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Lesson 3 TAKS Grade 8 Objective 1 (8.1)(C) Working with Irrational Numbers Irrational numbers are numbers that cannot be written in the form a b where a is any integer and b is any nonzero integer. Together, rational and irrational numbers form the set of real numbers. Approximating Irrational Numbers Pi (p) is the special name for the ratio of a circle’s circumference C to its diameter d. New Vocabulary • irrational numbers • real numbers • pi (p) pC d If you solve this equation for C, you get C pd, a formula for the circumference of a circle. Approximate values for p are 22 7 and 3.14. The word rational has the word ratio in it. The word irrational means “not rational.” Example: If d 5 cm, then C 5p cm. Using 3.14 for p, C 5 3.14 cm 15.7 cm. If a number is not a perfect square, you can estimate its square root to the nearest integer. Example: !26 is close to !25. Since !25 5, !26 5. Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. EXAMPLE 1 Estimate the value of !53 to the nearest integer. Step 1 Find the two perfect squares with 53 between them. 49 72 and 64 82 are the closest perfect squares to 53. Step 2 Place them on a number line. 49 72 53 The symbol means “is approximately equal to.” 64 = 82 Step 3 Decide the integer to which !53 is the closest. The number 53 is closer to 49 than to 64, so !53 is closer to !49. Therefore, 7 is the closest integer to !53. Quick Check 1 1a. Estimate the value of !115 to the nearest integer. TAKS Review and Preparation Workbook 1b. What is the approximate circumference of a circle with a radius of 6.1 inches? Use 3 for p. LESSON 3 ■ Working with Irrational Numbers 7 TAKS Objective 1 (8.1)(C) LESSON 3 Problems That Involve Irrational Numbers Solve problems with irrational numbers just as you solve any other math problem. EXAMPLE 2 The area of a square is 139 square centimeters. What is the approximate perimeter of the square? Step 1 Use the formula for the area of a square to find the approximate length of a side. A s2 139 s2 !139 s Write the formula. Substitute. Simplify. The number 139 is not a perfect square, so !139 is an irrational number. Approximate the value of !139 by using a calculator. Thus, !139 11.8. Step 2 Find the approximate perimeter of the square. The length of a side of the square is approximately 11.8 cm. Perimeter 4s Write the formula. 4(11.8 cm) Substitute. 47.2 cm Simplify. The approximate perimeter of the square is 47.2 cm. 2a. The area of a square is 127 square centimeters. What is the approximate length of a side of the square? 8 LESSON 3 ■ Working with Irrational Numbers 2b. The area of a square is 131 square meters. What is the approximate perimeter of the square? TAKS Review and Preparation Workbook Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. Quick Check 2 Name__________________________Class____________Date________ 1 Which point on this number line best represents !12? W X Y 4 The area of a square is 151 square meters. Which best represents the perimeter of the square? Z F 12.3 m 2 3 4 5 6 G 49.2 m A W H 24.6 m B X J 151 m C Y D Z 2 The area of a square is 149 square feet. Which best represents the length of a side of the square? 5 If the circumference of a circle is 16 meters, what is the approximate area of the circle? Use 3 for p. A 21.33 m2 B 24 m2 F 12.2 feet C 64 m2 G 13.4 feet D 85.33 m2 H 37.3 feet Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. J 74.5 feet 3 Chase has to choose the irrational number listed below that was closest to 5. Which number should Chase choose? 6 If a circle has a radius of 8.3 inches, what is the approximate circumference of the circle? Use 3 for p. F 49.8 in. A !10 G 24.9 in. C !28 J 16.6 in. B !15 H 206.7 in. D !32 TAKS Review and Preparation Workbook LESSON 3 ■ Working with Irrational Numbers 9