1.2 THE REAL NUMBERS. section. The Rational Numbers

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1 .2 The Real Numbers (-7) E F 58. E F, 2, 3,, 5, 6, 8 2, 59. (D E ) F 60. (D F) E 2, 3,, 5 2, 6. D (E F) 62. D (F E) 2, 3,, 5, 7 2, 3,, 5, (D F) (E F) 6. (D E ) (F E) 2, 3,, 5 2, 65. (D E) (D F) 66. (D F) (D E) 2, 3,, 5, 7 2, 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 D D 7. D E 72. D E D 73. D F F 7. 3 E F 75. E E F 76. E E F 77. D F F D 78. E F F E List the elements in each set. 79. x x is an even natural number less than 20 2,, 6,..., x x is a natural number greater than 6 7, 8, 9, x x is an odd natural number greater than 3, 5, 7, x x is an odd natural number less than, 3, 5,..., x x is an even natural number between and 79 6, 8, 0,..., x x is an odd natural number between 2 and 57 3, 5, 7,..., 55 Write each set using set-builder notation ,, 5, 6 x x is a natural number between 2 and 7 86., 3, 5, 7 x x is an odd natural number less than , 7, 9,,... x x is an odd natural number greater than 88., 5, 6, 7,... x x is a natural number greater than , 8, 0, 2,..., 82 x x is an even natural number between 5 and ,, 3, 5,..., 5 x x is an odd natural number between 8 and 52 GETTING MORE INVOLVED 9. Discussion. If A and B are finite sets, could A B be infinite? Explain. No 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? A B, A B 93. Discussion. What is wrong with each statement? Explain. a) 3, 2, 3 a) 3, 2, 3 b) 3, 2, 3 b) 3, 2, 3 c) c) 9. Exploration. There are only two possible subsets of, namely, and. a) List all possible subsets of, 2. How many are there?,, 2,, 2 b) List all possible subsets of, 2, 3. How many are there? 8 c) Guess how many subsets there are of, 2, 3,. Verify your guess by listing all the possible subsets. 6 d) How many subsets are there for, 2, 3,..., n? 2 n In this section.2 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 Graphing on the Number Line The Irrational Numbers 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

2 8 (-8) Chapter The Real Numbers helpful hint A negative number can be used to represent a loss or a debt. The number 0 could represent a debt of $0, a temperature of 0 below zero, or an altitude of 0 feet below sea level. 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. integers. We use the letters N, W, and J to name these sets: N, 2, 3,... The natural numbers W 0,, 2, 3,... The whole numbers J..., 3, 2,, 0,, 2, 3,... 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: a Q a and b are integers, with b 0 The rational numbers b Examples of rational numbers are ,,, 0,,,, and Note that the rational numbers are the numbers that can be expressed as a ratio (or quotient) of integers. The integer 7 is rational because we can write it as 7. 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. The following examples show some rational numbers and their equivalent decimal forms: Terminating decimal 00.0 Terminating decimal Terminating decimal The single digit 6 repeats. The pair of digits 25 repeats. The pair of digits 9 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 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 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.

3 .2 The Real Numbers (-9) 9 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. 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... 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. unit Origin unit FIGURE. 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..5. The three dots to the right and left on the number line indicate that the integers go on indefinitely in both directions FIGURE.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 b) a a is an integer between 3 and 9 c) y y is an integer greater than 3 a) The whole numbers less than are 0,, 2, and 3. Figure.6 shows the graph of this set FIGURE.6 b) The integers between 3 and 9 are, 5, 6, 7, and 8. The graph is shown in Fig FIGURE.7 c) The integers greater than 3 are 2,, 0,, and so on. To indicate the continuing pattern, we use a series of dots on the graph in Fig FIGURE.8

4 0 (-0) Chapter The Real Numbers When we draw a number line, we might label only the integers. But every point on the number line corresponds to a number. The set of all of these numbers is called the set of real numbers, R. It is easy to locate rational numbers on the number line. For example, the number 2 corresponds to a point halfway between 0 and, and 3 2 corresponds to a point halfway between and 2, as shown in Fig..9. However, on the number line, there are points such as 2 that do not correspond to rational numbers. These points correspond to the set of irrational numbers, I. The rational numbers Q and the irrational numbers I have no numbers in common, and together they form the set of real numbers R calculator close-up A calculator gives a 0-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. π = Circumference Diameter Circumference Diameter π = FIGURE.0 C D The Irrational Numbers FIGURE.9 Irrational numbers are those real numbers that cannot be expressed as a ratio of integers. To see an example of an irrational number, consider the positive square root of 2 (in symbols 2 ). The square root of 2 is a number that you can multiply by itself to get 2. So we can write (using a raised dot for times) If we look for 2 on a calculator or in Appendix B, we find.. But if we multiply. by itself, we get (.)(.) So 2 is not equal to. (in symbols, 2.). The square root of 2 is approximately. (in symbols, 2.). There is no terminating or repeating decimal that will give exactly 2 when multiplied by itself. So 2 is an irrational number. It can be shown that other square roots such as 3, 5, and 7 are also irrational numbers. It is easy to write examples of irrational numbers in decimal form because as decimal numbers they have digits that neither repeat nor terminate. Each of the following numbers has a continuing pattern that guarantees that its digits will neither repeat nor terminate: 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 767. 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..0. The value of rounded to nine decimal places is When using in computations, we frequently use the rational number 3. as an approximate value for. Figure. illustrates the subsets of the real numbers that we have been discussing.

5 .2 The Real Numbers (-) study tip 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, 3 5, 7 55, 3 Integers (J) 5.26, Whole numbers (W),, 2,, 0,, 2, 3, Real numbers (R) Counting numbers (N) Irrational numbers (I) 2, 6, 7, π FIGURE. E X A M P L E 3 Classifying real numbers Determine which elements of the set 7,, 0, 5,,.6, 2 are members of each of the following sets. a) Real numbers b) Rational numbers c) Integers a) All of the numbers are real numbers. b) The numbers, 0,.6, and 2 are rational numbers. c) The only integers in this set are 0 and 2. E X A M P L E Subsets of the real numbers Determine whether each of the following statements is true or false. a) 7 Q b) J W c) I Q d) 3 N e) J I f) Q R g) R N h) R a) False b) False c) True d) False e) True f) True g) False h) True WARM-UPS True or false? Explain your answer.. The number is a rational number. False 2. The set of rational numbers is a subset of the set of real numbers. True 3. Zero is the only number that is a member of both Q and I. False. The set of real numbers is a subset of the set of irrational numbers. False

6 2 (-2) Chapter The Real Numbers WARM-UPS (continued) 5. The decimal number is a rational number. True 6. The decimal number is a rational number. False 7. Every irrational number corresponds to a point on the number line. True 8. If a real number is not irrational, then it is rational. True 9. The set of counting numbers from through 5 trillion is an infinite set. False 0. There are infinitely many rational numbers. True. 2 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What are the integers? The integers consist of the positive and negative counting numbers and zero. 2. What are the rational numbers? The rational numbers consist of all numbers that can be expressed as a ratio of integers. 3. What kinds of decimal numbers are rational numbers? The repeating or terminating decimal numbers are rational numbers.. What kinds of decimal numbers are irrational? Decimals that neither repeat nor terminate are irrational numbers. 5. What are the real numbers? The set of real numbers is the union of the rational and irrational numbers. 6. What is the ratio of the circumference and diameter of any circle? The ratio of the circumference and diameter of any circle is. Determine whether each statement is true or false. Explain your answer. See Example Q True 8. 2 Q 7 True 9. 0 Q False 0. 0 N True Q True Q False 3. N Q True. Q J False List the elements in each set and graph each set on a number line. See Example x x is a whole number smaller than 6 0,, 2, 3,, 5 6. x x is a natural number less than 7, 2, 3,, 5, 6 7. a a is an integer greater than 5, 3, 2,, 0,, 8. z z is an integer between 2 and 2 3,, 5, 6, 7, 8, 9, 0, 9. w w is a natural number between 0 and 5, 2, 3, 20. y y is a whole number greater than 0, 2, 3,, 5, 2. x x is an integer between 3 and 5 2,, 0,, 2, 3, 22. y y is an integer between and 7 3, 2,, 0,, 2, 3,, 5, 6 Determine which elements of the set A 0, 3,, 0.025, 0, 2, 3, are members of the following sets. See Example Real numbers All 2. Natural numbers Whole numbers 0, Integers 3, 0, Rational numbers 3, 5 2, 0.025, 0, 3 2, Irrational numbers 0, 2 Determine whether each statement is true or false. Explain. See Example. 29. Q R True 30. I Q False 3. I Q 0 False 32. J Q True

7 .3 Operations on the Set of Real Numbers (-3) I Q R True 3. J Q False Q Q False True I I False True I False Q True. I True 2. Q False Place one of the symbols,,, or in each blank so that each statement is true. 3. N W. J Q 5. J N 6. Q W 7. Q R 8. I R 9. I 50. Q 5. N R 52. W R J 5. 6 J Q Q R I I Q 6. 2, 3 Q 62. 0, N 63. 3, 2 R 6. 3, 2 Q GETTING MORE INVOLVED 65. Writing. What is the difference between a rational and an irrational number? Why is 9 rational and 3 irrational? 66. 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 and 00 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. 67. Exploration. Find the decimal representations of ,,,,,, and a) What do these decimals have in common? b) What is the relationship between each fraction and its decimal representation? In this section Absolute Value Addition Subtraction Multiplication Division Division by Zero.3 OPERATIONS ON THE SET OF REAL NUMBERS Computations in algebra are performed with positive and negative numbers. In this section we will extend the basic operations of arithmetic to the negative numbers. Absolute Value The real numbers are the coordinates of the points on the number line. However, we often refer to the points as numbers. For example, the numbers 5 and 5 are both five units away from 0 on the number line shown in Fig..2. A number s distance from 0 on the number line is called the absolute value of the number. We write a for the absolute value of a. Therefore 5 5 and units 5 units FIGURE.2 E X A M P L E Absolute value Find the value of,, and 0. Because both and are four units from 0 on the number line, we have and. Because the distance from 0 to 0 on the number line is 0, we have 0 0.