1. Copyright © Cengage Learning. All rights reserved. CHAPTER 5 Extending the Number System

2. Copyright © Cengage Learning. All rights reserved. SECTION 5.1 Integers

3. 3 What Do You Think? Why is it that the sum of two negative numbers is a negative number, but the product of two negative numbers is a positive number? How do operations with integers connect with whole-number operations? How do they differ? Why is –8 less than –7?

4. 4 Integers Our first extension of the set of whole numbers is the set of integers, which is simply the union of the set of positive integers, the set of negative integers, and zero.

5. 5 Integer Connections

6. 6 Before we examine operations with integers, let us take a little time to see how integers connect to the set of whole numbers we have been working with up to now. We began our study of numbers with the set of natural numbers, N. N: {1, 2, 3, 4,...} With the invention of zero, we have the set of whole numbers, W. W: {0, 1, 2, 3, 4,...}

7. 7 Integer Connections With the invention of negative numbers, we have the set of integers, I. I: {... –4, –3, –2, –1, 0, 1, 2, 3, 4,...} Figure 5.1 shows a Venn diagram illustrating the notion of extending our set of numbers as our ancestors invented new kinds of numbers. Each set of numbers contains the previous set. Figure 5.1

8. 8 Integer Connections Integers in our world People use negative numbers both on and off the job. For instance: Businesses use negative numbers to indicate a business deficit, or “negative profit.” We often use negative numbers when describing change. For example, graphs often have negative numbers. We use negative numbers to indicate temperatures below zero.

9. 9 Representing Integers

10. 10 Representing Integers There are many models for representing integers. We will focus on the number line model because it is the model to which most real-life applications connect. Number lines can be represented horizontally or vertically, as shown in Figure 5.3. Figure 5.3

11. 11 Operations with Integers

12. 12 Operations with Integers In this section, we will examine the four operations with negative numbers. addition: combine, increase subtraction: take away, comparison, missing addend multiplication: repeated addition, rectangular array, Cartesian product division: partitioning, repeated subtraction, missing factor

13. 13 Understanding Addition with Integers

14. 14 Understanding Addition with Integers We will use a problem-solving tool from higher mathematics to develop algorithms for adding integers: We will examine all possible combinations (cases) and then look for patterns. The four cases are represented below:

15. 15 Understanding Addition with Integers Language You may have noticed that sometimes the + and – signs have been raised (printed as superscripts) and sometimes not. This convention allows us to avoid meaningless sentences. For example, we will read the problem –6 – (–8) as “negative 6 minus negative 8” instead of “minus 6 minus minus 8.” The words plus and minus will be used to refer to the operations of addition and subtraction.

16. 16 Understanding Addition with Integers The words positive and negative will be used to refer to the value of the number. Let us now examine the first two cases. Case 1: Both numbers are positive. Case 2: One number is positive, one number is negative, and the magnitude of the negative number is greater than the magnitude of the positive number.

17. 17 Understanding Addition with Integers Number line model Consider the problem 3 + (–4). That is, we begin at zero and move 3 units to the right, then we move 4 more units to the right; thus we find that 3 + 4 = 7. If a positive number is represented by an arrow pointing to the right (the positive direction), then a negative number is represented by an arrow pointing to the left (the negative direction).

18. 18 Understanding Addition with Integers Thus the problem 3 + (–4) can be represented as shown at the right in Figure 5.4. That is, we begin at zero and move 3 units to the right and then move 4 units to the left; thus we find that 3 + (–4) = –1. Figure 5.4

19. 19 Absolute Value

20. 20 Absolute Value One way to illustrate the concept of absolute value is to consider how far the number is from zero (the origin). The absolute value tells us the distance of the number from zero. For example, the numbers +5 and –5 are both the same distance from zero, so they both have the same absolute value, which is 5 (Figure 5.5). Figure 5.5

21. 21 Absolute Value With notation, we say Similarly, In English, we say that the absolute value of negative 5 is 5, and the absolute value of positive 5 is also 5. There is also language for referring to pairs of numbers whose absolute values are equal: We say they are opposites or negatives of each other. Thus the opposite of +6 is –6. Using another meaning of negative, we say that the negative of +6 is –6 and the negative of –6 is +6.

22. 22 Absolute Value As you can readily see, when we add any integer and its opposite, the result is zero; that is, a + (–a) = 0 Thus we can say that every integer has an additive inverse. In this text, we will use the term additive inverse instead of negative for two reasons. First, the term negative often creates a false impression. For example, if x = –6, then –x = + 6.

23. 23 Absolute Value In other words, when we are working with variables, the value of –x is often positive. Second, when working with fractions, we will develop a similar concept with a similar term: the multiplicative inverse.

24. 24 The Other Two Cases of Integer Addition

25. 25 The Other Two Cases of Integer Addition Case 3: One number is positive, one number is negative, and the magnitude of the positive number is greater than the magnitude of the negative number (an example is 4 + (–3)). When we look carefully at all integer addition problems that fall in this category, we find that the generalization stated in the preceding paragraph applies to this case too. In other words, the similarity between Case 2 problems (3 + (–4), 2 + (–8), –7 + 3, –9 + 4) and Case 3 problems (4 + (–3), 7 + (–2), –6 + 9, –1 + 5) is that one number is positive and one number is negative.

26. 26 The Other Two Cases of Integer Addition Regardless of which number has the larger magnitude, we can use the same procedure to determine the answer. Case 4: In Case 4, both numbers are negative. Let us examine a specific problem in detail and then look for generalizations. Figure 5.7 shows a representation of –3 + (–4) on a number line. Figure 5.7

27. 27 Understanding Subtraction with Integers

28. 28 Investigation A – Subtraction with Integers Do the following subtraction problems yourself before reading on. As you work, check to make sure that you are using your understanding of integers rather than just guessing. 1. 14 – (–25) = 2. –5 – 17 = 3. –6 – (–8) = 4. –12 – 5 = Discussion: We can define subtraction of negative numbers in terms of addition, which most people understand more easily.

29. 29 Investigation A – Discussion To make the connection stronger, let us examine how we might use what we know about subtraction to determine the answer to 4 – 6. In one sense, we cannot “take away” 6. However, if we think in terms of a checking account, if we take away 6 from 4, we will have a deficit of 2; that is, we have –2. When we subtract one number from another, we move to the left. cont’d

30. 30 Investigation A – Discussion When we begin at the point 4 and move 6 units to the left, we end up at the point negative 2, as shown in Figure 5.8. Thus we can conclude that 4 – 6 = –2. However, this is similar to the addition problem 4 + (–6) = –2. cont’d Figure 5.8

31. 31 Investigation A – Discussion We use this knowledge to define subtraction of integers formally in terms of addition: a – b = a + (–b) That is, subtracting is equivalent to adding the additive inverse. cont’d

32. 32 Connecting Whole-Number Subtraction to Integer Subtraction

33. 33 Connecting Whole-Number Subtraction to Integer Subtraction Let us compare the two definitions: a – b = c iff there is a number c such that c + b = a a – b = a + (–b) If you examine the actual problems involved in subtracting a positive number from a positive number, you find that we are not adding the opposite as much as we are taking away a positive amount or comparing the size of two sets (each with positive values).

34. 34 Connecting Subtraction Contexts to Algorithms

35. 35 Connecting Subtraction Contexts to Algorithms Problem 1 Denine realized that she had overdrawn her checking account by $60, and she was fined $15 for a returned check. What is her present balance? If we translate this problem into mathematical language, we find that we need to take away $15 from negative $60. Thus the problem is –60 – 15. Using our understanding of subtraction, we translate this subtraction problem into the following addition problem: –60 + (–15).

36. 36 Connecting Subtraction Contexts to Algorithms Applying our understanding of integer addition, we have an answer of –75; that is, her present balance is negative $75 (she is $75 in the red). Problem 2 On one day the high temperature in Nome, Alaska, was –6 degrees. On that same day, the high temperature at the North Pole was –64 degrees. How much warmer was Nome than the North Pole? If we translate this problem to mathematical language, we find that we are using the comparison model of subtraction.

37. 37 Connecting Subtraction Contexts to Algorithms Thus the problem is –6 – (–64), which we can translate as –6 + (+64), and the answer is 58; that is, it was 58 degrees warmer in Nome. We could also have interpreted this as a missing-addend problem: –64 + x = –6. Using algebra, x = –6 – (–64) = 58.

38. 38 Understanding Multiplication with Integers

39. 39 Investigation B – The Product of a Positive and a Negative Number Consider the following problem: 5  (–3). Can we apply our understanding of multiplication with positive numbers to determine the product? Discussion: Applying the repeated-addition model of multiplication, 5  (–3) literally means to add –3 five times, that is, –3 + (–3) + (–3) + (–3) + (–3). We know from integer addition that the sum must be –15. That is, we can deduce that 5  (–3) = –15.

40. 40 Investigation B – Discussion What about –3  5? Trying to apply repeated addition here is problematical; we must add 5 negative 3 times. However, the commutative property makes the task easier: –3  5 = 5  (–3) = –15. Two negative numbers What about –3  (–5)? None of the models for positive whole-number multiplication adapt nicely to this problem, and the commutative property does us no good here. cont’d

41. 41 Investigation B – Discussion However, we can use our knowledge that a positive times a negative is a negative and make use of patterns: We know that 4  (–3) = –12 Thus 3  (–3) = –9 cont’d

42. 42 Understanding Division with Integers

43. 43 Understanding Division with Integers We can now understand integer division by applying our knowledge of the relationship between division and multiplication. From an intuitive perspective, you may sense that the multiplication rules translate quite directly into division: The quotient of a positive and a negative number is a negative number. The quotient of two negative numbers is a positive number.

44. 44 Understanding Division with Integers We can apply the missing-factor model of division (y  n = x iff x  n = y) to verify these rules. For example, consider the problem –12  4. There is little doubt that the quotient is either –3 or + 3. Many students simply guess, but we can apply this model to help us find the correct answer. Applying our definition of division, we can say that –12  4 = x iff x  4 = –12.

45. 45 Understanding Division with Integers What number times 4 is equal to negative 12? We know from multiplication that this number must be –3.Therefore, –12  4 must equal –3.