1.2 REAL NUMBERS. 10 Chapter 1 Basic Concepts: Review and Preview

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1 10 Chapter 1 Basic Concepts: Review and Preview (b) Segment of a circle of radius R, depth R 2: A 4 R 2 (c) Frustum of cone: V 1 h R2 Rr r 2 R r R R 2 Conversion between fluid ounces and cubic inches: 1quart 2 ounces cubic inches 46. The height h and diameter d of a cylindrical can of pineapple juice are measured: h 6 inches, d inches. Find the volume in cubic inches and its equivalent in fluid ounces. Use the formula for frustum of a cone with r R. The label on the can indicates 46 ounces of pineapple juice. What is the difference between your answer and 46 ounces? Explain. 47. Forasoftdrinkcupthatissupposedtohold44ounces, the top diameter is 4 and the bottom diameter is. 8 8 Theheightofthecupismeasuredas6 4. If all measurements are accurate to the nearest 1 8, find the largest and smallest possible values for the volume. Is it reasonable to call the cup as 44-ounce cup? 48. Asoftdrinkcupismadeintheshapeofafrustumofa cone. If the cup is to have an upper diameter of 4 and h the lower diameter of,whatshouldtheheightbeifit is to hold 2 ounces? 49. A direct mail catalog features an Oriental wok in the shape of a section of a sphere. The catalog gives dimensions that indicate R 6 in., d in.andclaimsthat the wok holds qts. Assuming that the measurements are accurate to the nearest 1 8 in., find the volume corresponding to (a) R 5 7 in. d 2 7 in. 8 8 (b) R 6 1 in. d 1 in. 8 8 On the basis of your results in parts (a) and (b), isthe catalog claim of qts reasonable? Explain. 50. A metal barrel 18 in diameter and 0 long is cut in half to make a trough 9 deep and 0 long. (a) Find the volume (in cubic inches) of the resulting trough. (b) If the diameter and length are measured accurate to the nearest quarter-inch, find the largest and smallest possible values for the volume (see Example 1). 51. SupposethetroughinExercise50iscutdowntomake atroughofdepth4.5. What percent of the volume of the original is now in the shallower trough? 52. The box Decimal Parts of a Mile gives some familiar comparison measurements for decimal parts of a mile. Complete a similar chart for decimal parts of a kilometer. 0.1 km 0.01 km km km km km 1.2 REAL NUMBERS The complexities of modern science and modern society have created a need for scientific generalists, for men (and women as well) trained in many fields of science. The habits of mind and not the subject matter are what distinguish the sciences. Mosteller, Bode, Tukey, Winsor Numbers occur in every phase of life. It is impossible to imagine how anyone could function in a civilized society without having some familiarity with numbers. We recognize that you have had considerable experience working with numbers, and we also assume that you know something about the language and notation of sets.

2 1.2 Real Numbers 11 Subsets of Real Numbers We denote the set of real numbers by R. We make no attempt to develop the properties and operations of R; this is reserved for more advanced courses. Several subsets of the set of real numbers are used so frequently that we give them names. Most of these sets are familiar. The set of natural numbers is also called the set of positive integers or counting numbers. A prime is a positive integer greater than 1 that is divisible only by 1 and itself. The table lists the most commonly encountered subsets of R. Subset Natural numbers Whole numbers Integers Even integers Odd integers Prime numbers Rationals Irrationals Subsets of R Symbol and Elements N 1, 2,,... W 0, 1, 2,,... I..., 1,0,1,2,,... E..., 2,0,2,4,6,... O...,, 1,1,,5,... P 2,,5,7,11,1,... Q p q p, q I, q 0 H x x R and x Q I had such an amazingly deprived high school education. There wasn t a useful math book in the library. Bill Gosper Figure 4 indicates schematically that some of the sets listed are subsets of others. For example, P N, N W,andW I. The sets E and O together make up I, so we can write E O I. Further, for any p I, since p p 1, every integer is also a rational number, so I Q. The existence of some irrational numbers has been known since at least the time of the ancient Greeks, who discovered that the length of the diagonal of a R Real numbers Q Rational numbers H Irrational numbers I Integers W Whole numbers E Even integers O Odd integers N Natural numbers P Primes FIGURE 4 Subsets of the real numbers.

3 12 Chapter 1 Basic Concepts: Review and Preview square is not a rational multiple of the length of the sides (see Develop Mastery Exercise 8). The length of the diagonal of a unit square is the irrational number 2,andwerecognizemanyotherssuchas 1and2 7and.Theratio of the circumference of any circle to its diameter is the number (pi), approximately (See the earlier Historical Note, The Number Pi. ) Although most of this book (and most of calculus as well) involves only real numbers, we also make use of the set of complex numbers (see Section 1.), especially in Chapters and 7. EXAMPLE 1 Set notation Determine whether the statement is true. (a) N Q (b) I H 0 (c) 5 Q (d) 64 H (e) 41 P (f) 87 / P (a) True; every natural number is rational. (b) True; every integer is rational and hence not in H. (c) False; 5 is an irrational number. (d) False; 64 8 and is not irrational. (e) True; 41 is a prime number. (f) True; 87 29, so 87 is not a prime number. Strategy: Think about the meaning of each set (in words). For given numbers, decide if each fits the description of the indicated set. EXAMPLE 2 Union and intersection Simplify: (a) P N (b) W Q (c) Q H (a) P N P; every prime number is also a natural number. (b) W Q W; every whole number is also a rational number. (c) Q H R; every real number is rational or irrational. Decimal Representation of Numbers Every real number also has a decimal name. For instance, the rational number 4 can also be written as 0.75, which is called a terminating decimal. To get the decimal representation for the rational number 5,wedivide5by11andgetthe 11 repeating (nonterminating) decimal ,which we write as The bar notation indicates that the block under the bar, in this instance 45, repeats forever. A terminating decimal can also be considered as repeating. For instance, 4 can be named by 0.75, or by 0.750, or even by (see Example ). An irrational number such as 2 has a nonterminating and nonrepeating decimal representation. The distinction between repeating and nonrepeating decimals distinguishes the rational numbers from the irrationals. Approximating Pi As indicated in Section 1.1, the important number occurs in problem-solving applications as well as theoretical mathematics. In recent years sophisticated techniques have allowed computer evaluation of to billions of decimal places, but there is still no way to express the decimal representation of exactly. See the Historical Note, Approximating the Number.

4 1.2 Real Numbers 1 HISTORICAL NOTE APPROXIMATING THE NUMBER People continued to be fascinated by even after it was shown to be irrational. In 1844 Johann Dase, who could multiply 100 digit numbers in his head, took months to compute to 205 digits. The champion at hand calculating must be William Shanks, who spent 20 years to grind out 707 digits. His record stood until 1945, when D. W. Ferguson used a mechanical calculator to find an error in Shanks 528th digit. No further search for accuracy can be justified for practical purposes of distance or area computation. An approximation to 45 digits would measure the circumference of a circle encompassing the entire A computer can calculate these first 00 digits of in a fraction of a second. The same calculation by hand requires months of work. universe with an error less than the radius of a single electron. People have found many other reasons, in addition to the sheer fascination of knowing, for computing the digits of. Computers brought a new era. In 1949, a machine called ENIAC, composed of rooms full of vacuum tubes and wires, in 70 hours computed 207 digits of. More recent milestones are listed below. Remarkably, the last record was achieved on a home-built super computer. You can read more in Ramanujan and Pi, Scientific American (Feb. 1988), and in The Mountains of Pi, The New Yorker (Mar. 12, 1992). 197 Jean Guilloud, M. Bouyer CDC million digits 1985 R. William Gosper Symbolics 17 million digits 1986 David H. Bailey Cray-2 29 million digits 1987 Yasumasa Kanada NEC SX-2 14 million digits 1989 D. and G. Chudnovsky 480 million digits 1990 Yasumasa Kanada NEC SX-2 1 billion digits 1991 D. and G. Chudnovsky M Zero 2.26 billion digits The rational number 22 7 is sometimes used as an approximation to, but it is. Other rational number approxi- (see Develop Mastery Exercises 9 and 40). important to understand that is not equal to 22 7 mations of include, ,and208, 41 66,17 Characterizing real numbers A real number is rational if and only if its decimal representation repeats or terminates. A real number is irrational if and only if its decimal representation is nonterminating and nonrepeating. From Decimal Representations to Quotient Form Finding a decimal representation for a given rational number is simply a matter of division; going the other way is more involved but is not difficult. Some graphing calculators have built-in routines to convert decimals to fractions. Such programs are limited because calculators must work with truncated (cut-off, finite) decimals.

5 14 Chapter 1 Basic Concepts: Review and Preview There is no way to tell a calculator that a given decimal repeats (infinitely). If we know that a given number x has a repeating decimal representation, these steps will give the desired rational number as a quotient. 1. Multiply x by an appropriate power of 10 to move the decimal point to the beginning of the repeating block. 2. Multiply x byanotherpowerof10tomovethedecimalpointtothe beginning of the next block.. The difference between these two multiples of x is an integer, which allows us to solve for x. EXAMPLE From decimal to quotient Express each number as a quotient of integers in lowest terms. (a) 0.74 (b) 0.74 (c) (a) From the meaning of decimal notation, , which reduces to. Thus represents the rational number (b) With a repeating block, we follow the procedure outlined above. Let x The decimal point is already at the beginning of the block, so multiply by 100 to move the decimal point to the beginning of the next block. 100x x x 74, from which x Thus 0.74 represents the rational number 74.Youmaywishtoverifythisby 99 dividing 74 by 99. (c)let y 0.749, multiply by 1000, then by 100, and take the difference: 1000y y y 675, from which y Hence represents the rational number, which says that 4 4 has two different decimal names, and Actually, every rational number that can be written as a terminating decimal has two representations. Note that the procedure outlined above involves subtracting repeating decimals as if they were finite decimals. We justify such operations in Section 8.. Exact Answers and Decimal Approximations When we use a calculator to evaluate a numerical expression, in most cases the answer is a decimal approximation oftheexactanswer.whenweaskforafour decimal place approximation, we mean round off the calculator display to four decimal places. EXAMPLE 4 Calculator evaluation Useacalculatortogetafourdecimal place value. Is the value exact or an approximation? (a) (b) (c) 2

6 1.2 Real Numbers 15 (a) ; exact decimal value. (b) ; approximation. (c) ; approximation. Square Roots and the Square Root Symbol There are two numbers whose square is 2. That is, the equation x 2 2 has two roots. We reserve the symbol 2forthepositive root, so the roots of the equation are 2 and 2, which we often write as 2. For every positive x, the calculator will display a positive number when we press 57 x, and we use x to denote the positive number whose square is x. EXAMPLE 5 Calculators and rounding off Find an approximation rounded off to four decimal places. (a) 1 (b) 1 (a) Using a calculator, we get (b) After evaluating 1, take the square root to get EXERCISES 1.2 Check Your Understanding Exercises 1 5 True or False. Give reasons. 1. The number is equal to The integer 119 is a prime number.. The intersection of the set of rational numbers and the set of irrational numbers is the empty set. 4. The set of prime numbers is a subset of the set of odd numbers. 5. The sum of any two odd numbers is an odd number. Exercises 6 10 Fill in the blank so that the resulting statement is true. 6. The product of two odd numbers is an number. 7. When 5 7 is expressed as a repeating decimal, the eighth digit after the decimal is. 8. Of the numbers , 17, 25 9, 64 14, the one that is irrational is. 9. Of the four numbers, 64 16, 0.564, 5 2, the one that is rational is. 10. Of the four numbers 8, , 0.714, the smallest one is. Develop Mastery Exercises 1 8 Subsets of Real Numbers Determine whether each statement is true or false. Refer to the subsets of R listed in this section. 1. (a) 0 N (b) 17 / P 2. (a) 5 / N (b) 5 I. (a) 4, I (b) 7, 81 P 4. (a) 4, 5 H (b) 0.5, 0.7 Q 5. (a) I N I (b) I W W 6. (a) P I P (b) Q I Q 7. (a) Q H (b) H I H 8. (a) P Q Q (b) I Q I Exercises 9 10 Indicate which of the subsets P, N, I, O, E, Q, and H contain each number. For instance, 17 belongs to P, N, I, O, and Q. 9. (a) 29 (b) 16 (c) 2 (d) (a).27 (b) 29 (c) (d) 2 1

7 16 Chapter 1 Basic Concepts: Review and Preview Exercises Fraction to Decimal Express each as a terminating decimal, or as a repeating decimal using the bar notation. 11. (a) (a) (a) (a) 16 5 (b) 5 12 (b) 25 (b) 10 1 (b) Exercises Decimal to Fraction Express each as a fraction (quotient of integers) in lowest terms. 15. (a) 0.6 (b) (a) 1.45 (b) (a) 0.8 (b) (a) 1.6 (b) Exercises Give a decimal approximation rounded off to three decimal places (a) (a) (a) (b) (b) 1 2 (b) Exercises 22 0 Decimal Approximations Give decimal approximations rounded off to six decimal places. Do the numbers appear to be equal? 22. 8; ; ; ; ; ; ; ; ; What is the smallest nonprime positive integer greater than 1 that has no factors less than 12? 2. What is the smallest prime number that divides ? Exercises 4 True or False. Give reasons.. (a) The sum of any two odd numbers is an odd number. (b) The product of any two odd numbers is an odd number. (c) The product of any two consecutive positive integers is an even number. 4. (a) The sum of three consecutive even numbers is an odd number. (b) If a positive even integer is a perfect square, then it is the square of an even number. (c) If the sum of two integers is even, then both must be even. 5. Give an example of irrational numbers for x and y that satisfy the given condition. (a) x y is irrational. (b) x y is rational. (c) x y is rational. (d) x y is rational. 6. If x , determine whether x is rational or irrational. (Hint: Evaluate x 2.) 7. If x 2 2, determine whether x is rational or irrational. (Hint: Evaluate x 2.) 8. Prove that 2 is not a rational number. (Hint: Suppose 2 b, where b, c N and b c c is in lowest terms. Then b 2 2c 2.Explainwhybmust be even. Then also explain why c must be even. This would contradict the assumption that b c is in lowest terms.) Exercises 9 40 Approximations for Refer to the number, whose decimal form is nonterminating and nonrepeating. Rounded off to 24 decimal places, The following rational numbers are used as approximations of. Use your calculator to evaluate and compare each result with the given decimal approximation of. (a) 22 7 (b) 106 (c) The rational number 208,41 66,17 is an excellent approximation of. Evaluate it to at least 12 decimal places and compare the result with the approximation given above. 41. In 1991 the Chudnovsky brothers used a supercomputer they built to compute more than 2.26 billion digits of.countthenumberofsymbolsinanaveragelineof this book and estimate how long a line of type (measured in miles) the Chudnovsky result would give. (See the Historical Note, Approximating the Number. )

California 5 th Grade Standards / Excel Math Correlation by Lesson Number

California 5 th Grade Standards / Excel Math Correlation by Lesson Number (Activity) L1 L2 L3 Excel Math Objective Recognizing numbers less than a million given in words or place value; recognizing addition and subtraction fact families; subtracting 2 threedigit numbers with