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The Real Number System 1.1 So Many Numbers, So Little Time Number Sort................................................................................ 3 1.2 Is It a Bird or a Plane? Rational Numbers. ......................................................... 11 1.3 Sew What? Irrational Numbers........................................................ 19 1.4 Worth 1000 Words Real Numbers and Their Properties. .............................. 29 © Carnegie Learning Pi is probably one of the most famous numbers in all of history. As a decimal, it goes on and on forever without repeating. Mathematicians have already calculated trillions of the decimal digits of pi. It really is a fascinating number. And it's delicious! 1 462351_CL_C3_CH01_pp001-042.indd 1 28/08/13 2:12 PM © Carnegie Learning 2 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 2 28/08/13 2:12 PM So Many Numbers, So Little Time Number Sort Learning Goals In this lesson, you will: Review and analyze numbers. Determine similarities and differences among various numbers. Sort numbers by their similarities and rationalize the differences between the groups of numbers. I f someone were to ask you to define the word “number”, could you do it? Could you then think of the different types of numbers and try to organize them? It’s actually not as easy as it sounds. Throughout history, the task of categorizing numbers has caused a lot of headaches. For example, the Greeks didn’t use negative numbers, and for a long time people didn’t recognize 0 as a number. Even if mathematicians could agree on which numbers should be included, they struggled with how to group them. How many types of numbers are there? Should we define them as positive or negative? What about zero? Should fractions be separate? What about decimals? These are the types of questions mathematicians have taken very seriously over © Carnegie Learning the years. In fact, the history of numbers probably has caused more anger and drama than a reality TV show. If you doubt this, consider the Greek mathematician Hippasus. He was the first to claim that some numbers exist which go on forever and never repeat (gasp!). He was allegedly drowned as punishment for this statement! We have a system now for classifying numbers. It isn’t necessarily the “right” way, or even the only way—it’s just a way the mathematical community has agreed to group numbers. Can you think of other systems that people have developed for classifying things? 1.1 Number Sort • 3 462351_CL_C3_CH01_pp001-042.indd 3 28/08/13 2:12 PM Problem 1 Ready, Set, Sort! Mathematics is the science of patterns and relationships. Looking for patterns and sorting objects into different groups can provide valuable insights. In this lesson, you will analyze many different numbers and sort them into various groups. 1. Cut out the thirty numbers on the following page. Then, analyze and sort the numbers into different groups. You may group the numbers in any way you feel is appropriate. However, you must sort the numbers into more than one group. In the space provided, record the information for each of your groups. • Name each group of numbers. • List the numbers in each group. © Carnegie Learning • Provide a rationale for why you created each group. 4 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 4 28/08/13 2:12 PM p 0.25 3 2 __ 8 2101 20% |23| 26.41 0.91 √ 100 627,513 0.001 2 2 __ 3 0 √ 2 3.21 3 1012 1,000,872.0245 42 0.5% ____ ___ __ __ 2√9 __ 2√2 __ 20.3 1.523232323. . . 21 ___ 1.0205 3 10 1 6 __ 4 223 √ 9 ___ 16 _____ √ 0.25 © Carnegie Learning 212% 100 ____ 11 |2| 1.1 Number Sort • 5 462351_CL_C3_CH01_pp001-042.indd 5 28/08/13 2:12 PM © Carnegie Learning 6 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 6 28/08/13 2:12 PM 2. Compare your groupings with your classmates’ groupings. Create a list of the different types of numbers you noticed. © Carnegie Learning Are any of the types of numbers shared among your groups? Or, are they unique to each group? 1.1 Number Sort • 7 462351_CL_C3_CH01_pp001-042.indd 7 28/08/13 2:12 PM Problem 2 Let’s Take a Closer Look 1. Lauren grouped these numbers together. __ __ __2 ___ 100 0.91 , 2 , , 1.523232323. . ., 20.3 3 11 Why do you think Lauren put these numbers in the same group? 2. Zane and Tanya provided the same rationale for one of their groups of numbers. However, the numbers in their groups were different. Zane √ |23|, 1 0 0, 627,513, 3.21 3 1012, Tanya √ 20%, 100, 627,513, 3.21 3 1012, 42, |2| 42, |2|, 212% When I simplify each number, Each of these numbers it is a positive integer. represents a positive integer. Who is correct? Explain your reasoning. © Carnegie Learning 8 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 8 28/08/13 2:12 PM 3. Tim grouped these numbers together. __ __ 3 2 2 __8 , 2101, 26.41, 2__ 3 , 2√9 , 21, 20.3 What rationale could Tim provide? 4. Isaac used the reasoning shown when creating one of his groups of numbers. The numbers are between 0 and 1. Identify all of the numbers that satisfy Isaac’s reasoning. 5. Lezlee grouped these numbers together. 100 , 1.523232323. . ., 212%, 6 _1 26.41, ___ 11 4 What could Lezlee name the group? Explain your reasoning. © Carnegie Learning Clip all your numbers together and keep them. You’ll need them later in this chapter. Be prepared to share your solutions and methods. 1.1 Number Sort • 9 462351_CL_C3_CH01_pp001-042.indd 9 28/08/13 2:12 PM © Carnegie Learning 10 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 10 28/08/13 2:12 PM Is It a Bird or a Plane? Rational Numbers Learning Goals Key Terms In this lesson, you will: Use a number line to compare and order rational numbers. Learn about types of numbers and their properties. natural numbers (counting numbers) whole numbers integers closed rational numbers Perform operations with rational numbers. T here are all kinds of different numbers, which have been given some strange names. It seems that mathematicians can come up with an infinite number of different kinds of numbers. A repunit number is an integer with all 1’s as digits. So, 11, 111, and so on, are all repunit numbers. Pronic numbers are numbers that are the products of two consecutive numbers. The numbers 2, 6, and 12 are the first pronic numbers (1 3 2 5 2, 2 3 3 5 6, and 3 3 4 5 12). To find the numbers that some call “lucky” numbers, first start with all the © Carnegie Learning counting numbers (1, 2, 3, 4, and so on). Delete every second number. This will give you 1, 3, 5, 7, 9, 11, and so on. The second number in that list is 3, so cross off every third number remaining. Now you have 1, 3, 7, 9, 13, 15, 19, 21, and so on. The next number that is left is 7, so cross off every seventh number remaining. Can you list all the “lucky” numbers less than 50? 1.2 Rational Numbers • 11 462351_CL_C3_CH01_pp001-042.indd 11 28/08/13 2:12 PM Problem 1 A Science Experiment Your science class is conducting an experiment to see how the weight of a paper airplane affects the distance that it can fly. Your class is divided into two groups. Group 1 uses a yard stick to measure the distances that an airplane flies, and Group 2 uses a meter stick. Group 2 then takes their measurements in Because paper is typically sold in 500-sheet quantities, a paper's weight is determined by the weight of 500 sheets of the paper. So, 500 sheets of 20-pound paper weighs 20 pounds. meters and converts them to feet. The results of the experiment are shown in the table. Type of Paper Group 1 Measurements Group 2 Converted Measurements 20-pound paper 7 13 __ feet 8 13.9 feet 28-pound paper 3 14 __ feet 8 14.4 feet 1. Your science class needs to compare the Group 1 measurement to the Group 2 converted measurement for each type of paper. 3 as a decimal. 7 as a decimal. b. Write 14 __ a. Write 13 __ 8 8 2. On the number line shown, graph the Group 1 measurements 13.5 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5 3. Use the number line to determine which group’s flight traveled farther for the 20-pound paper and for the 28-pound paper. Write your answers using © Carnegie Learning written as decimals and the Group 2 converted measurements. complete sentences. 12 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 12 28/08/13 2:12 PM Problem 2 Natural Numbers, Whole Numbers, and Integers 1. The first set of numbers that you learned when you were very young was the set of counting numbers, or natural numbers. Natural numbers consists of the numbers that you use to count objects: {1, 2, 3, 4, 5, …}. a. How many counting numbers are there? In the set {1, 2, 3, 4, 5, . . .} the dots at the end of the list mean that the list of numbers goes on without end. b. Does it make sense to ask which counting number is the greatest? Explain why or why not. c. Why do you think this set of numbers is called the natural numbers? You have also used the set of whole numbers. Whole numbers are made up of the set of natural numbers and the number 0, the additive identity. 2. Why is zero the additive identity? 3. Other than being used as the additive identity, how else is zero used © Carnegie Learning in the set of whole numbers? 4. Explain why having zero makes the set of whole numbers more useful than the set of natural numbers. 1.2 Rational Numbers • 13 462351_CL_C3_CH01_pp001-042.indd 13 28/08/13 2:12 PM Another set of numbers is the set of integers, which is a set that includes all of the whole numbers and their additive inverses. 5. What is the additive inverse of a number? 6. Represent the set of integers. Remember to use three dots to show that the numbers go on without end in both directions. Use brackets to represent sets. 7. Does it make sense to ask which integer is the least or which integer is the greatest? Explain why or why not. When you perform operations such as addition or multiplication on the numbers in a set, the operations could produce a defined value that is also in the set. When this happens, the set is said to be closed under the operation. The set of integers is said to be closed under the operation of addition. This means that for every two integers a and b, the sum a 1 b is also an integer. 8. Are the natural numbers closed under addition? Write an example to support 9. Are the whole numbers closed under addition? Write an example to support your answer. © Carnegie Learning your answer. 10. Consider the operation of subtraction. Are the natural numbers closed under subtraction? Write an example to support your answer. 14 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 14 28/08/13 2:12 PM 11. Are the whole numbers closed under subtraction? Write an example to support your answer. 12. Are the integers closed under subtraction? Write an example to support your answer. 13. Are any of these sets closed under multiplication? Write examples to support your answers. 14. Are any of these sets closed under division? Write examples to support your answer. You have learned about the additive inverse, the multiplicative inverse, the additive identity, and the multiplicative identity. © Carnegie Learning 15. Which of these does the set of natural numbers have, if any? Explain your reasoning. 16. Which of these does the set of whole numbers have, if any? Explain your reasoning. 1.2 Rational Numbers • 15 462351_CL_C3_CH01_pp001-042.indd 15 28/08/13 2:12 PM 17. Which of these does the set of integers have, if any? Explain your reasoning. Problem 3 Rational Numbers a , where a and b are 1. A rational number is a number that can be written in the form __ b both integers and b is not equal to 0. a. Does the set of rational numbers include the set of whole numbers? Write an example to support your answer. b. Does the set of rational numbers include the set of integers? Write an example to support your answer. c. Does the set of rational numbers include all fractions? Write an example to support your answer. d. Does the set of rational numbers include all decimals? Write an example to 2. Is the set of rational numbers closed under addition? Write an example to support your answer. © Carnegie Learning support your answer. 3. Is the set of rational numbers closed under subtraction? Write an example to support your answer. 16 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 16 28/08/13 2:12 PM 4. Is the set of rational numbers closed under multiplication? Write an example to support your answer. 5. Is the set of rational numbers closed under division? Write an example to support your answer. 6. Does the set of rational numbers have an additive identity? Write an example to support your answer. 7. Does the set of rational numbers have a multiplicative identity? Write an example to support your answer. 8. Does the set of rational numbers have an additive inverse? Write an example to © Carnegie Learning support your answer. 9. Does the set of rational numbers have a multiplicative inverse? Write an example to support your answer. 1.2 Rational Numbers • 17 462351_CL_C3_CH01_pp001-042.indd 17 28/08/13 2:12 PM 10. You can add, subtract, multiply, and divide rational numbers in much the same way that you did using integers. Perform the indicated operation. a. 1.5 1 (28.3) 5 b. 212.5 2 8.3 5 1 3 5 c. 2 __ 2 __ 2 4 7 5 d. 2 __ 1 1 23 __ 2 8 e. 22.0 3 (23.6) 5 f. 6.75 3 (24.2) 5 2 3 __ 3 5 g. 2 __ 3 8 3 5 3 3 22 __ h. 23 __ 5 4 i. 21.5 4 4.5 5 j. 22.1 4 (23.5) 5 2 k. 2__ 4 ___ 3 5 5 10 3 4 22 __ 2 5 l. 21 __ 5 8 Remember, it is a good idea to estimate first! ( ) ( ) © Carnegie Learning ( ) Be prepared to share your solutions and methods. 18 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 18 28/08/13 2:12 PM Sew What? Irrational Numbers Learning Goals Key Terms In this lesson, you will: Identify decimals as terminating or repeating. Write repeating decimals as fractions. Identify irrational numbers. irrational number terminating decimal repeating decimal bar notation I n 2006, a 60-year-old Japanese man named Akira Haraguchi publicly recited the first 100,000 decimal places of p from memory. The feat took him 16 hours to accomplish—from 9 a.m. on a Tuesday morning to 1:30 a.m. the next day. Every one to two hours, Haraguchi took a break to use the restroom and have a snack. And he was videotaped throughout the entire process—to make sure he © Carnegie Learning didn’t cheat! 1.3 Irrational Numbers • 19 462351_CL_C3_CH01_pp001-042.indd 19 28/08/13 2:12 PM Problem 1 Repeating Decimals You have worked with some numbers like p that are not rational numbers. For example, __ __ √ 2 and √ 5 are not the square roots of perfect squares and cannot be written in the form __ a , where a and b are both integers. b Even though you often approximate square roots using a decimal, most square roots are irrational numbers. Because all rational numbers can be written as __ a where a and b are b integers, they can be written as terminating decimals (e.g. __ 1 5 0.25) or repeating decimals 4 1 5 0.1666...). Therefore, all other decimals are irrational numbers because these (e.g., __ 6 decimals cannot be written as fractions in the form __ a where a and b are integers and b is b not equal to 0. 1. Convert the fraction to a decimal by dividing the numerator by the denominator. Continue to divide until you see a pattern. _________ 1 5 3) 1 __ 3 2. Describe the pattern that you observed in Question 1. 3. Order the fractions from least to greatest. Then, convert each fraction to a decimal by dividing the numerator by the denominator. Continue to divide until you see a pattern. ________ __________ 5 56 5 ) a. __ 6 b. __ 2 59 ) 2 9 ________ ) c. ___ 9 511 9 11 _ __________ d. ___ 3 5 22 ) 3 22 © Carnegie Learning 20 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 20 28/08/13 2:12 PM 4. Explain why these decimal representations are called repeating decimals. A terminating decimal is a decimal that has a last digit. For instance, the decimal 5 __ 1 . 1 divided by 8 is equal to 0.125. 0.125 is a terminating decimal because _____ 125 1000 8 A repeating decimal is a decimal with digits that repeat in sets of one or more. You can use two different notations to represent repeating decimals. One notation shows one set of digits that repeat with a bar over the repeating digits. This is called bar notation. __ ___ 7 5 0.318 1 5 0.3 ___ __ 3 22 Another notation shows two sets of the digits that repeat with dots to indicate repetition. You saw these dots as well when describing the number sets in the previous lesson. 7 5 0.31818… 1 5 0.33… ___ __ 3 22 5. Write each repeating decimal from Question 2 using both notations. 5 5 a. __ 6 2 5 b. __ 9 9 5 c. ___ 11 3 5 d. ___ 22 2 , and __ 1 , and are used Some repeating decimals represent common fractions, such as __ 1 , __ 3 3 6 often enough that you can recognize the fraction by its decimal representation. For most repeating decimals, though, you cannot recognize the fraction that the decimal represents. For example, can you tell which fraction is represented by the repeating decimal ___ © Carnegie Learning 0.44… or 0.09 ? 1.3 Irrational Numbers • 21 462351_CL_C3_CH01_pp001-042.indd 21 28/08/13 2:12 PM You can use algebra to determine the fraction that is represented by the repeating decimal 0.44… . First, write an equation by setting the decimal equal to a variable that will represent the fraction. w 5 0.44… Next, write another equation by multiplying both sides of the equation by a power of 10. The exponent on the power of 10 is equal to the number of decimal places until the decimal begins to repeat. In this case, the decimal begins repeating after 1 decimal place, so the exponent on the power of 10 is 1. Because 101 5 10, multiply both sides by 10. 10w 5 4.4… Then, subtract the first equation from the second equation. 10w 5 4.44… 2w 5 0.44… 9w 5 4 Finally, solve the equation by dividing both sides by 9. 6. What fraction is represented by the repeating decimal 0.44...? ___ 8. Repeat the procedure above to write the fraction that represents each repeating decimal. a. 0.55… 5 ___ c. 0.12 5 b. 0.0505… 5 ___ © Carnegie Learning 7. Complete the steps shown to determine the fraction that is represented by 0.09 . d. 0.36 5 22 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 22 28/08/13 2:12 PM Problem 2 Nobody’s Perfect . . . Unless They’re a Perfect Square Recall that a square root is one of two equal factors of a given number. Every positive number has two square roots: a positive square root and a negative square root. For instance, 5 is a square root of 25 because (5)(5) 5 25. Also, 25 is a square root of 25 because (25)(25) 5 25. The positive square root is called the principal square root. In this course, you will only use the principal square root. The symbol, , is called a radical and it is used to indicate square roots. The radicand is the quantity under a radical sign. radical √25 radicand This is read as “the square root of 25,” or as “radical 25.” Remember that a perfect square is a number that is equal to the product of a distinct factor multiplied by itself. In the example above, 25 is a perfect square because it is equal to the product of 5 multiplied by itself. 1. Write the square root for each perfect square. __ a.√1 5 ___ d.√ 16 5 ___ g.√ 49 5 ____ j.√100 5 ____ m.√169 5 __ b.√4 5 ___ e.√ 25 5 h.√64 5 ___ ____ k.√121 5 n.√196 5 ____ __ c.√9 5 ___ f.√ 36 5 i.√ 81 5 ___ ____ l.√144 5 o.√225 5 ____ __ © Carnegie Learning 2. What do you think is the value of √ 0 ? Explain your reasoning. 1.3 Irrational Numbers • 23 462351_CL_C3_CH01_pp001-042.indd 23 28/08/13 2:12 PM 3. Notice that the square root of each expression in Question 1 resulted in a rational number. Do you think that the square root of every number will result in a rational number? Explain your reasoning. 4. Use a calculator to evaluate each square root. Show each answer to the hundredthousandth. ___ √25 5 __ √5 5 _____ √0.25 5 _____ √ 225 5 _____ √ 2500 5 _____ √ 6.76 5 ____ √ 676 5 _____ √ 67.6 5 ____ √250 5 ___ √2.5 5 _____ 5 √ 6760 ______ 5 √26.76 5. What do you notice about the square roots of rational numbers? 7. Is the square root of a decimal always an irrational number? © Carnegie Learning 6. Is the square root of a whole number always a rational number? 24 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 24 28/08/13 2:12 PM 8. Consider Penelope and Martin’s statements and reasoning, which are shown. Penelope ___ I know that 144 is a perfect square, and so √ 144 is a rational ____ and number. I can move the decimal point to the left and √ 14.4 ____ √ 1.44 will also be rational numbers. ____ Likewise, I can move the decimal point to the right so √ 1440 and ______ √14,400 will also be rational numbers. Martin ___ I know that 144 is a perfect square, and so √ 144 is a rational number. I can move the decimal point two places to the right or left to get another perfect square rational number. For instance, ____ ______ √ 1.44 and √ 14,400 will also be rational numbers. Moving the decimal two places at a time is like multiplying or dividing by 100. The square root of 100 is 10, which is also a rational number. Who is correct? Explain your reasoning. © Carnegie Learning 1.3 Irrational Numbers • 25 462351_CL_C3_CH01_pp001-042.indd 25 28/08/13 2:12 PM The square root of most numbers is not an integer. You can estimate the square root of a number that is not a perfect square. Begin by determining the two perfect squares closest to the radicand so that one perfect square is less than the radicand, and one perfect square is greater than the radicand. Then, use trial and error to determine the best estimate for the square root of the number. “It might be helpful to use the grid you created in Question 1 to identify the perfect squares.” ___ To estimate √ 10 to the nearest tenth, identify the closest perfect square less than 10 and the closest perfect square greater than 10. The closest The closest perfect square The square root perfect square less than 10: you are estimating: greater than 10: ___ 9 √10 You know: __ 16 ___ √ 9 3 √16 4 ___ ___ √ 10 So, 10 __ is___ between √ 90 and √ 10 16 . Why can't I say__ it's between ___ √ 0 10 ? 1 and √ 25 This means the estimate of √ 10 is between 3 and 4. Next, choose decimals between 3 and 4, and calculate the square of each number to determine which one is the best estimate. Consider: (3.1)(3.1) 5 9.61 (3.2)(3.2) 5 10.24 ___ So, √10 3.2 The symbol < means approximately equal to. ___ The location of √ 10 is closer to 3 than 4 when plotted on a √10 0 1 2 3 4 5 6 7 8 9 10 © Carnegie Learning number line. 26 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 26 28/08/13 2:12 PM 9. Identify the two closest perfect squares, one greater than the radicand and one less than the radicand. __ a.√8 ___ b.√ 45 ___ c.√ 70 ___ d.√ 91 10. Estimate the location of each square root in Question 9 on the number line. Then, plot and label a point for your estimate. 0 1 2 3 4 5 6 7 8 9 10 © Carnegie Learning 1.3 Irrational Numbers • 27 462351_CL_C3_CH01_pp001-042.indd 27 28/08/13 2:12 PM 11. Estimate each radical in Question 9 to the nearest tenth. Explain your reasoning. __ a. √8 ___ b. √45 ___ c. √70 ___ © Carnegie Learning d. √91 Be prepared to share your solutions and methods. 28 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 28 28/08/13 2:12 PM Worth 1000 Words Real Numbers and Their Properties Learning Goals Key Terms In this lesson, you will: real number Venn diagram closure Classify numbers in the real number system. Understand the properties of real numbers. T he word zero has had a long and interesting history so far. The word comes from the Hindu word sunya, which meant "void" or "emptiness." In Arabic, this word became sifr, which is also where the word cipher comes from. In Latin, it was changed to cephirum, and finally, in Italian it became zevero or zefiro, which was shortened to zero. The ancient Greeks, who were responsible for creating much of modern formal © Carnegie Learning mathematics, did not even believe zero was a number! 1.4 Real Numbers and Their Properties • 29 462351_CL_C3_CH01_pp001-042.indd 29 28/08/13 2:12 PM Problem 1 Picturing the Real Numbers 1. A wrestler uses a scale at home and at work to monitor his weight for a week. He records the following weights: 3 lbs, 152.0 lbs, 151 __ 1 lbs, 151.6 lbs 1 lbs, 151.8 lbs, 152.1 lbs, 151 __ 152 __ 2 4 2 Write his weights in order from least to greatest. In the first lesson of this chapter, you cut out 30 real numbers and sorted them. Now, let’s create a Venn diagram to organize this set of numbers. 2. Write the 30 numbers from the first lesson in order from least to greatest. The Venn diagram was introduced in 1881 by John Venn, British philosopher and mathematician. Combining the set of rational numbers and the set of irrational numbers produces the set of real numbers. You can use a Venn diagram to represent how the sets within the set of real numbers are related. 3. On the next page, create a Venn diagram to show the relationship between the six sets of numbers shown. Then, write each of the the Venn diagram. integers irrational numbers natural numbers rational numbers real numbers whole numbers © Carnegie Learning 30 numbers from Question 2 in the appropriate section of 30 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 30 28/08/13 2:12 PM 4. Use your Venn diagram to decide whether each statement is true or false. Explain your reasoning. a. A whole number is sometimes an irrational number. © Carnegie Learning b. A real number is sometimes a rational number. c. A whole number is always an integer. 1.4 Real Numbers and Their Properties • 31 462351_CL_C3_CH01_pp001-042.indd 31 28/08/13 2:12 PM d. A negative integer is always a whole number. e. A rational number is sometimes an integer. f. A decimal is sometimes an irrational number. 5. Omar A square root is always an irrational number. Explain to Omar why he is incorrect in his statement. 6. A fraction is never an irrational number. © Carnegie Learning Robin Explain why Robin’s statement is correct. 32 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 32 28/08/13 2:12 PM Problem 2 Properties of Real Numbers The real numbers, together with their operations and properties, form the real number system. You have already encountered many of the properties of the real number system in various lessons. Let’s review these properties. Closure: A set of numbers is said to be closed under an operation if the result of the operation on two numbers in the set is a defined value also in the set. For instance, the set of integers is closed under addition. This means that for every two integers a and b, the sum a 1 b is also an integer. 1. Is the set of real numbers closed under addition? Write an example to support your answer. 2. Is the set of real numbers closed under subtraction? Write an example to support your answer. 3. Is the set of real numbers closed under multiplication? Write an example to support your answer. 4. Is the set of real numbers closed under division? Write an example to support your answer. Additive Identity: An additive identity is a number such that when you add it to a second © Carnegie Learning number, the sum is equal to the second number. 5. For any real number a, is there a real number such that a 1 (the number) 5 a? What is the number? 6. Does the set of real numbers have an additive identity? Write an example to support your answer. 1.4 Real Numbers and Their Properties • 33 462351_CL_C3_CH01_pp001-042.indd 33 28/08/13 2:12 PM Multiplicative Identity: A multiplicative identity is a number such that when you multiply it by a second number, the product is equal to the second number. 7. For any real number a, is there a real number such that a 3 (the number) 5 a? What is the number? 8. Does the set of real numbers have a multiplicative identity? Write an example to support your answer. Additive Inverse: Two numbers are additive inverses if their sum is the additive identity. 9. For any real number a, is there a real number such that a 1 (the number) 5 0? What is the number? 10. Does the set of real numbers have an additive inverse? Write an example to support your answer. Multiplicative Inverse: Two numbers are multiplicative inverses if their product is the multiplicative identity. You have been using these properties for a long time, moving forward you now know that they hold true for the set of real numbers. 11. For any real number a, is there a real number such that a 3 (the number) 5 1? What is 12. Does the set of real numbers have a multiplicative inverse? Write an example to support your answer. © Carnegie Learning the number? 34 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 34 28/08/13 2:12 PM Commutative Property of Addition: Changing the order of two or more addends in an addition problem does not change the sum. For any real numbers a and b, a 1 b 5 b 1 a. 13. Write an example of the property. Commutative Property of Multiplication: Changing the order of two or more factors in a multiplication problem does not change the product. For any real numbers a and b, a 3 b 5 b 3 a. 14. Write an example of the property. Associative Property of Addition: Changing the grouping of the addends in an addition problem does not change the sum. For any real numbers a, b and c, (a 1 b) 1 c 5 a 1 (b 1 c). 15. Write an example of the property. Associative Property of Multiplication: Changing the grouping of the factors in a multiplication problem does not change the product. For any real numbers a, b, and c, (a 3 b) 3 c 5 a 3 (b 3 c). © Carnegie Learning 16. Write an example of the property. Reflexive Property of Equality: Symmetric Property of Equality: For any real number a, a 5 a. For any real numbers a and b, if a 5 b, 17. Write an example of the property. then b 5 a. 18. Write an example of the property. Transitive Property of Equality: For any real numbers a, b, and c, if a 5 b and b 5 c, then a 5 c. 19. Write an example of the property. 1.4 Real Numbers and Their Properties • 35 462351_CL_C3_CH01_pp001-042.indd 35 28/08/13 2:12 PM Talk the Talk For each problem, identify the property that is represented. 1. 234 1 (2234) 5 0 2. 24 3 (3 3 5) 5 (24 3 3) 3 5 3. 224 3 15 224 4. 267 3 56 5 56 3 (267) 5. 2456 1 34 5 34 1 (2456) 6. 4 3 0.25 5 1 7. If 5 5 (21)(25) then (21)(25) 5 5. 8. If c 5 5 3 7 and 35 5 70 4 2, then c 5 70 4 2. ( )( ) 9. a 1 (4 1 c) 5 (a 1 4) 1 c 3 4 10. 2__ 2__ 5 1 4 3 3 3 3 1 5 22 __ 11. 22 __ 4 4 3 3 4 1 __ 4 1 5 5 2__ 1 __ 1 5 12.2__ 4 4 3 3 ( ) ( ) ( ) © Carnegie Learning 13. Order each set of real numbers from least to greatest. __ __ 4 , 22.8, √ a. __ 3 , 2√5 , 22% b. A baker measures the following amounts 3 of whole wheat flour for several recipes: 1 cup, 2.5 cups, 0.75 cup, __ 2 cup, __ 3 4 1 __ 1 cups, 0.2 cup 3 Be prepared to share your solutions and methods. 36 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 36 28/08/13 2:12 PM Chapter 1 Summary Key Terms Properties natural numbers Additive Identity (1.4) Multiplicative (counting numbers) (1.2) whole numbers (1.2) integers (1.2) closed (1.2) rational numbers (1.2) irrational number (1.3) terminating decimal Identity (1.4) Additive Inverse (1.4) Multiplicative repeating decimal (1.3) bar notation (1.3) Addition (1.4) Associative Property of Multiplication (1.4) Reflexive Property of Equality (1.4) Inverse (1.4) Commutative Property of Symmetric Property of Equality (1.4) Addition (1.4) Commutative Property of (1.3) Associative Property of Transitive Property of Equality (1.4) Multiplication (1.4) real number (1.4) Venn diagram (1.4) closure (1.4) Providing Rationale for Groupings of Numbers Numbers can be grouped in a variety of ways according to their characteristics. Numbers can be identified as whole numbers or integers, fractions or decimals, rational or irrational. Sometimes a number may fit into multiple groupings. When providing a rationale, all of the numbers in the group must fit that rationale. © Carnegie Learning Example 3 p, 3.25, 2__ , 65%, 215, 2.52, |28|, ___ 20 , 3.9 3 102 7 5 The numbers 215, |28|, ___ 20 , and 3.9 3 102 can be grouped together and identified as 5 integers because each of these numbers can be written as an integer. Chapter 1 Summary • 37 462351_CL_C3_CH01_pp001-042.indd 37 28/08/13 2:12 PM Comparing and Ordering Rational Numbers Using a Number Line Fractions and decimals can be compared and ordered by converting fractions to decimals and plotting on a number line. Example 5 . 5.3. The fraction 5 ___ 5 is equal to 5.3125. The number line is used to show 5 ___ 16 16 5.3 5.28 5.29 5.30 5.3125 5.31 5.32 5.33 5.34 5.35 Performing Operations with Rational Numbers A rational number is a number that can be written in the form __ a , where a and b are both b integers and b is not equal to 0. You can add, subtract, multiply, and divide rational numbers in much the same way that you do using integers. Example 19 ___ 3 3 6 __ 4 5 2 ___ 22 __ 3 58 8 8 9 9 1102 5 2 _____ 72 22 5 215 ___ 72 11 5 215 ___ 36 Identifying Terminating and Repeating Decimals A terminating decimal is a decimal that has a last digit. A repeating decimal is a decimal with digits that repeat in sets of one or more. Two different notations are used to represent repeat, and place a bar over the repeating digits. Another notation is to write the decimal, including two sets of the digits that repeat, and using dots to indicate repetition. Examples __ 7 is a terminating decimal: 8 4 is a repeating decimal: and __ 9 0.875 ______ © Carnegie Learning repeating decimals. One notation is to write the decimal, including one set of digits that 0.444 ______ 4 5 9)4.000 __ __ 7 5 8)7.000 8 9 __ 4 5 0.4 __ 7 5 0.875 __ 8 9 38 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 38 28/08/13 2:12 PM Writing Repeating Decimals as Fractions Some repeating decimals represent common fractions, such as 0.33… 5 __ 1 , and are 3 used often enough that we recognize the fraction by its decimal representation. However, there are decimals in which it is difficult to determine the fractional equivalent. To determine the fraction for the decimal, first, write an equation by setting the decimal equal to a variable that will represent the fraction. Next, write another equation by multiplying each side of the equation by a power of 10. The exponent on the power of 10 is equal to the number of decimal places until the decimal begins to repeat. Then, subtract the first equation from the second equation. Finally, solve the equation. Example ___ The repeating decimal 0.15 is equal to the fraction ___ 5 . 33 ___ 100w 5 15.15 ___ 15 5 0. ____________ 2w 99w 99w 55 1515 w 5 ___ 15 99 w 5 ___ 5 33 Identifying Irrational Numbers Decimals that do not repeat and do not terminate are said to be irrational numbers. An irrational number is a number that cannot be written in the form __ a , where a and b are both b integers and b fi 0. Example ___ An example of an irrational number is √ 11 because it is a square root that is not a perfect © Carnegie Learning square and therefore has no repeating patterns of digits. Chapter 1 Summary • 39 462351_CL_C3_CH01_pp001-042.indd 39 28/08/13 2:12 PM Estimating Square Roots A square root is one of two equal factors of a nonnegative number. Every positive number has two square roots, a positive square root (called the principal square root) and a negative square root. To determine a square root that is not a whole number, identify the two closest perfect squares, one greater than and one less than the radicand. Then, estimate the square root to the nearest tenth. Example ___ Estimate √ 23 to the nearest tenth. ___ Twenty-three is between the two perfect squares 16 and 25. This means that √ 23 is between 4 and 5, but closer to 5. ___ 23 is approximately 4.8. Because 4.72 5 22.09 and 4.82 5 23.04, √ ___ The location of √ 23 is closer to 5 than 4 when plotted on a number line. √23 1 2 3 4 5 6 7 8 9 10 © Carnegie Learning 0 40 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 40 28/08/13 2:12 PM Classifying Numbers in the Real Number System Combining the set of rational numbers and the set of irrational numbers produces the set of real numbers. Within the set of rational numbers, a number can be or not be an integer, whole number, natural number, or some combination. You can use a Venn diagram to represent how the sets within the set of real numbers are related. Real Numbers Rational Numbers 2 3 , – , – 0.3, 1.0205 10–23, 3 8 9 , 0.001, 0.5%, 20%, 0.25, 0.25, 16 100 , 1,000,872.0245 0.91, 1.523232323…, 212%, 6 1 , 4 11 – 6.41, – Irrational Numbers – 2, 2, Integers – 101, – 9 , –1 Whole Numbers 0 Natural Numbers |2|, |–3|, 100, 42, 627,513, 3.21 1012 Examples π is an irrational number. 28 is a rational number and an integer. 23 is a natural number, whole number, integer, and rational number. __ 1 is a rational number. © Carnegie Learning 4 Chapter 1 Summary • 41 462351_CL_C3_CH01_pp001-042.indd 41 28/08/13 2:12 PM Understanding the Properties of Real Numbers The real numbers, together with their operations and properties, form the real number system. The properties of real numbers include: Closure: A set of numbers is said to be closed under an operation if the result of the operation on two numbers in the set is another member of the set. Additive Identity: An additive identity is a number such that when you add it to a second number, the sum is equal to the second number. Multiplicative Identity: A multiplicative identity is a number such that when you multiply it by a second number, the product is equal to the second number. Additive Inverse: Two numbers are additive inverses if their sum is the additive identity. Multiplicative Inverse: Two numbers are multiplicative inverses if their product is the multiplicative identity. Commutative Property of Addition: Changing the order of two or more addends in an addition problem does not change the sum. Commutative Property of Multiplication: Changing the order of two or more factors in a multiplication problem does not change the product. Associative Property of Addition: Changing the grouping of the addends in an addition problem does Hopefully you didn't become irrational during this chapter! Remember keep a positive attitude_it makes a difference! not change the sum. Associative Property of Multiplication: Changing the grouping of the factors in a multiplication problem does not change the product. Reflexive Property of Equality: For any real number a, a 5 a. Symmetric Property of Equality: For any real numbers a and b, if a 5 b, then b 5 a. Transitive Property of Equality: For any real numbers a, b, and c, Examples 128 1 (2128) 5 0 shows the additive inverse. 13 3 (27) 5 27 3 13 shows the commutative property of multiplication. © Carnegie Learning if a 5 b and b 5 c, then a 5 c. 89 3 1 5 89 shows the multiplicative identity. (31 3 x) 1 y 5 31 1 (x 1 y) shows the associative property of addition. If x 5 7 1 y and 7 1 y 5 21, then x 5 21 shows the transitive property of equality. 42 • Chapter 1 The Real Number System 462351_CL_C3_CH01_pp001-042.indd 42 28/08/13 2:13 PM