of each of the following numbers. 3 b 7 c 0.25 a 5 6 Solution 3 1 5 a c 1 4 and 0.25 = = 4 1 5 3 0.25 4 1 7 1 6 b = 6 7 Exercise 1.9 1 Compute each of the following. 4 12 13 177 14 15 4 ÷ 24 4 12 2 1 12 7 14 7.5 4 5 2 1 a × 5 ÷c) × dd)× 4 8 × ×÷ 3 e) 4 × e)×4 f )×42÷gc) × g)h)1 2 5 7 9 63 5 37 3 33 8 9 163 2 3 3 4 12 13 13 17717 14 41515 2 1 1 1 2 27 5 5 × b c) ÷ × ÷ e) 4 × 24 ×d) d) ÷ ÷e24 f)2f4) 4 ÷ ÷g) × h)1 h3h)1.5.5 ÷ ÷ 5 7 5 5 4 84 9 36 6 3 3 3 3 3 316 2 2 4 12 13 17 7 14 154 2 11 2 7 5 × ÷ cc) × d) × ÷ 24 e) 4 f ×f )42 ÷ g) h)1 3÷ × .5 5 7 5 4 8 9 63 3 33 3 16 2 28 4 2 If the product of two numbers is - and one of the number is - , then find the other number. 27 9 27 Grade 8 MATHEMATICS 3 Compute each of the following 7 13 17 2 1 3 5 a 16 + 5 × 20 b × c 1.5 ÷ 2 × 0.2 0.3 + 0.7 3 2 2 4 Find the multiplicative inverse of each of the following numbers. 9 25 c 1.35 d −21.3 a b 11 29 5 Five students divided 1.45 meters long sugarcane into five equal parts. How many meters of sugarcane does each student get? 6 A photograph measuring 8cm by 4cm is enlarged by a ratio of 11:4. What are the dimensions of the new photo? 1.7 Applications of Rational Numbers in Calculating Interest and Loans There are different real life situations involving the set of rational numbers. Some of these situations are related to shares, interests, loans, etc. In this section, you will learn some applications of rational numbers, simple interest and compund interest. 1.7.1 Simple Interest Amount of money that a person or a company borrows from a bank or a financial institution to his/her needs is called a loan. Some examples of loan are home loans, car loans, education loans, personal loans etc. A loan is required to be returned by the person or the company that borrows it to the financial institution on time with an extra amount, which is called the interest of the loan. The interest could be simple interest; it is an amount that will be paid based on a certain rate only on the principal amount borrowed from the given institution. In simple interest, the principal amount is always the same. Interest could also be paid for individuals for saving money in banks or financal institutions. Activity 1.13 Suppose your father saves 1000 Birr every month with a rate of 7% per year in a certain microfinance. If the interest is calculated only for the principal amount; a what will be the total interest in one year? b what will be the total amount at the end of two years? 28 Unit 1፡Rational numbers From your responses in Activity 1.13, observe that, to determine the amount of interest for a given principal amount, you have to multiply the given principal amount with the interest rate and with the given amount of time. Definition 1.9 If a principal amount P is invested with a simple interest rate R in T years, then the total interest I after T years is given by: I = PRT The total amount A after T years is the principal plus the interest and it is given by: A=I+P Example 1.26 A textile factory takes a loan of 100,000 Birr from microfinance for a period of 2 years with a simple interest rate of 10% per year. Find the total interest and the total amount the factory has to pay at the end of the two years. Solution The principal amount is P = 100,000 Birr, the Interest Rate is R = 10% and the number of years is T = 2 Years. The amount of interest that has to be paid in 2 years is given by I = PRT = 100, 000Birr × 10% per Year × 2 Years =100, 000 × 0.1 × 2 = 20, 000 Birr The total amount that the factory las to play to the microfinance at the end of two years will be A=P+I = 100, 000 Birr + 20, 000 Birr =120, 000 Birr. Example 1.27 A farmer paid a total amount of 9100 Birr to the amount 7000 Birr which he borrowed from a certain microfinance with a simple interest for 2 years. Find the rate of interest. Solution Given A = 9100 Birr, P = 7000 Birr and T = 2 years. You are required to find I and R 29 Grade 8 MATHEMATICS I = A – P = 9100 Birr 7000 Birr = 2100 Birr And, I = PRT implies I 2100 Birr R= = × 100% PT 7000 Birr × 2 2100 = × 100% = 15% 14000 Therefore, the rate of the simple interest was 15%. Exercise 1.10 1 Find the principal which earns Birr 115.38 in 8 years at a rate 4% simple interest per year. 2 Find the time in which Birr 1680 will earn simple interest of Birr 290 at a rate of 5% per year. 3 Find the rate per year at which Birr 380 earns simple interest of birr 128.25 in 7 years and 6 months. 4 A private limited company borrows Birr 800,000 for 2 years at a simple interest rate of 12%. What is the total amount that must be repaid at the end of two years? 1.7.2 Compound Interest Activity 1.14 Suppose your mother saved 3000 Birr in a bank with interest rate 7% per year. a Calculate the simple interest after 2 years. b If the interest is calculated for both saved amount and interest added on each year, calculate the interest after 2 years. From your responses in Activity 1.14, Question (b), observe that, interest is paid for the principal amount and interest earned during the previous period; the interest paid in such cases is called compound interest. Definition 1.10 If P amount is invested at a rate of r% compounded annually, the compound interest at the end of the nth year is computed by r nt A P 1 n where A is the total amount, p is principal, r is interest rate, t is time and n is number of times interest is compounded per unit time. 30 Unit 1፡Rational numbers Example 1.28 Find the amount when Birr 2000 is invested for 3 years with an interest of 6% compounded annually. Solution Since it is a compound interest, we must find the interest at the end of each year and add it to the principal before computing the interest for the next years. 1st year interest I = PRT = 2000 × 0.06 = 120 Birr The Amount at the end of the 1st year is 2000 Birr + 120 Birr= 2120 Birr 2nd year Interest I = PRT = 2120 Birr × 0.06 =127.20 Birr The Amount at the end of the 2nd year is 2247.20 Birr 3rd year interest I = PRT = 2447.20 Birr × 0.06 =134.80 Birr Therefore, the Amount at the end of the 3rd year is 2382 Birr Using the formula for compound interest, this can be calculated as: nt æ rö A = P çç1 + ÷÷÷ çè nø 1×3 6 = 2000 Birr 1 + 100 × 1 = 2000 Birr 1 + 0.06 3 = 2000 Birr 1.06 » 2382Birr 3 Example 1.29 Find the amount at the end of 5th year if 2000 Birr is borrowed at 5% interest compounded annually. Solution Given P = 2000 Birr, r = 5%, t = 5 years, n = 1 Required A = ? nt 5 r 5 A = P 1 + = 2000 Birr 1 + n 100 = 2000 Birr 1 + 0.05 5 31 Grade 8 MATHEMATICS =2000 Birr 1.05 =2552.56 Birr 5 Therefore, the amount at the end of 5th year is 2552.56 Birr. Exercise 1.11 1 Find the amount of 50,000 Birr at a rate of 12% compounded annually at the end of 3 years. 2 A company increases its capital by 10% each year. If it starts with 100,000 Birr capital, how much Birr will the company have at the beginning of the fourth year? 3 A small scale enterprise borrows 100,000 Birr to start business. The enterprise borrows the money at 15% interest and repays it in full after three years. How much interest will the enterprise pay? 4 W/ro Emebet saved 1000 Birr in Commercial bank of Ethiopia at 7% interest rate compounded annually. How much money will she get at the end of 5 years? 32 Unit 1፡Rational numbers Unit Summary 1 Terminating and repeating decimals are rational numbers. 2 Negative rational numbers are located to left of zero on the number line, whereas positive rational numbers are located to right of zero on the number line. 3 is closed under the operations "+", " " & "×" , but not "÷". 4 is closed under the operations "+", " " & "×" , but not "÷". 5 The set of nonzero rational numbers is closed under the operations "+", " " , "×" & "÷". 6 x = x for all rational numbers. x Î a c e 7 For any rational numbers , , , b d f a c ad + cb a c ac a b + d = bd c b × d = bd a c ad cb a c a d ad = b b d bd d b ÷ d = b × c = bc 8 The sum of two opposite rational numbers is 0 a c e 9 For any rational numbers , , Î when b, d , f ¹ 0 , we have b d f a c c a a + = + b d d b a c c a b b × d = d × b a c e a c a e c × + = × + × b d f b d b f a c e a c e d b + d + f = b + d + f a c e a c e e × × = × × b d f b d f 33 Grade 8 MATHEMATICS 10 Rules of signs for Addition: Let a and b are rational numbers, then a a+b> 0 if either both are positive or one of the summands is greater than opposite of the other b a + b < 0. if either both are negative or one of the summands is less than opposite of the other. 11 Rules of signs of multiplication: Let a and b are rational numbers, then a If a > 0 and b > 0 ,then ab > 0 c If a > 0 and b < 0 , then ab < 0 b If a 0 and b 0 , then ab < 0 d If a < 0 and b < 0 , then ab > 0 12 Rules of signs of division: Let a and b b ¹ 0 are rational numbers, then a a If a > 0 and b > 0, then >0 b a b If a > 0 and b < 0, then b < 0 a c If a < 0 and b > 0, then < 0 b a d If a < 0 and b < 0 then >0 b a c 13 For any two rational numbers and b d a c = if ad = bc b d 14 The simple interest I = PRT is computed for only saved or deposited amount, R nT whereas the compound interest A=P is computed for both saved or 1+ n deposited amount and interest added, where P is principal, R is interest rate, T is time and n is number of times the interest is compounded per unit time. 34 Unit 1፡Rational numbers Review Exercises 1 Convert each of the following decimals to fraction form. aa) 56. 42358 a) 56. 42358 b) 6.73 c)1.3256 cc)1.3256 b) 6.73 d) ( 2 ( 2 + 0.2 d) + 0.2 3 3 2 2 a) 56. 42358 a) 56. 42358 bb) 6.73 b) 6.73 c)1.3256 c)1.3256 d)d ( + 0.2 d) ( + 0.2 3 3 2 Locate each of the following rational numbers on the number line. 7 a c 1.2 3 2 d 1.6 b - 5 3 Arrange the following sets of rational numbers in ascending order. 2 27 5 1 aa) , , , 0, 3 9 3 2 3 23 23 17 bb)1.25, , 1.25 , , 2 30 5 6 c 6.0, 0.6, 0.66, 0.606, −0.06, −6.6, 6.606 4 Insert at least three rational numbers between the of the following pairs of rational numbers. 5 7 56 75 64 53 4 3 aa) and a)b) and and c) b) and and c c) and 13 8 13 11 87 11 7 75 7 5 5 7 6 5 4 3 a) and b b) 11 and 7 c) and 13 8 7 5 5 Evaluate each of the following operations. 5 8 dd) + 4 0 – 5 – 4 – 812 5 aa) 4 5 36÷4c) 3 1 + 7b)1 9 4 5 36÷4 bb)10 – 5 – 4 – 8 2 e 1.4 + 0.5 1 7 1 5 c c) 3 + d) + 5 8 4 9 35 Grade 8 MATHEMATICS 1 1 1 2 3 6 6 Find the simplified form of 2 + 3 × ÷ 4 + ÷ 4 12 5 7 In a Physics text book, 35% of the pages are colored. If there are 98 colored pages, how many pages are there in the whole text book? 2a 3 7 8 If 5 and are equal rational numbers, then what is the value of a ? 8 9 Suppose you want to borrow 5,000 Birr at 15% interest per year from Abay Bank for 6 months. How much interest would you pay to the bank? How much is the total money you pay to the bank at the end of 6 months? 10 W/ro. Abeba borrowed 6000Birr from a certain Micro-finance at the rate of 15% compounded annually. How much will she pay at the end of the fourth year? 36 Unit 2 SQUARES, SQUARE ROOTS, CUBES AND CUBE ROOTS Learning outcomes: After completing this unit, you will be able to: ª understand the notion of square, square root, cube and cube root. ª determine the square and cube of rational numbers. ª determine the square roots and cube root of rational numbers. ª approximate square/cube roots of rational numbers by using table values and scientific calculators. ª apply the concept of squares, square roots, cubes and cube roots in the real-life problems. Key terms square cube perfect square perfect cube square roots cube root approximate value closest whole number table value scientific calculator radical sign base decimal place exponent Grade 8 MATHEMATICS Introduction In your previous grades you have been working with numbers. When working on numbers you have seen how to multiply a number by itself, which is squaring the number; finding a number whose square is a certain given number, finding a square root. You have also learned also how to multiply a number by itself three times, which is called cubing the number and finding a number whose cube is a certain given number. In this unit you will learn squaring a number, finding square root of a number, cubing a number and finding cube root of a given number. Squares, square roots, cubes and cube roots of numbers are commonly used in banking, physics and geometry. 2.1 Squares and Square Roots 2.1.1 Square of a Rational Number Activity 2.1 A farmer has planted avocado trees in a square pattern as shown in Figure 2.1 below. Figure 2.1. If there are 20 rows and 20 avocado trees in each row, find the total number of avocado trees that the farmer has planted. In Activity 2.1, to determine the total number of avocado trees that the farmer has planted, you have to multiply the number of rows, which is 20, with the number of avocado trees in each row, which is again 20. In the process, you have to multiply 20 by 20, multiplying 20 by itself. This process is called squaring a number. 38 Unit 2: Squares, square roots, cubes and cube roots Definition 2.1 The process of multiplying a number by itself is called squaring the number. The product of a number y and itself is called the square of the number, denoted by y´y=y2 and read as “y squared” or “y to the power of 2”. In the expression below, for two rational numbers a and r if is the square of a is r, then is called the base (the number to be multiplied with itself) and 2 is called an exponent (number of times that the number appears in the product). Base Exponent r = a ´ a = a2 Standard Factor Power numeric form form form Example 2.1 Show that each of the following numbers is a square number: 0, 1, 4, 9, 16, 25. Solution 02 = 0 × 0 =0 12 = 1 × 1=1, 22 = 2 × 2 = 4 32 = 3 × 3= 9 42 = 4 × 4 = 16 52= 5 × 5 = 25 This implies, 0, 1, 4, 9, 16, 25 are square numbers. Activity 2.2 Complete the following table. y 1 3 4 5 6 7 9 12 13 16 20 y+y 2 6 8 10 12 y ´y 1 9 16 25 36 Explain the differences between y + y and y × y. 39 Grade 8 MATHEMATICS From your responses in Activity 2.2, observe that for each of the given numbers y + y and y × y are different. When a number is multiplied by itself we get another number which is the product of two equal numbers, for example 3 × 3=32=9. On the other hand when a number is added onto itself, we get another number which is the sum of two equal numbers, for example, 3 + 3=2 × 3=6. Note In general, for a number y, y + y =2y and y × y = y2 are different Definition 2.2 A whole number y is called a perfect square or a square number if it is the square of a certain whole number x, that is y and x are whole numbers and y =x2 Example 2.2 Show that the whole numbers 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 are perfect squares and the whole numbers 12, 23 and 232 are not perfect squares. Solution 02 = 0 × 0 =0, 12 = 1 × 1=1, 22 = 2 × 2 = 4 32 =3 × 3= 9, 42 = 4 × 4 = 16, 52=5 × 5=25, 62 = 6 × 6 = 36, 72 = 7 × 7= 49, 82 = 8 × 8=64 92 = 9 × 9 = 81, 102 =10 × 10 = 100, 112=11 × 11= 121, 12 =12 × 12 =144 13 = 13 × 13 = 169. 2 2 This implies the numbers 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 are perfect squares. i. 32 = 9, 42=16 and 9 < 10 < 16 implies there is no whole number whose square is 12. ii. 42=16, 52 =25 and 16 < 23 < 25 implies there is no whole number whose square is 23. iii. 152=225, 162 =256 and 225 < 232 < 256 implies there is no whole number whose square is 232. Therefore, the whole numbers 12, 23 and 232 are not perfect squares. Use of Prime Factorization to Determine Perfect Squares Activity 2.3 1 Find the prime factorization of each of the following numbers. a 36 c 400 b 194 d 1000 2 Which of the numbers in Question 1 above can be written as a product of two identical sets of prime factors? 40 Unit 2: Squares, square roots, cubes and cube roots In your responses in Activity 2.3, observe that 36 can be written as a product of two identical sets of prime factors and 400 also can be written as a product of two identical sets of prime factors. The numbers 36 and 400 are both perfect squares. You can use prime factorization of a given whole number to determine whether the given whole number is a perfect square or not. The following steps can be used for this purpose. Step 1: First find the prime factorization of the number; Step 2: if possible, arrange the factors so that the number is a product of two identical sets of prime factors; Step 3: if Step 2 is possible, then the given number is a perfect square and if Step 2 is not possible, then the number is not a perfect square. Example 2.3 Use prime factorization to determine if the following numbers are perfect squares. a 144 b 1250 c 62500 Solution a 144 = 2 × 2 × 2 × 2 × 3 × 3 = (2 × 2 × 3) × (2 × 2 × 3) =12 × 12 =144. Since the factors can be arranged so that 144 is a product of two identical set of prime factors, (2 × 2 × 3) × (2 × 2 × 3), we conclude that 144 is a perfect square. b 1250 = 5 × 5 × 5 × 5 × 2. Since the factors cannot be arranged as a product of two identical set of prime factors, 1250 is not a perfect square. 1250 is not a perfect square 62500 = 5 × 5×5×2 × 5× 5×5×2 = 250×250 = 250. Since the 2 c factors can be arranged as a product of two identical set of prime factor, 62500 = 5 × 5×5×2 × 5× 5×5×2 the 250 is a perfect square. 2 number=62500 = 250×250 Note 1 If a given whole number is a square number, then its unit’s digit is either 0,1,4,5,6 or 9. 2 No square whole number ends with an odd number of zeros, for example 10, 1000, and 100000 are not square numbers. 41 Grade 8 MATHEMATICS The Square of a Rational Number in Fraction Form a In Unit 1, you have learned that, rational numbers can be written in the form , where a a b and b are integers and b ¹ 0. Therefore, squaring the rational number is: b a 2 a a a×a a 2 = × = = 2. b b b b×b b Example 2.4 Find the square of each of the following rational numbers. 1 6 10 20 a b - c d 3 5 11 19 Solution 1 2 12 1 10 2 102 100 a = 2 = c = 2= 3 3 9 11 11 121 6 2 -6 36 2 20 2 202 400 b - = = d = 2 = 19 19 361 5 52 25 The Square of a Rational Number in Decimal Form If a rational number is written in decimal form you can find the square of the number by multiplying the number by itself. Example 2.5 Find 3.24 23.7 2 2 a b Solution a 3.24 b 23.7 × 3.24 × 23.7 1296 1659 648 711 972 474 10.4976 561.69 Therefore, 3.24 =10.4976 and 23.7 =561.69. 2 2 42 Unit 2: Squares, square roots, cubes and cube roots Exercise 2.1 1 Find the square of each of the following numbers. a 17 c 0.9 b -8 1 d 6 2 Express the following numbers in power form and identify the base and exponent. a 36 c 0.49 b 64 1 d 100 3 Which of the following numbers are perfect squares? a 0.09 e 0.25 i 0.3 b 0.9 f -4 j 0.04 c 0.81 g 2 d 8 h 81 4 Find two whole numbers whose squares have a sum of 45. 5 List all square numbers less than 100. 6 Show that the difference between the 7th square whole number and the 4th square whole number is a multiple of 3. 7 Show that the difference between any two consecutive square natural numbers is an odd natural number. Use of Table of Squares In the previous discussion, you have seen that the square of a rational number is a nonnegative rational number and you can determine the square of a rational number by applying the usual procedure of multiplication. Applying the usual procedure of multiplication to find the square of a rational number is sometimes tedious and time consuming. For this reason, tables of squares of numbers are prepared. In such Tables of squares find the page with the formula y = x2. The first column is headed by “x”, which lists the numbers from 1.0 to 9.9 and the first row contains numbers from 0 to 9. Now if you want to determine the square of a number from the Tables of squares, the procedures are illustrated with the following example. 43 Grade 8 MATHEMATICS Example 2.6 Find 3.85 from the Table of squares. 2 Solution Step 1: In the first column which starts with x find the row which starts with 3.8; Step 2: In the first row, which starts with x, find the column which starts with 5; Step 3: Read the number in the intersection of the row which starts with 3.8 and the column which starts with 5, that is 14.82, hence 3.85 14.82. 2 x 0 1 2 3 4 5 6 8 9 1.0... 3.8 14.82... 9.9 Figure 2.2. Table of Squares Note 1 The steps 1 to 3 can be shortened by “3.8 under 5; 2 The values you get from table of squares are mostly approximates. 3 In the Table of squares, only the squares of numbers from 1.00 to 9.99 are given. As you have learned in Unit 1, a rational number can be written as a number between 1 and 10 and powers of 10, for example 2345 = 2.345 × 1000. To find the square of a number that is not in this range by using Table of squares first write the number as product of a number between 1 and 10 and a power of 10. 44 Unit 2: Squares, square roots, cubes and cube roots Example 2.7 Complete each of the following using Table of squares. 32.4 567 2 2 a b Solution a 32.4 =3.24 ´10 and from the Table of squares 3.24 » 10.50. 2 Therefore, 32.4 = 3.24 ×102 » 10.50×100=1050. 2 2 567 = 5.67 ×100 and from Table of squares, 5.67 » 32.149. 2 2 2 b 2 Therefore, 567 = 5.67 ×100 » 32.149×10000=321490. 2 2 2 Use of Scientific Calculators You can use scientific calculators to find squares of numbers. Consider a scientific calculator as shown in Figure 2.3. Steps on how to use a scientific calculator to determine square of a given number: Step 1: Write the number on a scientific calculator, by pressing the digits on the calculator; Step 2: Press the square sign x2 on the calculator; Figure 2.3. Scientific Calculator Step 3: Round the result to the desired decimal places to approximate the given square number whenever necessary. Example 2.8 Approximate each the following numbers to two decimal places using scientific calculators. 2 2 a 3.15 c 3.132 2 2 b 3 Solution Using your scientific calculator, first press the digits of the given number and then the square sign (x2) gives the required result and finally approximate the given number to the nearest two decimal places. 3.15 2 a =9.9225 » 9.92 2 2 b First, change to decimal form: =0.666…. 3 3 Then 0.666… =0.444…» 0.44 2 c 3.132 2 =9.809424 » 9.81 45 Grade 8 MATHEMATICS Exercise 2.2 1 Find each of the following numbers using Table of squares. a (8.54)2 c (0.151)2 b (35.42)2 2 Approximate each of the following numbers using a scientific calculator. a (3.58)2 c (9230)2 b (14.68)2 2.1.2 Squa
