* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download the real numbers
Abuse of notation wikipedia , lookup
Ethnomathematics wikipedia , lookup
Law of large numbers wikipedia , lookup
History of logarithms wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Location arithmetic wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Infinitesimal wikipedia , lookup
Bernoulli number wikipedia , lookup
Non-standard analysis wikipedia , lookup
Surreal number wikipedia , lookup
Positional notation wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Hyperreal number wikipedia , lookup
Large numbers wikipedia , lookup
Real number wikipedia , lookup
dug46476_ch01.qxd 12/2/02 11:17 AM Page 7 1.2 58. E F 59. (D E ) F 60. (D F) E 61. D (E F) 62. D (F E) 63. (D F) (E F) 64. (D E ) (F E) 65. (D E) (D F) 66. (D F) (D E) 86. 1, 3, 5, 7 Use one of the symbols , , , , or in the blank of each statement to make it correct. 67. D _____ x x is an odd natural number 68. E _____ x x is an even natural number smaller than 9 3 _____ D D _____ E D F _____ F E E _____ F D _____ F F D 70. 72. 74. 76. 78. 3 _____ D D E _____ D 3 E _____ F E E _____ F E _____ F F E List the elements in each set. 79. x x is an even natural number less than 20 80. x x is a natural number greater than 6 81. x x is an odd natural number greater than 11 82. x x is an odd natural number less than 14 83. x x is an even natural number between 4 and 79 84. x x is an odd natural number between 12 and 57 In This Section ● The Rational Numbers ● Graphing on the Number Line ● The Irrational Numbers ● Intervals of Real Numbers 1.2 (1-7) 7 Write each set using set-builder notation. Answers may vary. 85. 3, 4, 5, 6 57. E F 69. 71. 73. 75. 77. The Real Numbers 87. 5, 7, 9, 11, . . . 88. 4, 5, 6, 7, . . . 89. 6, 8, 10, 12, . . . , 82 90. 9, 11, 13, 15, . . . , 51 GET TING MORE INVOLVED 91. Discussion. If A and B are finite sets, could A B be infinite? Explain. 92. Cooperative learning. Work with a small group to answer the following questions. If A B and B A, then what can you conclude about A and B? If (A B) (A B), then what can you conclude about A and B? 93. Discussion. What is wrong with each statement? Explain. a) 3 1, 2, 3 b) 3 1, 2, 3 c) 94. Exploration. There are only two possible subsets of 1, namely, and 1. a) List all possible subsets of 1, 2. How many are there? b) List all possible subsets of 1, 2, 3. How many are there? c) Guess how many subsets there are of 1, 2, 3, 4. Verify your guess by listing all the possible subsets. d) How many subsets are there for 1, 2, 3, . . . , n? THE REAL NUMBERS The set of real numbers is the basic set of numbers used in algebra. There are many different types of real numbers. To understand better the set of real numbers, we will study some of the subsets of numbers that make up this set. The Rational Numbers We have already discussed the set of counting or natural numbers. The set of natural numbers together with the number 0 is called the set of whole numbers. The whole numbers together with the negatives of the counting numbers form the set of dug46476_ch01.qxd 8 (1-8) 12/2/02 11:17 AM Chapter 1 Helpful Hint Page 8 The Real Numbers integers. We use the letters N, W, and J to name these sets: A negative number can be used to represent a loss or a debt.The number 10 could represent a debt of $10, a temperature of 10° below zero, or an altitude of 10 feet below sea level. N 1, 2, 3, . . . W 0, 1, 2, 3, . . . J . . . , 3, 2, 1, 0, 1, 2, 3, . . . The natural numbers The whole numbers The integers Rational numbers are numbers that are written as ratios or as quotients of integers. We use the letter Q (for quotient) to name the set of rational numbers and write the set in set-builder notation as follows: Q a a and b are integers, with b 0 b The rational numbers Examples of rational numbers are 7, 9 , 4 17 , 10 0, 0 , 4 3 , 1 47 , 3 and 2 . 6 Note that the rational numbers are the numbers that can be expressed as a ratio (or 7 quotient) of integers. The integer 7 is rational because we can write it as . 1 Another way to describe rational numbers is by using their decimal form. To obtain the decimal form, we divide the denominator into the numerator. For some rational numbers the division terminates, and for others it continues indefinitely. These examples show some rational numbers and their equivalent decimal forms: Calculator Close-Up Display a fraction on a graphing calculator, then press ENTER to convert to a decimal. The fraction feature converts a repeating decimal into a fraction. Try this with your calculator. 26 0.26 100 4 4.0 1 1 0.25 4 2 0.6666 . . . 3 25 0.252525 . . . 99 4177 4.2191919 . . . 990 Terminating decimal Terminating decimal Terminating decimal The single digit 6 repeats. The pair of digits 25 repeats. The pair of digits 19 repeats. Rational numbers are defined as ratios of integers, but they can be described also by their decimal form. The rational numbers are those decimal numbers whose digits either repeat or terminate. E X A M P L E 1 Subsets of the rational numbers Determine whether each statement is true or false. a) 0 W b) N J c) 0.75 J d) J Q Solution a) True, because 0 is a whole number. b) True, because every natural number is also a member of the set of integers. c) False, because the rational number 0.75 is not an integer. ■ d) True, because the rational numbers include the integers. dug46476_ch01.qxd 12/2/02 11:17 AM Page 9 1.2 Calculator Close-Up These Calculator Close-ups are designed to help reinforce the concepts of algebra, not replace them. Do not rely too heavily on your calculator or use it to replace the algebraic methods taught in this course. (1-9) 9 The Real Numbers Graphing on the Number Line To construct a number line, we draw a straight line and label any convenient point with the number 0. Now we choose any convenient length and use it to locate points to the right of 0 as points corresponding to the positive integers and points to the left of 0 as points corresponding to the negative integers. See Fig. 1.4. The numbers corresponding to the points on the line are called the coordinates of the points. The distance between two consecutive integers is called a unit, and it is the same for any two consecutive integers. The point with coordinate 0 is called the origin. The numbers on the number line increase in size from left to right. When we compare the size of any two numbers, the larger number lies to the right of the smaller one on the number line. 1 unit 4 1 unit Origin 3 2 1 0 1 2 3 4 FIGURE 1.4 It is often convenient to illustrate sets of numbers on a number line. The set of integers, J, is illustrated or graphed as in Fig. 1.5. The three dots to the right and left on the number line indicate that the integers go on indefinitely in both directions. …4 3 2 1 0 1 2 3 4 … FIGURE 1.5 E X A M P L E 2 Study Tip Take notes in class. Write down everything you can. As soon as possible after class, rewrite your notes. Fill in details and make corrections. Make a note of examples and exercises in the text that are similar to examples in your notes. If your instructor takes the time to work an example in class,it is a“good bet”that your instructor expects you to understand the concepts involved. Graphing on the number line List the elements of each set and graph each set on a number line. a) x x is a whole number less than 4 b) a a is an integer between 3 and 9 c) y y is an integer greater than 3 Solution a) The whole numbers less than 4 are 0, 1, 2, and 3. Figure 1.6 shows the graph of this set. 3 2 1 0 1 2 3 4 5 FIGURE 1.6 b) The integers between 3 and 9 are 4, 5, 6, 7, and 8. The graph is shown in Fig. 1.7. 1 2 3 4 5 6 7 8 9 FIGURE 1.7 c) The integers greater than 3 are 2, 1, 0, 1, and so on. To indicate the continuing pattern, we use a series of dots on the graph in Fig. 1.8. 5 4 3 2 1 0 FIGURE 1.8 1 2 3 … ■ dug46476_ch01.qxd 10 12/2/02 (1-10) 11:17 AM Chapter 1 Page 10 The Real Numbers The Irrational Numbers Some numbers can be expressed as ratios of integers and some cannot. Numbers that cannot be expressed as a ratio of integers are called irrational numbers. To better understand irrational numbers consider the positive square root of 2 (in sym). The square root of 2 is a number that you can multiply by itself to get 2. bols 2 So we can write (using a raised dot for times) Calculator Close-Up A calculator gives a 10-digit rational approximation for 2. Note that if the approximate value is squared, you do not get 2. The screen shot that appears on this page and in succeeding pages may differ from the display on your calculator. You may have to consult your manual to get the desired results. Circumference Diameter Circumference Diameter FIGURE 1.9 C D 2 2. 2 If we look for 2 on a calculator or in Appendix B, we find 1.414. But if we multiply 1.414 by itself, we get (1.414)(1.414) 1.999396. is not equal to 1.414 (in symbols, 2 1.414). The square root of 2 is apSo 2 proximately 1.414 (in symbols, 2 1.414). There is no terminating or repeating decimal that will give exactly 2 when multiplied by itself. So 2 is an irrational , 5, and 7 are also number. It can be shown that other square roots such as 3 irrational numbers. In decimal form the rational numbers either repeat or terminate. The irrational numbers neither repeat nor terminate. Examine each of these numbers to see that it has a continuing pattern that guarantees that its digits will neither repeat nor terminate: 0.606000600000600000006 . . . 0.15115111511115 . . . 3.12345678910111213 . . . So each of these numbers is an irrational number. Since we generally work with rational numbers, the irrational numbers may seem to be unnecessary. However, irrational numbers occur in some very real situations. Over 2000 years ago people in the Orient and Egypt observed that the ratio of the circumference and diameter is the same for any circle. This constant value was proven to be an irrational number by Johann Heinrich Lambert in 1767. Like other irrational numbers, it does not have any convenient representation as a decimal number. This number has been given the name (Greek letter pi). See Fig. 1.9. The value of rounded to nine decimal places is 3.141592654. When using in computations, we frequently use the rational number 3.14 as an approximate value for . The set of irrational numbers I and the set of rational numbers Q have no numbers in common and together form the set of real numbers R. The set of real numbers can be visualized as the set of all points on the number line. Two real numbers are equal if they correspond to the same point on the number line. See Fig. 1.10. 2.99 3 –– √ 2 2 1 1 — 3 1 — 2 0 –– √2 1 –– √5 2 3 FIGURE 1.10 Figure 1.11 illustrates the relationship between the set of real numbers and the various subsets that we have been discussing. dug46476_ch01.qxd 12/2/02 11:17 AM Page 11 1.2 Study Tip The Real Numbers 1-11) 11 Real numbers (R) Start a personal library. This book as well as other books from which you study should be the basis for your library. You can also add books to your library at garage sale prices when your bookstore sells its old texts. If you need to reference some material in the future, it is much easier to use a book with which you are familiar. Rational numbers (Q) 2, — 5 155 — —, — —, 3 7 13 Irrational numbers (I) 5.26, 0.37373737… –– –– –– √2 , √6 , √7 , Integers (J) 0.5656656665… Whole numbers (W) Counting numbers (N) …, 3, 2, 1, 0, 1, 2, 3, … FIGURE 1.11 E X A M P L E 3 Classifying real numbers Determine which elements of the set 7, 4, 0, 5, , 4.16, 12 1 are members of each of these sets. a) Real numbers b) Rational numbers c) Integers Solution a) All of the numbers are real numbers. 1 b) The numbers 4, 0, 4.16, and 12 are rational numbers. c) The only integers in this set are 0 and 12. E X A M P L E 4 ■ Subsets of the real numbers Determine whether each of these statements is true or false. Q b) J W c) I Q d) 3 N a) 7 e) J I f) Q R g) R N h) R Solution a) False e) True b) False f) True c) True g) False d) False h) True ■ Intervals of Real Numbers (2, 3) 0 1 2 3 4 5 FIGURE 1.12 An interval of real numbers is the set of real numbers that lie between two real numbers, which are called the endpoints of the interval. Interval notation is used to represent intervals. For example, the interval notation (2, 3) is used to represent the real numbers that lie between 2 and 3 on the number line. The graph of (2, 3) is shown in Fig. 1.12. Parentheses are used to indicate that the endpoints do not belong dug46476_ch01.qxd 12 12/2/02 (1-12) 11:17 AM Chapter 1 The Real Numbers to the interval, whereas brackets are used to indicate that the endpoints do belong to the interval. The interval [2, 3] consists of the real numbers between 2 and 3 including the endpoints. It is graphed in Fig. 1.13. The infinity symbol may also be used as an endpoint, even though it does not represent a number. The interval (3, ) consists of the real numbers greater than 3 and extending infinitely far to the right on the number line. See Fig. 1.14. The interval (, 3) consists of the real numbers less than 3, as shown in Fig. 1.15. The entire set of real numbers is written in interval notation as (, ) and graphed as in Fig. 1.16. Parentheses are always used next to and . [2, 3] 0 1 2 3 4 Page 12 5 FIGURE 1.13 (3, ) (, 3) 0 1 2 3 4 5 0 FIGURE 1.14 E X A M P L E 5 1 2 3 4 (, ) 32 1 0 1 2 3 5 FIGURE 1.15 FIGURE 1.16 Interval notation Write each interval of real numbers in interval notation and graph it. a) The set of real numbers greater than or equal to 2 b) The set of real numbers less than 3 c) The set of real numbers between 1 and 5 inclusive d) The set of real numbers greater than or equal to 2 and less than 4 Solution a) The set of real numbers greater than or equal to 2 includes 2. It is written as [2, ) and graphed in Fig. 1.17. b) The set of real numbers less than 3 does not include 3. It is written as (, 3) and graphed in Fig. 1.18. c) The set of real numbers between 1 and 5 inclusive includes both 1 and 5. It is written as [1, 5] and graphed in Fig. 1.19. d) The set of real numbers greater than or equal to 2 and less than 4 includes 2 but not 4. It is written as [2, 4) and graphed in Fig. 1.20. (, 3) [2, ) 0 1 2 3 4 5 FIGURE 1.17 5 4 3 2 1 FIGURE 1.18 [2, 4) [1, 5] 0 1 2 3 4 5 6 FIGURE 1.19 0 1 2 3 4 5 6 FIGURE 1.20 ■ Intervals are sometimes graphed using a hollow circle to indicate that an endpoint is not in the interval and a solid circle to indicate that an endpoint is in the interval. The advantage of using parentheses and brackets on the graph is that they match the interval notation and the interval notation looks like an abbreviated version of the graph. dug46476_ch01.qxd 12/2/02 11:17 AM Page 13 1.2 (1-13) 13 The Real Numbers The intersection of two intervals is the set of real numbers that belong to both intervals. The union of two intervals is the set of real numbers that belong to one, or the other, or both of the intervals. E X A M P L E 6 Combining intervals Write each union or intersection as a single interval. a) (2, 4) (3, 6) b) (2, 4) (3, 6) c) (1, 2) [0, ) d) (1, 2) [0, ) Solution a) Graph (2, 4) and (3, 6) as in Fig. 1.21. The union of the two intervals consists of the real numbers between 2 and 6, which is written as (2, 6). b) Examining the graphs in Fig. 1.21, we see that only the real numbers between 3 and 4 belong to both (2, 4) and (3, 6). So the intersection of (2, 4) and (3, 6) is (3, 4). c) Graph (1, 2) and [0, ) as in Fig. 1.22. The union of these intervals consists of the real numbers greater than 1, which is written as (1, ). d) Examining the graphs in Fig. 1.22, we see that the real numbers between 0 and 2 belong to both intervals. Note that 0 also belongs to both intervals but 2 does not. So the intersection is [0, 2). Intersection: (3, 4) Intersection: [0, 2) 1 2 3 4 5 6 7 2 1 0 1 2 3 4 1 2 3 4 5 6 7 2 1 0 1 2 3 4 Union: (2, 6) FIGURE 1.21 Union: (1, ) FIGURE 1.22 WARM-UPS True or false? Explain your answer. 1. The number is a rational number. 2. The set of rational numbers is a subset of the set of real numbers. 3. Zero is the only number that is a member of both Q and I. 4. The set of real numbers is a subset of the set of irrational numbers. 5. The decimal number 0.44444 . . . is a rational number. 6. The decimal number 4.212112111211112 . . . is a rational number. 7. Every irrational number corresponds to a point on the number line. 8. The intervals (2, 6) and (3, 9) both contain the number 6. 9. (1, 3) [3, 4) (1, 4) 10. (1, 5) [2, 8) (2, 8) ■ dug46476_ch01.qxd 14 12/2/02 (1-14) 1. 2 11:17 AM Chapter 1 Page 14 The Real Numbers EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are the integers? 20. y y is a whole number greater than 0 21. x x is an integer between 3 and 5 2. What are the rational numbers? 22. y y is an integer between 4 and 7 3. What kinds of decimal numbers are rational numbers? 4. What kinds of decimal numbers are irrational? 5. What are the real numbers? Determine which elements of the set 5 1 8 A 10 , 3, , 0.025, 0, 2 , 3 , 2 2 2 are members of these sets. See Example 3. 23. Real numbers 6. What is the ratio of the circumference and diameter of any circle? 24. Natural numbers 25. Whole numbers Determine whether each statement is true or false. Explain your answer. See Example 1. 2 7. 6 Q 8. Q 7 9. 0 Q 10. 0 N 11. 0.6666 . . . Q 12. 0.00976 Q 13. N Q 14. Q J List the elements in each set and graph each set on a number line. See Example 2. 15. x x is a whole number smaller than 6 26. Integers 27. Rational numbers 28. Irrational numbers Determine whether each statement is true or false. Explain. See Example 4. 29. Q R 30. I Q 31. I Q 0 32. J Q 33. I Q R 34. J Q 35. 0.2121121112 . . . Q 36. 0.3333 . . . Q 16. x x is a natural number less than 7 37. 3.252525 . . . I 38. 3.1010010001 . . . I 17. a a is an integer greater than 5 39. 0.999 . . . I 41. I 40. 0.666 . . . Q 42. Q 18. z z is an integer between 2 and 12 19. w w is a natural number between 0 and 5 Place one of the symbols , , , or in each blank so that each statement is true. 43. N ___ W 44. J ___ Q 45. J ___ N 46. Q ___ W 47. Q ___ R 48. I ___ R 49. ___ I 50. ___ Q 51. N ___ R 52. W ___ R 53. 5 ___ J 54. 6 ___ J dug46476_ch01.qxd 12/2/02 11:17 AM Page 15 1.2 55. 57. 59. 61. 63. 7 ___ Q 2 ___ R 0 ___ I 2, 3 ___ Q 3, 2 ___ R 56. 58. 60. 62. 64. 8 ___ Q 2 ___ I 0 ___ Q 0, 1 ___ N 3, 2 ___ Q Write each interval of real numbers in interval notation and graph it. See Example 5. 65. The set of real numbers greater than 1 66. The set of real numbers greater than 2 (1-15) 15 The Real Numbers 77. 4321 0 1 2 78. 79. 3 4 5 6 7 8 9 40 50 60 70 80 90 100 80. 81. 3 4 5 6 7 8 9 9 8 7 6 5 4 3 82. 121110 9 8 7 6 67. The set of real numbers less than 1 68. The set of real numbers less than 5 69. The set of real numbers between 3 and 4 70. The set of real numbers between 1 and 3 71. The set of real numbers between 0 and 2 inclusive Write each union or intersection as a single interval. See Example 6. 83. (1, 5) (4, 9) 84. (1, 2) (0, 8) 85. (0, 3) (2, 8) 86. (1, 8) (2, 10) 87. (2, 4) (0, ) 88. (, 4) (1, 5) 89. (, 2) (0, 6) 90. (3, 6) (0, ) 91. [2, 5) (4, 9] 92. [2, 2] [2, 6) 93. [2, 6) [2, 8) 94. [1, 5] [2, 9] 72. The set of real numbers between 1 and 1 inclusive GET TING MORE INVOLVED 73. The set of real numbers greater than or equal to 1 and less than 3 95. Writing. What is the difference between a rational and an irrational number? Why is 9 rational and 3 irrational? 74. The set of real numbers greater than 2 and less than or equal to 5 96. Cooperative learning. Work in a small group to make a list of the real numbers of the form n , where n is a natural number between 1 and 100 inclusive. Decide on a method for determining which of these numbers are rational and find them. Compare your group’s method and results with other groups’ work. Write the interval notation for the interval of real numbers shown in each graph. 75. 76. 97. Exploration. Find the decimal representations of 2 , 9 2 , 99 23 , 99 23 , 999 234 , 999 23 , 9999 and 1234 . 9999 3 4 5 6 7 8 9 3 4 5 6 7 8 9 a) What do these decimals have in common? b) What is the relationship between each fraction and its decimal representation?