Chapter 10 Expanding Our Number System


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1 Chapter 10 Expanding Our Number System Thus far we have dealt only with positive numbers, and, of course, zero. Yet we use negative numbers to describe such different phenomena as cold temperatures and debt. Negative numbers have been used for many centuries. Even though negative numbers did not come into worldwide acceptance until much later, the Chinese, in about 200 B.C., used red rods to represent positive numbers and black rods to represent negative numbers. This method is just the opposite of how we sometimes represent numbers today: we use black ink to indicate credits and red ink to indicate debits. What do you suppose running in the red means? In this chapter we extend our number system to include negative numbers and how to calculate with them. We review the properties of operations on numbers and once again discuss rational numbers and real numbers. Another section introduces some curious, but mathematically useful, number systems, and the final section gives a bit more history Ways of Thinking About Signed Numbers Positive and negative numbers are sometimes called signed numbers because of the + sign (for a positive number) or the sign (for a negative number) that may introduce the symbol, as in +2 or + 2 or 2 or 2. Except for emphasis, the + sign is often omitted, suggesting that the numbers we have used thus far can be regarded as special signed numbers. It is not uncommon to place the negative sign higher so that it is not confused with subtraction, until addition and subtraction of signed numbers are well understood: 3 may be written 3. Later, when there is no fear of confusion, we may use the symbol for negative numbers and for subtraction. One reason that signed numbers are important is that they have many applications, and so there are many ways of thinking about them. Here are some. Discussion 1 Other Ways of Representing Signed Numbers How could each of the following be used to think about signed numbers? Describe what positive and negative numbers, and zero, would mean. Continue on the next page. 201
2 202 Chapter 10 Expanding Our Number System 1. Financial matters like bank balances, profit/loss, paycheck/bill, income/debt, credit cards, etc. 2. Temperature changes 3. Sea levels 4. Sports settings like football and golf 5. Diets 6. Atomic charges (although atomic charges may not be part of the K 6 curriculum) 7. Games in which you can go in the hole We will focus now on two other ways to represent signed numbers: chips of two colors and the number line. Chips of two colors are an adaptation of the ancient Chinese method of 200 B.C. We will use white for positive and black for negative. For example, three white chips can represent + 3 (or 3), and 4 black chips can represent 4. Of course, any two colors can be used, so long as it is clear which represents positive numbers and which negative numbers. represents + 3 represents 4 Just as a gain of $2 can be cancelled by a loss of $2, giving a zero change in finances, a chip representation for 0 can occur in many forms. Each of these drawings is a way of showing 0: or or. Notice that there is a degree of abstraction here, requiring an understanding of the representation. Two white and two black chips represent zero, even though there are four chips involved. Two white chips have, in a sense, a canceling effect on two black chips (or two black chips have the opposite effect of two white chips). It is natural to think of addition as describing putting two white chips with two black ones, and the canceling effect gives the important numerical result, = 0. With the canceling idea in mind, chip drawings like each of the following can be interpreted as having the value 3. Do you see why? but also or These more complicated ways of showing an integer can be handy in subtraction, as we will see.
3 Section 10.1 Ways of Thinking About Signed Numbers 203 THINK ABOUT... What are several ways of showing + 2 with the chips, besides with two white chips? Why are these ways more abstract than showing just two white chips? The use of the number line to illustrate signed numbers is likely familiar to you and builds on work with whole numbers and fractions. If we use the usual number line as a model for the numbers, then we can answer the question, What s to the left of 0? by describing the numbers to the left of zero as negative numbers. For the time being and with the usual elementary school curriculum in mind, we will first restrict ourselves to those numbers that are the opposites of the whole numbers, 1 is the opposite of 1, 1 is the opposite of 1, 2 is the opposite of 2, and so on. When we combine the set of whole numbers with their opposites, including zero, we obtain a set of numbers we call integers. That is, I = {... 3, 2, 1, 0, 1, 2, 3,...}. These numbers can be represented on the number line as follows: We say that 6 and 6 are opposites or additive inverses of one another because their sum is 0. Generally speaking, the opposite of a is also denoted by a, whether a is positive or negative, and a + a = 0. THINK ABOUT... What is the opposite of 2? What is the opposite of 2? What is the opposite of the opposite of 2? What is the opposite of the opposite of 2? What is the opposite of 0? Is a always a negative number? What are the different meanings of the sign in 5 ( 2)? Although 6 can be read as either negative six or the additive inverse of six or the opposite of six, a should be read as the additive inverse of a or the opposite of a, but not as negative a, because a can be positive. Keep in mind that the labeling of the points for numbers on a number line comes about because of their distances from 0, a starting point. But + 2, say, could be thought of as any jump of length 2 units toward the right but starting anywhere (not necessarily at the 0 point), just as 3 could be thought of as any jump of length 3 units, but toward the left and starting anywhere. Thinking of a signed number as a description of a jump size rather than only as a point on the number line has value in working with addition and subtraction of signed numbers
4 204 Chapter 10 Expanding Our Number System As with colored chips, 0 can be represented with a number line in a variety of ways, such as the following: 1 +2 (0 units) or or Consistent with the points for whole numbers and fractions, on the usual number line the point for a smaller number is to the left of the point for a larger number. Hence, for example, 100 is less than 5, or 100 < 5. Thinking of debts and worse off for < helps to make such inequalities believable. We have focused primarily on the integers because they are the first of the signed numbers to appear in the usual elementary school curriculum. But the familiar fractions and decimal numbers can also have opposites (also called additive inverses). For example, ( 3) is a negative number between 4 1 and 0 on the number line. Its additive inverse is 3 4. (We have used parentheses here simply to show that the negative sign is for the entire fraction, not just the numerator.) Signed numbers include all positive and negative integers, positive and negative fractions, and positive and negative repeating or terminating decimal numbers, that is, all rational numbers, extending the term introduced in Chapter 6 for nonnegative rational numbers. Similarly, as with 3, there are negative irrational numbers (numbers that have nonterminating and nonrepeating decimals). Because the rational numbers and irrational numbers together are called the real numbers, these show that every real number has an additive inverse. Just as every integer can be matched to a point on the number line, so can every real number be matched to a point on the number line. And every point on a number line corresponds to some real number. With two or more number lines arranged to give the familiar xy coordinate system that you studied in algebra, this match of numbers and geometry allows many geometric shapes to be studied with algebra and many algebraic topics to be represented geometrically. Discussion 2 Between Any Two Rational Numbers 13 and. Find another rational num Think of any two rational numbers, such as 7 12 ber between the two numbers. 2. Find another rational number between 7 12 This will give a second number between 7 12 and the number you found in part a. and How many rational numbers are there between 7 12 and in all?
5 Section 10.1 Ways of Thinking About Signed Numbers 205 Your answer to Question 3 in Discussion 2 is perhaps a surprising consequence of the line of reasoning for Question 1. The mathematical term for this phenomenon is called the density property of rational numbers. The rational numbers are said to be dense. A set of numbers is dense if, for every choice of two different numbers from the set, there is always another number from the set that is between them (the density property). THINK ABOUT... How does the density property assure that there are infinitely many rational numbers, not just one, between every two different rational numbers? Hence, with the number line in mind, one might think that the points for the rational numbers completely fill up the line. But, as you know, the irrational numbers also have points on the number line. Even though the set of rational numbers is dense, there are still empty spaces for the irrational numbers. TAKEAWAY MESSAGE... There are many ways of thinking about integers and other signed numbers. Two of these involve chips of two colors and the number line, which can illustrate the opposing effects of a number and its additive inverse or opposite. The key feature of the additive inverse of a number a is that a + a = 0. All the rational and irrational numbers make up the real numbers, with every real number corresponding to a point on a number line, and vice versa. The rational numbers are dense, meaning that there is always another rational number between two given rational numbers; indeed, there are infinitely many. Learning Exercises for Section Make drawings of chips to show the following. Use white for positive and black for negative, for consistency. a. 5 b Make drawings of chips to show zero in at least four ways different from those in the text. Again, use white for positive and black for negative, for consistency. 3. Give the single integer that each of the following chip drawings can represent. Be ready to explain your thinking. (White positive, black negative) a. b. c. d. e. f.
6 206 Chapter 10 Expanding Our Number System 4. Reorder each group of numbers from smallest to largest. a. + 50, 3, 3 22,, , 2 3, 1, 1 b. 3.1, 5, , 4 0.9, 9, 0.5, 13 10, 4 9, 0.1, a. What is ( 8)? Explain why you think your answer is correct. b. Zero is regarded as neither positive nor negative, but one occasionally runs into 0 in calculations. What is 0? Explain. c. Is a always negative? Explain. 6. In each part, what number is being described? a. the additive inverse of the additive inverse of the additive inverse of 9 b. the additive inverse of the additive inverse of negative 9 c. the additive inverse of the additive inverse of 6 d. ( ( ( ( 10.3)))) 7. A jump on a number line may be followed by another jump that starts where the first jump ends. What single integer describes the net result in each of the following? a. + 3 b. + 2 c d. + 3 e. 4 f Give, if possible, an example of each type of number. If it is not possible, explain why. a. a negative real number that is not rational b. a negative real number that is not irrational c. a negative integer that is not real d. a negative real number that is not an integer 9. Interpret + 10 and 4 in these settings. a. a financial situation of some sort b. a sport c. a temperature change d. a temperature
7 Section 10.2 Adding and Subtracting Signed Numbers For = n, a child might draw a number line like the following. What might the child be thinking? Sometimes the negative sign is not raised, so instead of writing 3 we write 3. Why do you think this is done? What are the pros and cons of writing negative numbers this way? 12. The set of real numbers is dense. What does that mean? 13. In each part, tell whether the set of numbers is dense. Justify your answers. a. the set of integers b. the set of positive rational numbers c. the set of negative rational numbers 14. Give eight fractions between each pair. 7 a. and 9 b. 4 5 ( 2) and 3 ( 31) Adding and Subtracting Signed Numbers Just as there are several ways in which to think about signed numbers, there are several ways to think about adding and subtracting them. Here we will focus first on adding and subtracting integers via colored chips, the number line, and a money argument. Then we summarize the rules symbolically, using absolutevalue language. The rules that arise with integers are applicable to all signed numbers. We will use white chips to indicate positive integers, and black ones to indicate negative integers. We can think of the chips as positive and negative charges that cancel out one another, if there is the same number of whites as blacks. If we begin with three white chips and add three black chips, the chips cancel each other out, and we have zero: 3 + ( 3) = 0, incorporating the important additive inverse property into this model. Again, notice that although there are 6 chips visible, the meaning attached to them allows one to say 0 for this arrangement, much like having a check for $3 and a bill for $3 gives, in effect, $0. Using an optional box to surround the work is occasionally useful, especially when the sum is zero.
8 208 Chapter 10 Expanding Our Number System With the chips, adding integers with the same sign is straightforward and involves showing each addend with chips and then counting the total = n = n = = 7 If the signs of the addends differ, we can use the additive inverse feature for integers. For example, for = n, we begin with three white chips and add five black chips, but three of the black chips cancel out the three white chips, and we are left with two black chips: = 2 For = n, two black chips cancel two of the six white chips, leaving four white chips, so = = 4 Notice that in effect, when the signs differ, just finding the difference in the numbers of chips for the addends and then giving that difference the sign of the larger number of chips, yields the sum. You may have learned something like that as a rule for adding numbers with different signs. Let us turn to subtraction of integers with the chips. Discussion 3 Subtracting with the Chip Model How can you use the takeaway interpretation for subtraction to show each of the following with chips? ( 2) As you probably noticed, some subtractions are very easy and can be shown in ways similar to those used for whole numbers. For example, for 4 3 = 1, we could show the following Or, for 5 2 = 3,
9 Section 10.2 Adding and Subtracting Signed Numbers 209 In Discussion 3, the last problem raises the question, What if there are not enough chips to remove, as with 4 7 = n? Two ways may have arisen, each adding equal numbers of both negative and positive chips in effect, adding 0 and then subtracting. 4 4, so 7 whites can be taken away after 7 whites removed 4 7 = 3 You may also have thought, Why not just put in 3 more whites and 3 blacks; then you could take away 7 whites? The answer is, You could. The way illustrated, however, suggests at the third step, 4 7 = 4 + 7, and you may recognize in that equation the basic rule for subtracting signed numbers: Change the sign of the subtrahend (the number being subtracted), and add. It may be instructive to see the symbolic form for the steps in the work for 2 ( 3) = n. 2 ( 3) = [2 + 0] ( 3) = [ ( 3)] ( 3) = [5 + ( 3)] ( 3) = 5. Replace 0 with 3 + (  3). Notice that 2 ( 3) gives the same result as 2 + 3, supporting the usual rule for subtraction. You can even see the in the last drawing. Let us turn to the number line. Addition and subtraction of signed numbers on a number line may already be familiar to you. Activity 1 Hopping on the Number Line Using the number line, find the following, in a way that makes sense Then check your answers by reading the next paragraphs. If you have forgotten how to use the number line for these sums, here is some help. (Draw a number line, and read the instructions slowly.) To find 3 + 4, start at 0, move 3 units to the right, then move 4 more units to the right, to 7. (Starting at 0 allows one to read the answer from the numberline markings.) Think of a positive addend as moving to the right and a negative addend as moving to the left.
10 210 Chapter 10 Expanding Our Number System To subtract 6 5 using the number line, begin at 0, move to the right 6 units, and then take away 5 units to the left, to 1. Continuing on the number line you drew (or a copy), if you want to find 6 + ( 5), begin at 0 and move 6 units to the right, then move 5 units to the left. Note that 6 + ( 5) takes us to the same place on the number line as does 6 5, that is, 6 5 = 6 + ( 5). Similarly, if you want to find 6 + ( 9), you go from 0 to 6 and move 9 units to the left, ending at 3. The 6 5 = 6 + ( 5) equation suggests that 6 + ( 9) should be 6 9. And from the number line, 6 9 = 3. Again, notice that the equations 6 5 = and 6 9 = suggest the eventual rule for subtraction, even though that may seem irrelevant at this point. How would one show 6 ( 9) on the number line? Rather than introduce a new interpretation of subtraction (e.g., Do the opposite) as is often done, we can introduce 0 in a clever way, as was done with the chips, and continue with takeaway as the meaning for subtraction. First, show 6 as 6 + 0, in the form , so that the 9 can be taken away Once the 9 segment is taken away, we are left with 6 + ( + 9). So our illustration means that 6 ( 9) = 6 + ( + 9), an equation again supporting the eventual rule. As a final way of thinking about addition and subtraction of signed numbers, let us consider money, first looking at addition and the symbolic rules governing addition, and then similarly at subtraction. The symbolic rules use the idea of absolute value, so we will first review that topic. There are times when we are interested in a number s direction that is, whether its place on the number line is to the left or to the right of 0. At other times we are interested only in a number s distance away from 0 on the number line, and do not care in which direction we must go to arrive at the number. A number s distance from 0 on the number line is called the absolute value of the number, and we consider this value to be positive (or zero in the case of zero). We denote the absolute value of a number b as b. EXAMPLE 1 We can say that 6 is 6, and similarly, 6 is 6. Both 6 and 6 are 6 units away from zero. Opposite numbers always have the same absolute value. In terms of chips, absolute value can be interpreted as just how many uncanceled chips there are. For example, 3 = 3 because there are 3 black chips (or a surplus of 3 black chips), and + 9 = 9 because there are 9 white chips (or a surplus of 9 white chips).
11 Section 10.2 Adding and Subtracting Signed Numbers = 33 = 3 With absolute value in mind, we now return to the addition of signed numbers. Activity 2 Leading to the Rules for Addition Use examples from a money context to consider these situations. 1. Adding two signed numbers that have the same sign (for example, , and ). 2. Adding two signed numbers with different signs (for example, , and ). Your thinking in Activity 2 could be formalized using absolute value, as is often done in algebra classes. Addition of signed numbers when both numbers have the same sign: If both numbers are positive, then a + b = a + b. If both numbers are negative, then a + b = ( a + b ). Addition of signed numbers when one is positive and the other is negative: Consider a to be positive and b to be negative. If a > b then a + b = a b. If a < b then a + b = ( b a ). THINK ABOUT How is your thinking in Activity 2 reflected in the formal addition rules using absolute value? 2. Is a b = b a always? Is a + b = a + b always? Is a b = a b always?
12 212 Chapter 10 Expanding Our Number System Two special cases for addition, with the relevant vocabulary, should be highlighted. Special cases: If a = 0, then a + b = b and b + a = b. We call 0 the additive identity. If a = b then a + b = b + b = 0 (the additive identity). We call each of b and b the additive inverse of the other because their sum is 0. Notice that work with chips or the number line or some other representation of signed numbers can suggest the formal statements. Having a mental image of chips or a money situation, for example, can allow you to understand where the rules come from. Fortunately, we can use the rules for addition when we subtract, by changing from subtraction to addition according to the rule below. We noted this relationship in several of the previous calculations with the chips and the number line. Subtraction of signed numbers: If c and d are signed numbers, then c d = c + ( d). EXAMPLE 2 a. 5 2 = = 7 b = = 16 Activity 3 Can You Add and Subtract? Calculate, referring to the rules for addition and subtraction given on pages 209 and 210. (For each subtraction problem, first rewrite it as an addition problem.) Although we have used the takeaway view of subtraction with the chips and the number line to motivate the rule for subtraction of signed numbers, the missingaddend view of subtraction could also be used. Recall that the missingaddend approach for c d would ask: What can be added to d to get c? From c d = x, a symbolic line of reasoning could also result in the c d = c + d rule, as follows.
13 Section 10.2 Adding and Subtracting Signed Numbers 213 c d = x Then, thinking of missing addend for c d, x + d = c x + d + d = c + d (by adding d to both sides) x + 0 = c + d x = c + d So c d = x = c + d Activity 4 Does the MissingAddend View Work with Chips and the Number Line? Can the colored chips and the number line also be used with the missingaddend view of subtraction? Try them with the following (Think: What can be added to 2 to get 4?) We have used some properties of addition without comment. The properties reviewed earlier for whole numbers and rational numbers do continue to be true when negative and irrational numbers are involved, that is, when any real numbers come up. The commutative property of addition allows us to commute the order of the addends, for example, = The associative property of addition allows us to change the order in which we add. For example, (3 + 7) + 5 = 3 + ( 7 + 5). In checking that these two sums are equal, we have = 9 and = 9 for both sides of the equation. THINK ABOUT... Does the associative property of addition allow you to ignore the parentheses when only addition is involved? Why or why not? Rational numbers include all whole numbers, fractions, and repeating decimal numbers. Negative integers and negative fractions (or their decimal forms) are also rational numbers. In fact, when we add any two rational numbers, the sum is also a rational number. This is an example of what is called the closure property, a property that is not usually emphasized in grades K 6 but is useful in more advanced work. A set of numbers is closed under an operation if, when operating on every two numbers in the system, the result is also in the set of numbers.
14 214 Chapter 10 Expanding Our Number System EXAMPLE 3 When we add any two positive rational numbers, such as and 5, the sum, , is also a rational number. This property extends to include negative and positive numbers = 4 3 4, a rational number. In both cases, the sum of two rational numbers was another rational number, so we could say that this example illustrates that the set of rational numbers is closed under addition. Even though the full set of real numbers is not commonly encountered in grades K 6, it is also true that the set of real numbers is closed under addition. For example, and are real numbers. Finally, addition of signed numbers has two additional properties, noted earlier. The first is that 0 is the additive identity. The second is that every real number has an additive inverse. EXAMPLE 4 a = 3 and = 3, so 0 is the additive identity. b = 0, so 3 and 3 are additive inverses of one another. c. What is the additive inverse of 2 1 8? It is because = is also the additive inverse of d. Similarly, 7 and 7 are additive inverses of each other because their sum equals 0. THINK ABOUT... Suppose you are restricted to the set of integers. Do all five of the properties for addition closure, commutativity, associativity, additive identity, and existence of additive inverses hold true for all integers? The properties often (but not always) occur in mathematical situations, perhaps situations not even involving numbers. Mathematicians look for situations in which the properties do occur because the properties may lead to still other results that may be useful in the situations. TAKEAWAY MESSAGE Colored chips and the number line are just two methods of illustrating the addition of signed numbers. Their use can motivate the usual rules for adding and subtracting signed numbers. The rules apply to all the real numbers, irrational as well as rational. There are five properties for addition of rational numbers, all of which also are true for all real numbers: 1. The set of rational numbers is closed under addition. That is, for every two rational numbers a and b, a + b is also a rational number. 2. Addition is commutative. That is, for every two rational numbers a and b, a + b = b + a.
15 Section 10.2 Adding and Subtracting Signed Numbers Addition is associative. That is, for every three rational numbers a, b, and c, (a + b) + c = a + (b + c). 4. Existence of an additive identity. It is 0. That is, for every rational number a, a + 0 = 0 + a = a. 5. Every rational number has an additive inverse that is rational. That is, for any rational number a, there is another rational number a such that a + a = 0. Learning Exercises for Section Using drawings of two colors of chips, find the following. a b c. 4 2 d e. (4 + 3) + 1 f g. 4 3 h Make numberline drawings to find the following. a b c d e f g. 5 2 h Add and subtract these numbers using drawings of chips, the number line, or a money situation. a b. ( 5) + 5 c. 5 + ( 5) d. 5 5 e. 5 ( 5) f Calculate the following. a b c d e f g h. 3.5 ( 8) 5. a. Is the set of even integers closed under addition? Why or why not? (The even integers are..., 4, 2, 0, 2, 4,...) b. Is the set of multiples of 3 closed under addition? Why or why not? c. Is the set of odd numbers closed under addition? Why or why not? (The odd integers are... 3, 1, 1, 3, 5,....) d. Is the set of whole numbers closed under subtraction? Why or why not? e. Is the set of all integers closed under addition? Why or why not? f. Is the set of all integers closed under subtraction? Why or why not? g. Is the set of all positive rational numbers closed under subtraction? Why or why not?
16 216 Chapter 10 Expanding Our Number System 6. For each of the following, say whether or not the statement is true. If it is true, state the property that makes it true. a. (3 + 4) + 6 = ( 4 + 3) + 6 b. (3 + 4) + 6 = 3 + ( 4 + 6) c = 2 d = 0 e (4 + 4) = f (4 + 4) = g is a rational number. In fact, it is an integer. h. (3 + 4) + 6 = 6 + (3 + 4) i = j = 0 7. Provide the additive inverse for each of the following: a. 13 b c. 4 9 d In Learning Exercise 5 in Section 3.2 we considered families of addition and subtraction facts such as the following: = = = = 2 Complete these fact families for integers: a = = 2 b. 2 5 = = = 5 c = a. 3 =? b. 3 =? c =? d =? e =? f. 6 6 =? g. 6 6 =? 10. Temperature change is often used as a setting for adding and subtracting integers. Design some problems that you could use to teach someone else how to add and subtract integers. 11. Is it possible to use the comparison view of subtraction with the chip model with signed numbers? With the number line? With money? (The comparison for 7 2, for example, would tell how much greater 7 is than 2, or how much less 2 is than 7.) 12. For each story and with signed numbers, write an equation that describes the situation, and answer the question. a. An official from Company A said, Here s how we did last year. The first quarter we earned $57,000, and the second quarter, $35,000. But during each of the third and fourth quarters, we lost $16,000. How did Company A fare, for the whole year? (Remember to use signed numbers in your equation.)
17 Section 10.3 Multiplying and Dividing Signed Numbers 217 b. Company B reports, During the second quarter, we earned $92,000, so for the first two quarters, we have earned a total of $15,000. How did Company B do during the first quarter? c. Company B later reports, We lost $125,000 in the fourth quarter, so now we have earned only $11,000 in all, for the last two quarters. How much did the company gain or lose during the third quarter? d. How did Company B (parts b and c) do in all, for the first and fourth quarters only? 13. Write a story problem involving financial matters for an individual (paycheckbill, incomedebt, credit cards, etc.) that could be described by each of the following. Give the answers to your questions. a = n b = n 14. For each equation below, write a story problem involving football or golf or diets that could be described by the equation. a = n b = n 10.3 Multiplying and Dividing Signed Numbers Addition and subtraction of signed numbers usually first appear in grades 5 6, perhaps just with the integers. Multiplication and division of signed numbers are then treated in grades 6 7. You may have heard the rhyme: Minus times minus is plus, the reason for this we need not discuss. But here we do discuss it. If you have your own doubts about why multiplying two negative numbers gives a positive number, you are in good company. Until the mid1800s there was a great deal of resistance to that result even by many mathematicians (see Section 10.5 for some more history). We will offer different arguments as to what the sign of a product involving negative numbers should be. The first argument will assume commutativity of multiplication and consider a pattern. The second will show that the desirable properties of multiplication necessarily lead to negative times negative is positive. A final argument, using chips, is offered. The chips argument is given last because the usual elementary school curriculum does not include it. Multiplications involving positive numbers or zero are already familiar. The product of two positive numbers is positive, and if zero is a factor, the product is zero. The other cases involve multiplying numbers (a) of opposite signs and (b) when both are negative. We will focus on integers, although the same results will apply to all real numbers. Earlier we found that one way of thinking about multiplication is as repeated addition. Applying this view of multiplication to 4 2 gives 4 2 = = 8, suggesting that (positive) (negative) = (negative).
18 218 Chapter 10 Expanding Our Number System But it is not so easy to think about what 2 4 could mean as repeated addition. However, if multiplication of integers is to be commutative, then 2 4 must equal 4 2, which we have just shown is 8. In other words, 2 4 = 8, suggesting that (negative) (positive) = (negative). EXAMPLE 5 a. 3 4 = 12 b = 14 6 = 84 The only case left is that of having two factors that are both negative. The pattern in the next activity is suggestive (and is common in the elementary school curriculum). Activity 5 A Strange Rule? Using the results for multiplying a negative number and a positive number, complete the patterns in these two columns. (Even if you know the answers already, look for the patterns as you go down a column.) 4 2 = = = = = = = =? 0 2 = =? 1 2 =? 4 1 =? 2 2 =? 4 2 =? 3 2 =? 4 3 =? etc. etc. Although the result may seem counterintuitive, the pattern suggests that the product of two negative numbers must equal a positive number. Here is a summary of all the results from this first line of reasoning. (The results actually apply to all real numbers.) Multiplying two signed numbers. If the signs of the two numbers are the same, the product will be positive. If the signs of the two numbers are different, the product will be negative. EXAMPLE 6 a. 3 4 = 12 b = 0.08
19 Section 10.3 Multiplying and Dividing Signed Numbers 219 Discussion 4 Convinced? Did you find the pattern argument for the product of two negative numbers convincing? Do you think young students would find it convincing? (Mathematicians like patterns, but they do not trust them completely because sometimes the patterns can break down.) The second line of reasoning for the product of two negatives rests solely on properties of multiplication that we would want to continue to be true for signed numbers, so we will look at those properties. We have already used commutativity of multiplication, but there are other important properties as well. In the following activity and discussion, notice the parallels to the corresponding properties of addition. Many will be stated in terms of rational numbers, but they are also true for real numbers. Discussion 5 More Properties 1. Provide several examples to test whether multiplication of signed numbers is associative. That is, when three rational numbers are multiplied, does it matter which multiplication is done first: a(bc) = (ab)c, for every choice of rational numbers a, b, and c? For example, will 3. ( 2. 4) give the same result as ( 3. 2). 4? (Recall that the multiplication symbol is often replaced with.. In fact, when one or both factors are represented with letters, there oftentimes is no symbol between the letters if multiplication is intended: 2 b = 2. b = 2b, or a b = a. b = ab.) 2. Is the set of rational numbers closed under multiplication? That is, when every choice of two rational numbers are multiplied, is the product always a rational number? 3. Is there an identity for multiplication of rational numbers? That is, for each rational number a, is there a rational number x for which a. x = x. a = a? If so, what is it? 4. Does every rational number have a multiplicative inverse? That is, if c is any rational number, then does there exist a rational number d such that c. d = d. c = 1 (where 1 is the identity for multiplication)? 5. For rational numbers, is multiplication distributive over addition? That is, if a, b, and c are any rational numbers, is it true that a. (b + c) = a. b + a. c? Substitute numbers for a, b, and c and test whether or not this property appears to be true for all rational numbers. Commutativity of multiplication also gives (x + y). z = x. z + y. z as a form of this distributivity property. The multiplicative identity for the set of rational numbers is 1 because for every rational number a, 1. a = a, and a. 1 = a.
20 220 Chapter 10 Expanding Our Number System If the product of two numbers is 1, each number is the multiplicative inverse of the other number. If a is not 0, its multiplicative inverse is often written 1 a or even a 1. The multiplicative inverse of a (nonzero) fraction is sometimes called its reciprocal. THINK ABOUT... What is the reciprocal of m n? How does your answer satisfy the description above? Is there a multiplicative identity for the set of integers? Do integers have multiplicative inverses? Why doesn t 0 have a multiplicative inverse? A second way to show that defining the product of two negative numbers to be positive makes sense mathematically is illustrated in this example, which depends heavily on the distributive property. Suppose the product of concern is Start with 3. 0 = 0. Substitute for the first ( ) = 0 after the substitution. ( ) + ( 3. 2) = 0 using the distributive property. 6 + ( 3. 2) = 0 using the known = 6. So 3. 2 must be equal to + 6 to make the equation true. Basically, all this is saying is that if the product of two negative numbers were not defined to be a positive number, then at least some of the rules of numbers we so far know to be true would fail when negative numbers are included. The rules for multiplication of integers automatically provide us ways of dividing signed numbers, both positive and negative. The missingfactor way of thinking about division is useful. Recall that for 16 8, say, this view of division says to think, What times 8 gives 16? Because 2. 8 = 16, 16 8 = 2. In general, division can be defined as follows: If a, b, and c are real numbers and b is not 0, then c b = a if a. b = c.
21 Section 10.3 Multiplying and Dividing Signed Numbers 221 EXAMPLE 7 a = 3 because 3. 4 = 12 b = 3 because 3. 4 = 12 c = 3 because 3. 4 = 12 d = 3 because 3. 4 = 12 THINK ABOUT... Why can t 0 be the divisor in the definition of division (... and b is not 0... )? How does the missingfactor view lead to (negative) (negative) = (positive)? Multiplication and division of signed numbers: The product or quotient of two numbers with the same sign is positive. The product or quotient of two numbers with opposite signs is negative. The equality link between a b and a b extends to signed numbers. Hence, 17 5, 17 5, and ( 17 5 ) are all equal, because 17 5 = ( 17) 5 = ( 17 5 ) and 17 = 17 5 = ( 17). This equality between a 5 5 b defining the rational numbers in advanced work. and a b leads to a common way of A rational number is any number that can be expressed in the form integer nonzero integer You might wonder whether the chips of two colors can also be used to demonstrate multiplication of integers. Consider the repeated addition model of multiplication. We know that = 2 4 can be thought of as However, repeated addition cannot serve as a model for multiplication when the first factor is negative. Thinking about 2 4 as repeated addition does not make sense; how would one add 4 negative 2 times? Rather than appealing to commutativity of multiplication, the key is in recognizing that since repeated addition does not make sense for 2 4, then when the first factor is negative, we need a new interpretation. Since in we can think of adding 4 two times, it is not unnatural to think of 2 4 as subtracting 4, two times. But how do we get started? As with repeated addition, the answer is to start with a neutral amount (0), but here cleverly chosen so that the subtractions are possible. For 2 4, the work might proceed as follows: Starting with 0. Take away 2 fours; answer is 8.
22 222 Chapter 10 Expanding Our Number System For 2 4, we would take away two sets each with 4 black chips, so the answer is the 8 white chips left, or 2 4 = + 8. This process would lead us to the same rules for multiplying with signed numbers. Again, if division is considered in missingfactor terms, the rules for division of signed numbers would also continue to hold. TAKEAWAY MESSAGE... Multiplication and division of signed numbers were considered in this section. New are cases where one or both of two numbers are negative. In multiplying or dividing two positive numbers or two negative numbers, the answer is a positive number. If the two numbers have opposite signs, the answer is a negative number. These rules apply to all real numbers. The following five properties involving multiplication are true for the rational numbers and also for all the real numbers. 1. The set of rational numbers is closed under multiplication. That is, for every two rational numbers a and b, a. b is also a rational number. 2. Multiplication is commutative. That is, for every two numbers a and b, a. b = b. a. 3. Multiplication is associative. That is, for every three numbers a and b and c, (a. b). c = a. (b. c). 4. Multiplication has an identity. It is 1. That is, for every number a, a. 1 = 1. a = a. 5. Every nonzero number has a multiplicative inverse. That is, for each nonzero number a, there is another number b such that a. b = 1. The multiplicative inverse of a is sometimes called the reciprocal of a and denoted by 1 a or a 1. Finally, there is a sixth property that relates addition and multiplication: 6. For any numbers a, b, and c, multiplication is distributive over addition. That is, for rational numbers a, b, and c, a(b + c) = a. b + a. c. [Also useful is (x + y). z = x. z + y. z.] Learning Exercises for Section Use examples to test which property or properties of the five properties of addition, the five properties of multiplication, and the distributive property of multiplication over addition, do or do not hold for just the set of integers. 2. Use examples (including negative rational numbers) to illustrate that the eleven properties all hold for the set of rational numbers. 3. Does a fraction with a negative numerator and positive denominator have the same value as a similar fraction but this time with a positive numerator and
23 Section 10.3 Multiplying and Dividing Signed Numbers 223 negative denominator? That is, is to equal to 2 5? Is either or both of these equal? Explain in terms of a fraction representing a division. 4. Practice operations on signed numbers by completing the following computations. a b c. 7. (3 + 5) d e f. 7 4 g i j. k m. ( 1) 100 n. ( 1) Is there an identity for multiplication of integers? If so, what is it? h l. ( 8 + 2). 5 (two ways?) 6. a. Suppose you take any integer a. Can you always find another integer b such that a b = 1? (That is, does a have a multiplicative inverse in the set of integers?) b. Which two integers are their own multiplicative inverses? 7. Identify which of the eleven properties of addition and multiplication is exhibited in each of the following. Or, if a statement is not true, fix it so that it is, and tell which property you used. a. 7. (3 + 5) = (3 + 5). 7 b. 7. (3 + 5) = 7. ( 5 + 3) c. 7. (3 + 5) = d. 7. [(3 + 5) + 4] = 7. [3 + ( 5 + 4)] e. 7. [(3 + 5) + 4] = 7. (3 + 5) f. 7. [(3 + 5) + 4] = [(3 + 5) + 4]. 7 g = h = 1. 7 = 7 i. (3 + 0) + 4 = j. (3 + 5). 6 = Demonstrate, with drawings, how obtaining the following products could be demonstrated using two colors of chips. a. 3 5 b. 3 5 c. 2 3 d Give the line of reasoning similar to the one given for 3 2 = + 6 to show that 7 5 must be What is missing in each student s understanding? Ann: I had = + 6, and you marked it wrong. But you said when you multiply negatives, you get a positive. Bobo: You said two negatives make a positive, but when I did = + 5, my Mom said it wasn t correct.
24 224 Chapter 10 Expanding Our Number System 11. Using signed numbers, write an equation that describes each of these story problems. Little BoPeep loses 4 sheep every week from her very large flock, and they never come home! a. In 5 weeks, how will the number in her flock compare to the present number? (Remember to use signed numbers.) b. Six weeks ago, how did the number in her flock compare to the present number? Godzilla has been losing 30 pounds a month by watching his diet and by exercising. (Remember to use signed numbers.) c. If he continues at this rate, how will his weight in 6 months compare to his present weight? d. Three months ago, how did his weight compare to his present weight? 10.4 Some Other Number Systems In the summaries of Sections 10.2 and 10.3, five properties were listed as being true for addition, five for multiplication, and one property that connected multiplication and addition. When all eleven of these properties hold for addition and multiplication on any set of numbers, mathematicians call this set with its two operations a field. (This perhaps surprising term was given by a mathematician who had a wide view of numbers.) Discussion 6 What Makes a Mathematical Field? 1. Is the set of even integers (that is,... 6, 4, 2, 0, 2, 4, 6,....) with addition and multiplication defined as usual, a field? If not, which of the eleven properties fails? 2. Is the set of positive rational numbers a field? If not, which of the eleven properties fails? In an earlier chapter we also talked about irrational numbers, that is, numbers that cannot be expressed as a fraction or as a repeating decimal. Numbers such as π and 2 are irrational numbers. You learned that the set of rational numbers combined with the set of irrational numbers is called the set of real numbers. Although we will not spend more time here on real numbers, it suffices to say that (1) the real numbers, with operations of addition and multiplication, form a field, and (2) every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number.
25 Section 10.4 Some Other Number Systems 225 Students will encounter real numbers primarily when they reach algebra, and it will be necessary for them to use the field properties to operate with real numbers in algebra and beyond. Yet, some mathematically important number systems do not have infinitely many numbers. One such system is sometimes called clock arithmetic. Suppose you have a fivehour clock, that is, the numbers 0, 1, 2, 3, and 4 are evenly spaced around the clock, such as this: Sometimes 5 is used instead of 0, but you will soon see why 0 is used here. When one adds in this arithmetic, it is like going around the clock the number of hours indicated by the addends, starting at 0. Thus, begins at 0, goes two spaces to 2, and then goes clockwise four spaces, ending at 1. So = 1, a result that looks quite strange but makes sense in this number system. Also, begins at 0, moves to 4, then moves two spaces clockwise, landing on 1, so = 1. Likewise would mean beginning at 0, moving to 4, and then traveling 0 hours, so = 4. But would mean beginning at 0, going zero spaces, and then going four spaces, landing on 4, so = 4. Activity 6 Which Properties Hold? 1. Complete the following, using clock arithmetic with five numbers: 0, 1, 2, 3, and 4. a b c d e. 2 + (3 + 4) f. (2 + 3) + 4 g. (1 + 2) + 4 h. 1 + (2 + 4) i j Have you illustrated, in 1, any instances of the commutative property of addition? If so, which one(s)? Try some others. 3. Have you illustrated, in 1, any instances of the associative property of addition? If so, which one(s)? Try some others. 4. Is this set of numbers closed under addition? That is, for any two clock numbers in the set, would the sum be in the set? 5. Is there an additive identity in this system? If so, what is it? (Now you see why 0 rather than 5 was chosen.) Continue on the next page.
26 226 Chapter 10 Expanding Our Number System 6. Does each number have an additive inverse? That is, for any given clock number c, is there a number d such that c + d = 0? (Of course, this assumes that 0 is the additive identity.) We can also define multiplication in this system, using repeated addition. That is, 4 2 = = 3 because (starting at 2 and moving clockwise 2 places) is 4. Then is 1, and finally = 3. Also, 2 4 = = 3. Activity 7 Do the Field Properties Hold in Clock Arithmetic? 1. Fill in these two tables: Some results have been entered for you: Try several examples to illustrate that multiplication in the fivehour clock system is commutative and associative. 3. Is the set closed under multiplication? 4. Is there a multiplicative identity? If so, what is it? 5. Does each number in the system have a multiplicative inverse? 6. Finally, is multiplication distributive over addition? If all eleven properties hold, then clock arithmetic for 5 is a field. Is this a field? TAKEAWAY MESSAGE... The rational numbers, together with the operations of addition and multiplication, form what is called a field, as do the real numbers with addition and multiplication. A mathematical field is defined as a set of numbers with two operations for which the eleven properties we have discussed all hold. Some sets of numbers we have worked with, such as the set of whole numbers, do not form a field with addition and multiplication because one or more properties fail. The rational numbers and real numbers are both infinite number systems. Are there any finite number systems that form a field? You have found one: clock arithmetic for a clock with numbers 0, 1, 2, 3, and 4, with operations defined as in the tables. Are there other finite fields? That s an interesting question.
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