. Approximating Square Roots How can you find decimal approximations of square roots that are not rational? ACTIVITY: Approximating Square Roots Work with a partner. Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer. He tried to find a rational number whose square is. Two that he tried were 65 5 and 5 780. a. Are either of these numbers equal to? Explain. b. Use a calculator to approximate. Write the number on a piece of paper. Enter it into the calculator and square it. Then subtract. Do you get 0? What does this mean? c. The value of is between which two integers? square root key d. Tell whether the value of is between the given numbers. Explain your reasoning..7 and.8.7 and.7.7 and.7 ACTIVITY: Approximating Square Roots Geometrically Work with a partner. Refer to the square on the number line below. Square Roots In this lesson, you will define irrational numbers. approximate square roots. approximate values of expressions involving irrational numbers. 0 a. What is the length of the diagonal of the square? b. Copy the square and its diagonal onto a piece of transparent paper. Rotate it about zero on the number line so that the diagonal aligns with the number line. Use the number line to estimate the length of the diagonal. c. STRUCTURE How do you think your answers in parts (a) and (b) are related? 66 Chapter Real Numbers and the Pythagorean Theorem
Math Practice Recognize Usefulness of Tools Why is the Pythagorean Theorem a useful tool when approximating a square root? ACTIVITY: Approximating Square Roots Geometrically Work with a partner. a. Use grid paper and the given scale to draw a horizontal line segment unit in length. Label this segment AC. b. Draw a vertical line segment units in length. Label this segment DC. c. Set the point of a compass on A. Set the compass to units. Swing the compass to intersect segment DC. Label this intersection as B. d. Use the Pythagorean Theorem to find the length of segment BC. e. Use the grid paper to approximate to the nearest tenth. D B Scale: of a unit 0 C A. Compare your approximation in Activity with your results from Activity. 5. Repeat Activity for a triangle in which segment AC is units and segment BA is units. Use the Pythagorean Theorem to find the length of segment BC. Use the grid paper to approximate 5 to the nearest tenth. 6. IN YOUR OWN WORDS How can you find decimal approximations of square roots that are not rational? Use what you learned about approximating square roots to complete Exercises 5 8 on page 65. Section. Approximating Square Roots 67
. Lesson Lesson Tutorials Key Vocabulary irrational number, p. 68 real numbers, p. 68 A rational number is a number that can be written as the ratio of two integers. An irrational number cannot be written as the ratio of two integers. The square root of any whole number that is not a perfect square is irrational. The cube root of any integer that is not a perfect cube is irrational. The decimal form of an irrational number neither terminates nor repeats. Remember The decimal form of a rational number either terminates or repeats. Real Numbers Rational numbers and irrational numbers together form the set of real numbers. Rational Integer Whole Natural 0 0. Real Numbers.5 7 Irrational EXAMPLE Classifying Real Numbers Classify each real number. Study Tip When classifying a real number, list all the subsets in which the number belongs. a. b. c. d. e. Number Subset(s) Reasoning Irrational is not a perfect square. 0. 5 Rational 0. 5 is a repeating decimal. 9 Integer, Rational 9 is equal to. 7 π Natural, Whole, Integer, Rational Irrational 7 is equal to 8. The decimal form of π neither terminates nor repeats. Exercises 9 6 Classify the real number.. 0..... 96. 68 Chapter Real Numbers and the Pythagorean Theorem
EXAMPLE Approximating a Square Root Estimate 7 to the nearest (a) integer and (b) tenth. a. Make a table of numbers whose squares are close to 7. Number 7 8 9 0 Square of Number 9 6 8 00 The table shows that 7 is between the perfect squares 6 and 8. Because 7 is closer to 6 than to 8, 7 is closer to 8 than to 9. 9 6 7 8 00 7 8 9 0 So, 7 8. b. Make a table of numbers between 8 and 9 whose squares are close to 7. Study Tip You can continue the process shown in Example to approximate square roots using more decimal places. Number 8. 8. 8.5 8.6 Square of Number 68.89 70.56 7.5 7.96 Because 7 is closer to 70.56 than to 7.5, 7 is closer to 8. than to 8.5. 68.89 70.56 7 7.5 7.96 8. 8. 8.5 8.6 So, 7 8.. Exercises 0 5 Estimate the square root to the nearest (a) integer and (b) tenth.. 8 5. 6. 7. 0 EXAMPLE Comparing Real Numbers Which is greater, 5 or? Estimate 5 to the nearest integer. Then graph the numbers on a number line. 5.6 is to the right of 5. So, 9 is greater. Section. Approximating Square Roots 69
EXAMPLE Approximating the Value of an Expression The radius of a circle with area A is approximately A. The area of a circular mouse pad is 5 square inches. Estimate its radius to the nearest integer. A = 5 Substitute 5 for A. = 7 Divide. The nearest perfect square less than 7 is 6. The nearest perfect square greater than 7 is 5. 7 6 5 5 Because 7 is closer to 6 than to 5, 7 is closer to than to 5. So, the radius is about inches. EXAMPLE h 5 Real-Life Application The distance (in nautical miles) you can see with a periscope is.7 h, where h is the height of the periscope above the water. Can you see twice as far with a periscope that is 6 feet above the water than with a periscope that is feet above the water? Explain. Use a calculator to find the distances. Feet Above Water 6 Feet Above Water.7 h =.7 Substitute for h..7 h =.7 6.0 Use a calculator..87 You can see.87. times farther with the periscope that is 6 feet.0 above the water than with the periscope that is feet above the water. No, you cannot see twice as far with the periscope that is 6 feet above the water. Exercises 6 Which number is greater? Explain. 8. 5, 9. 0, 5 0.,. The area of a circular mouse pad is 6 square inches. Estimate its radius to the nearest integer.. In Example 5, you use a periscope that is 0 feet above the water. Can you see farther than nautical miles? Explain. 650 Chapter Real Numbers and the Pythagorean Theorem
. Exercises Help with Homework. VOCABULARY How are rational numbers and irrational numbers different?. WRITING Describe a method of approximating.. VOCABULARY What are real numbers? Give three examples.. WHICH ONE DOESN T BELONG? Which number does not belong with the other three? Explain your reasoning. 5.075 8. 9+(-6)= +(-)= +(-9)= 9+(-)= Tell whether the rational number is a reasonable approximation of the square root. 5. 559 50, 5 6. 0 50, 7. 678 50, 8 8. 677 50, 5 Classify the real number. 9. 0 0.. π 6..5. 5. 8 5. 9 6. 5 7. ERROR ANALYSIS Describe and correct the error in classifying the number. is irrational. 8. SCRAPBOOKING You cut a picture into a right triangle for your scrapbook. The lengths of the legs of the triangle are inches and 6 inches. Is the length of the hypotenuse a rational number? Explain. Rational Integer Whole Natural Real Numbers Irrational 9. VENN DIAGRAM Place each number in the correct area of the Venn Diagram. a. the last digit of your phone number b. the square root of any prime number c. the ratio of the circumference of a circle to its diameter Estimate the square root to the nearest (a) integer and (b) tenth. 0. 6. 685. 6. 05. 7 5. 5 Section. Approximating Square Roots 65
Which number is greater? Explain. 6. 0, 0 9., 7. 5,.5 6 8 0. 0.5, 0.5 8., 0. 8, 9 Use the graphing calculator screen to determine whether the statement is true or false.. To the nearest tenth, 0 =... The value of is between.7 and.75.. 0 lies between. and.6 on a number line. 5. FOUR SQUARE The area of a four square court is 66 square feet. Estimate the side length s to the nearest tenth of a foot. 6. CHECKERS A checkers board is 8 squares long and 8 squares wide. The area of each square is square centimeters. Estimate the perimeter of the checkers board to the nearest tenth of a centimeter. s s Approximate the length of the diagonal of the square or rectangle to the nearest tenth. 7. 8. 9. cm 6 ft 0 in. 8 cm 8 in. 6 ft 0. WRITING Explain how to continue the method in Example to estimate 7 to the nearest hundredth.. REPEATED REASONING Describe a method that you can use to estimate a cube root to the nearest tenth. Use your method to estimate to the nearest tenth.. RADIO SIGNAL The maximum distance (in nautical miles) that a radio transmitter signal can be sentis represented by the expression. h, where h is the height (in feet) above the transmitter. Estimate the maximum distance x (in nautical miles) between the plane that is receiving the signal and the transmitter. Round your answer to the nearest tenth. 65 Chapter ms_accel_pe_0.indd 65 x,000 ft Not drawn to scale Real Numbers and the Pythagorean Theorem //5 9:5:6 AM
. OPEN-ENDED Find two numbers a and b that satisfy the diagram. 9 a b 0 Estimate the square root to the nearest tenth.. 0.9 5..9 6..5 r 6.76 m 7. ROLLER COASTER The speed s (in meters per second) of a roller-coaster car is approximated by the equation s = 6r, where r is the radius of the loop. Estimate the speed of a car going around the loop. Round your answer to the nearest tenth. 8. STRUCTURE Is a rational number? Is 6 a rational number? Explain. 9. WATER BALLOON The time t (in seconds) it takes a water balloon to fall d meters is represented by the equation d t =. Estimate the time it takes the balloon to fall.9 to the ground from a window that is meters above the ground. Round your answer to the nearest tenth. 50. Determine if the statement is sometimes, always, ays, or never true. Explain your reasoning and give an example of each. a. A rational number multiplied by a rational number is rational. b. A rational number multiplied by an irrational number is rational. c. An irrational number multiplied by an irrational number is rational. Find the missing length of the triangle. (Section.) 5. m c m 5. 0 in. 6 in. b 5. cm a 5 cm 5. MULTIPLE CHOICE What is the ratio (red to blue) of the corresponding side lengths of the similar triangles? (Section.5) A : B 5: 8 0 9.5 0 C : D :5 Section. Approximating Square Roots 65