WNCP9_SE_U3_92-93.qxd 5/27/09 2:11 PM Page 92 Suppose you are ice fishing on Blachford Lake, NWT. The temperature at midnight is –12°C. At 6 A.M. the next day, the temperature is –11°C. What must the temperature have been at some time during the night? What You’ll Learn • Compare and order rational numbers. Why It’s • Solve problems by adding, Important subtracting, multiplying, and dividing rational numbers. You have learned ways to use • Explain and apply the order of positive fractions and decimals. operations with rational numbers, In some applications, such as with and without technology. temperature, finances, and graphing on a grid, negative fractions and negative decimals are required. 92 WNCP9_SE_U3_92-93.qxd 5/27/09 2:11 PM Page 93 Key Words • rational number • irrational number 93 WNCP9_SE_U3_94-105.qxd 5/29/09 12:24 PM Page 94 3.1 What Is a Rational Number? FOCUS • Compare and order rational numbers. The label on a package of frozen cranberries says that it must be stored at a temperature between Ϫ18°C and Ϫ22°C. Name some possible temperatures. How could these temperatures be shown on a number line? Investigate 2 ➤ Determine each quotient. 12 Ϫ12 12 2 2 Ϫ2 ➤ Use what you know about integer division to determine each quotient. 11 Ϫ11 11 2 2 Ϫ2 3 Ϫ3 3 5 5 Ϫ5 ➤ On a number line, mark a point for each quotient. How can you name the point another way? Reflect Compare your strategies and answers with those of Share & another pair of classmates. Use integer division to explain what each fraction means. How could you write each answer as a decimal? 94 UNIT 3: Rational Numbers WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 95 Connect We extend a number line to the left of 0 to show negative integers. We can also represent negative fractions on a number line. Ϫ3 is the same distance to the left of 0 as 3 is to the right of 0. 4 4 We use the same symbol to represent a negative number –1 – 3 – 1 0 1 3 1 4 4 4 4 as we use for subtraction. For every positive fraction, there is a corresponding negative fraction. Ϫ3 and 3 are opposites. 4 4 Any fraction can be written as a decimal; so, for every positive decimal there is a corresponding negative decimal. –1 –0.75 –0.25 0 0.25 0.75 1 0.25 and Ϫ0.25 are opposites. Any number that can be written as a fraction with an integer numerator and Ϫ a non-zero integer denominator is a rational number; for example,3 ,3 , 3 Ϫ 4 4 Ϫ4 To visualize 3 , use a number line and think of (Ϫ3) Ϭ 4. 4 – 3 – 3 – 3 – 3 4 4 4 4 –3 –2 –1 0 Each part is Ϫ3 . 4 Ϫ So,3 is the same as Ϫ3 . 4 4 The value of a fraction remains the same if its numerator and denominator are multiplied by the same non-zero number. Ϫ Ϫ 3 can be written as 3 1 3 Ϫ4 Ϫ4 Ϫ1 4 Ϫ Ϫ Ϫ Since 3 ϭ 3 and 3 ϭ Ϫ3, then 3 ϭϭ3 Ϫ3 Ϫ4 4 4 4 Ϫ4 4 4 3.1 What Is a Rational Number? 95 WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 96 A fraction can be written as a terminating or repeating decimal: –3– 7 –2 3 –1 013 2 7 3 3 – 2 2 3 –3–2.3 –2 –1.5 –1 011.5 2 2.3 3 Any mixed number can be written as an improper fraction: 1 13 1 13 3 ϭ and Ϫ3 ϭ Ϫ 4 4 4 4 1 1 –3 4 3 4 –4 –3 –2 –1 0 1 2 3 4 So, mixed numbers are rational numbers. Any integer can be written as a fraction with denominator 1; Ϫ for example, Ϫ12 ϭ 12 , so integers are rational numbers. 1 All these numbers are rational numbers: Ϫ3 , 0.5, Ϫ1.8, 0, Ϫ5,7 , 2, Ϫ3.3 , 13 4 3 4 – 3 13 7 –3.3 –1.8 4 0.5 4 3 –5 –4 –3 –2 –1 0 1 2 3 ◗ Definition of a Rational Number m A rational number is any number that can be written in the form n , where m and n are integers and n 0. Not all numbers can be written as fractions. For example, π and 22 are numbers that you have used in calculations but they cannot be written as fractions. These are irrational numbers. 96 UNIT 3: Rational Numbers WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 97 Example 1 Writing a Rational Number between Two Given Numbers Write 3 rational numbers between each pair of numbers. 1 1 1 1 a) 1.25 and Ϫ3.26 b) Ϫ0.25 and Ϫ0.26 c) Ϫ and d) Ϫ and Ϫ 2 4 2 4 ǠA Solution There are many rational numbers between any two given numbers. Sketch or visualize a number line in each case. a) 1.25 and Ϫ3.26 Label a number line with integers from Ϫ4 to 2. –3.26 –2.5 0.3 1.25 –4 –3 –2 –1 0 1 2 From the number line, 3 possible rational numbers are: Ϫ2.5, Ϫ1, and 0.3 b) Ϫ0.25 and Ϫ0.26 Label a number line with these rational numbers. Divide the line into 10 equal parts. –0.259 –0.255 –0.252 –0.26 –0.25 From the number line, 3 possible rational numbers are: Ϫ0.252, Ϫ0.255, and Ϫ0.259 1 1 c) Ϫ and 2 4 Label a number line from Ϫ1 to 1. Divide the line into quarters. 1 1 1 – 4 – 8 8 –1 1 011 1 – 2 4 2 From the number line, 3 possible rational numbers are: Ϫ1 , Ϫ1 , and 1 4 8 8 3.1 What Is a Rational Number? 97 WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 98 1 1 d) Ϫ and Ϫ 2 4 Label a number line from Ϫ1 to 0. Divide the line into quarters. –1 1 1 0 – 2 – 4 Write equivalent fractions for Ϫ1 and Ϫ1 with denominators of 8 2 4 to identify fractions between the two numbers. Ϫ1 ϭ Ϫ2 ϭ ϪϪ4 1 ϭ Ϫ2 2 4 8 4 8 Between Ϫ4 and Ϫ2 , there is only one fraction, Ϫ3 , with denominator 8. 8 8 8 So, write equivalent fractions with denominator 16: Ϫ1 ϭ Ϫ2 ϭ Ϫ4 ϭ ϪϪ8 1 ϭ Ϫ2 ϭ Ϫ 4 2 4 8 16 4 8 16 Divide the number line into sixteenths. 7 5 – 16 – 16 –1 1 6 1 0 – 2 – 16 – 4 From the number line, 3 possible rational numbers are: Ϫ 5 , Ϫ 6 , and Ϫ 7 16 16 16 Example 2 Ordering Rational Numbers in Decimal or Fraction Form a) Use a number line. Order these numbers from least to greatest. 0.35, 2.5, Ϫ0.6, 1.7, Ϫ3.2, Ϫ0.6 b) Order these numbers from greatest to least. Record the numbers on a number line. Ϫ3 ,,5 Ϫ10 , Ϫ1,1 7 ,8 8 9 4 4 10 3 ǠSolutions a) 0.35, 2.5, Ϫ0.6, 1.7, Ϫ3.2, Ϫ0.6 Mark each number on a number line. Ϫ0.6 ϭϪ0.666 666…; so, Ϫ0.6 ϽϪ0.6 –3.2 –0.6 0.35 2.51.7 –1–2–3–4 0 123 –0.6 For least to greatest, read the numbers from left to right: Ϫ3.2, Ϫ0.6 , Ϫ0.6, 0.35, 1.7, 2.5 98 UNIT 3: Rational Numbers WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 99 Method 1 3 5 10 1 7 8 b) Ϫ ,,Ϫ , Ϫ1, , 8 9 4 4 10 3 Visualize a number line. Consider the positive numbers:5 ,7 , 8 –3 –2 –1 0 1 2 3 9 10 3 Only 8 is greater than 1. 3 Both 5 and 7 are between 0 and 1. 9 10 To order 5 and 7 , write them with a common denominator: 9 10 9 ϫ 10 ϭ 90 5 ϭϭ50 7 63 9 90 10 90 Since 63 Ͼ 50 , then 7 Ͼ 5 90 90 10 9 Consider the negative numbers: Ϫ3 , Ϫ10 , Ϫ11 8 4 4 Ϫ11 is the improper fraction Ϫ5 , which is greater than Ϫ10 . 4 4 4 Ϫ3 is greater than Ϫ1.1 8 4 10 1 3 5 8 From greatest to least, the numbers are: – 4 –14 – 8 9 3 8 ,,,7 5 Ϫ3 , Ϫ1,1 Ϫ10 3 10 9 8 4 4 –3 –2 –1 07 1 2 3 10 Method 2 Ϫ3 ,,5 Ϫ10 , Ϫ1,1 7 ,8 8 9 4 4 10 3 Write each number as a decimal. Use a calculator when necessary. Ϫ3 ϭϪ0.375 5 ϭ 0.5 8 9 Ϫ10 ϭϪ2.5 Ϫ11 ϭϪ1.25 4 4 7 ϭ 0.7 8 ϭ 2.6 10 3 Mark each decimal on a number line. 0.5 –2.5 –1.25 2.6 Use the order of the decimals to order the fractions. –3 –2 –1 0 1 2 3 –0.375 0.7 From greatest to least, the numbers are: 8 ,,,7 5 Ϫ3 , Ϫ1,1 Ϫ10 3 10 9 8 4 4 3.1 What Is a Rational Number? 99 WNCP9_SE_U3_94-105.qxd 5/29/09 12:25 PM Page 100 Example 3 Ordering Rational Numbers in Fraction and Decimal Form Order these rational numbers from least to greatest.