5.4 The Irrational Numbers 255 Example 7 Estimating Square Roots The following is a sketch of a 16-in. ruler marked using 1 2 inches. 1 2 3 4 5 6 7 8 9 101112131415 Indicate between which two adjacent ruler marks each of the following irrational numbers will fall. a) 7 b) 89 Solution a) 4 7 9 4 7 9 2 7 3 < < < < < < Therefore, 7 is between 2 and 3. Furthermore, since 7 is closer to 9 than it is to 4, 7 is closer to 3 than it is to 2. Thus, 7 is between 2.5 and 3. b) 81 89 100 81 89 100 9 89 10 < < < < < < Therefore, 89 is between 9 and 10. Furthermore, since 89 is closer to 81 than it is to 100, 89 is closer to 9 than it is to 10. Thus, 89 is between 9 and 9.5. 7 Now try Exercise 69 TECHNOLOGY TIP Approximating Square Roots on a Scientific Calculator Consider the irrational number the square root of two. We use the symbol 2 to represent the exact value of this number. Although exact values are important, approximations are also important, especially when working with application problems. We can use a scientific calculator to obtain approximations for square roots. Scientific calculators generally have one of the following square root keys: x or For simplicity, we will refer to the square root key with the symbol. To approximate 2, perform the following keystrokes: 2 ENTER or, depending on your model of calculator, you may have to just enter the following: 2 The display on your calculator may read 1.414213562. Your calculator may display more or fewer digits. It is important to realize that 1.414213562 is a rational number approximation for the irrational number 2. The symbol means ≈ is approximately equal to, and we write 2 1.414213562 ↑ ↑ ≈ Exact value (irrational number) Approximation (rational number) Note: If your calculator has the symbol printed above the key instead of on the face of the key, you can access the square root function by first pressing the “2nd” or the “SHIFT” key. Instructor Resources for Section 5.4 in MyLab Math • Objective-Level Videos 5.4 • Animation: Simplifying Radicals • PowerPoint Lecture Slides 5.4 • MyLab Exercises and Assignments 5.4
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