Section 6.1 PRE-ACTIVITY PREPARATION Introduction to Negative Numbers and Computing with Signed Numbers In the previous five chapters of this book, your computations only involved zero and the whole numbers, decimal numbers, and fractions greater than zero (positive numbers). This chapter introduces numbers less than zero (negative numbers). Consider the example of how Maggie, a new customer service trainee, reported her cash drawer balances for her first six days behind the counter: over $1 (+$1), short $1.50 (–$1.50), +$1.25, –$0.50, neither over nor short (0), and –$0.95. By adding these six numbers, she knew that she was short $0.70 (–$0.70) for the week. Negative numbers are used in business applications to represent expenditures, debts, losses, year-end budget deficits, falling stock prices, and overdrawn checking accounts. They describe temperatures below zero, land below sea level, floors below street level on construction sites, and depths of submarines and scuba divers. Think of how often negative numbers appear even in leisure activities—points lost in card games, yardage lost in football games, and strokes under par in golf. Across the broad range of careers that require math competency, your understanding of signed numbers and how to compute with them will extend your ability to do practical applications beyond those that only involve numbers greater than or equal to zero. LEARNING OBJECTIVES • Recognize and distinguish between positive and negative numbers. • Order a set of signed numbers. • Master the addition and subtraction of signed numbers. • Master the multiplication and division of signed numbers. TERMINOLOGY PREVIOUSLY USED addend NEW TERMS number line TO LEARN absolute value opposite common denominator numerator additive inverse positive number difference order evaluate an expression positive sign + expression product integers signed number factor quotient negative number simplify less than symbol < sum negative sign – term 531 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 532 BUILDING MATHEMATICAL LANGUAGE Signed Numbers A positive number is a number greater than zero. For example, the number 7 is a positive number. It can be written with or without the positive number sign, + 7 or +7 is read “positive seven” or simply “seven.” A negative number is a number less than zero. For example, –7 is a negative number. A negative number must always be preceded by the negative number sign, – –7 is read, “negative seven.” The number zero is neither positive nor negative. Positive and negative numbers are referred to as signed numbers. Every signed number has an opposite. The opposite is the number that is the same distance from zero, but in the opposite direction. For example, the opposite of 3 is –3, and the opposite of –0.25 is + 0.25. The Inverse Property of Addition states that when you add a number to its opposite, the result is zero (0). For example, 3 + (–3) = 0 and –0.25 + 0.25 = 0. Because of this, the opposite of a number is sometimes referred to as its additive inverse. The set of numbers called integers is comprised of all the counting numbers (1, 2, 3, ….), all their opposites (–1, –2, –3,…), and zero; that is, {...–3, –2, –1, 0, 1, 2, 3,...}. Absolute Value The absolute value of a number is simply its distance from zero (0) on the number line. The symbol n indicates the absolute value of the number n. Every absolute value is positive: “The absolute value of fifteen is (equals) fifteen.” or 15 = 15 “The absolute value of positive fifteen is fifteen.” That is, the number 15 is fifteen units in the positive direction from 0 on the number line, so its distance from 0 is fifteen units. –25 and –15 = 15 –20 –15 –10 –5 0 5 10 15 20 25 “The absolute value of negative fifteen is (equals) fifteen.” That is, –15 is fifteen units in the negative direction from 0 on the number line, so its distance from 0 is also fifteen units. Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 533 Ordering Signed Numbers Below is a number line on which the integers from –7 through +7 are marked. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 As you move to the left on the number line, the numbers get smaller: –7 < –5 < –2 < 0 < 2 < 5 < 7 5 4 3 2 1 0 –1 –2 –3 –4 –5 A number line can also be drawn vertically. (Visualize a common weather thermometer.) As you move from top to bottom on the number line, the numbers get smaller. –5 < –1 < 0 < 2 < 4 To order a set of signed numbers, keep in mind that the negative number with the greatest absolute value is farthest to the left on the horizontal number line and therefore the smallest number. Example: Put the numbers 3, –0.5, –4, –6, 1.5, 7, and –3 in order from smallest to largest; that is, from most negative to most positive. VISUALIZE –0.5 –7 –6 –5 –4 –3 –2 –1 1.5 0 1 2 3 4 5 6 7 –6 < –4 < –3 < –0.5 < 1.5 < 3 < 7 Expressions and Terms An expression (refer to Section 4.2) is a mathematical symbol or combination of symbols that represents a value. You might think of an expression, with or without variables, as a problem to compute. For example, 5+3 16 – 9 2 × (–9) 28 + n – 6 + (–2) Chapter 6 — Signed Numbers, Exponents, and Order of Operations 534 Addition or subtraction signs separate the terms of an expression. In 28 + 3 – 6 + (–2), the terms are 28, 3, 6, and –2. In the expression − 2 2 + 3 2 −n + 4, the terms are − , 3 2 , n, and 4. 3 3 Mathematicians agree that two signs together, such as 4 + –7, may be unclear. Therefore, unless it is the first term in the expression (as in –7 +2), or the denominator of a division problem (as in 14 ), a negative −7 term is generally written within parentheses: 4 + (–7). To evaluate or to simplify an expression with no variables is to perform the stated operations and simplify the results to a single number answer. For the examples at the bottom of the previous page: Evaluating 5 + 3 results in 8 as the answer. Simplifying 16 – 9 results in 7 as the answer. Evaluating 2 × (–9) results in –18 as the answer. You cannot evaluate the fourth expression unless you substitiute a value for the variable. When n = 3, this expression becomes 28 + 3 – 6 + (–2) and simplifies to 23 as the answer. Reading Expressions Involving Signed Numbers Addition and Subtraction 3+2 “three plus two” 4 + (–7) “four plus negative seven” –3 + (–2) “negative three plus negative two” –11+ 7 “negative eleven plus seven” or “negative eleven plus positive seven” 5–3 “five minus three” or “five minus positive three” –12 – 4 “negative twelve minus four” or “negative twelve minus positive four” 13 – (–6) “thirteen minus negative six” –17 – (–12) “negative seventeen minus negative twelve” Multiplication and Division –15 × 2 “negative fifteen times two” or “negative fifteen times positive two” 2 • (–3) or 2 (–3) “two times negative three” or “positive two times negative three” –30 • (–8) or –30 (–8) 90 90 ÷ (–9) or −9 −12 –12 ÷ 4 or 4 –15 ÷ (–3) or −15 −3 “negative thirty times negative eight” “ninety divided by negative nine” or “positive ninety divided by negative nine” “negative twelve divided by four ” or “negative twelve divided by positive four” “negative fifteen divided by negative three” Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 535 Operations with Signed Numbers Think about the calculations you have done in the previous five chapters using only zero and the positive numbers. For each computation, your answer was always a number greater than or equal to zero. Before learning the methodologies for adding, subtracting, multiplying, and dividing signed numbers, you may find it helpful to consider some simple, real-life examples that use negative numbers. They are presented in table format before each methodology. Some of the answers, but not all, will be negative. Try to predict the answer to each problem before you look at the last two columns. You may find that some computations seem to come to you naturally—adding, for example, or multiplying and dividing with one positive and one negative number. On the other hand, some translations may seem unnatural—multiplying two negative numbers, dividing two negative numbers, or subtracting a negative number. For these problems, your tendency may be to think them through in terms of positive numbers because you are most comfortable in that frame of reference. In doing so, you might even predict the correct answers. However, the third column in the tables will demonstrate how to integrate negative numbers into your thought process. Adding Signed Numbers Predict the answers to the following examples: Example Your Prediction Translation into a Mathematical Expression You gained 4 pounds in March, lost 5 pounds in April, lost 2 pounds in May, and lost 1 pound in June. Think of gains as positive numbers and losses as negative numbers. a) What was your total weight change for March and April? 4 pounds + (–5) pounds 4 + (–5) b) What was your total weight change for May and June? –2 pounds + (–1) pounds –2 + (–1) The football team gained a total of 5 yards on its first two plays, but lost 13 yards on its third play. What was the total yardage on the three plays? Consider gains as positive numbers and losses as negative numbers. 5 yards + (–13) yards 5 + (–13) Answer –1 pound –3 pounds –8 yards (an 8 yard loss) The following two methodologies for adding signed numbers are based upon whether the signs of the numbers are the same or different. Chapter 6 — Signed Numbers, Exponents, and Order of Operations 536 METHODOLOGY Adding Numbers with the Same Sign ► ► Example 1: Evaluate –19 + (–24) Example 2: Evaluate (–26) + (–29) Try It! Steps in the Methodology Step 1 Example 1 Identify the terms and confirm they have the same sign. Identify terms. –19 and –24 Special Adding signed fractions Case: (see page 537, Model 3) Step 2 Determine absolute values. both negative Determine the absolute value of each term. −19 = 19 −24 = 24 Step 3 Add their absolute values. Add absolute values. In this step, compute only with absolute values, not signs. Step 4 To present your answer, attach the common sign of the terms to the sum. Present the answer. Example 2 19 +24 43 ???? –43 Why do you do Steps 3 and 4? ???? Why do you do Steps 3 and 4? It may be helpful to visualize the addition process on a number line—you are already quite familiar with adding two positive numbers for which the result is always positive, as in Example 1 below. Example 1: 5+3=8 –5 Example 2: –4 +3 Visualize the addition process: –3 –2 –1 –20 + (–30) = –50 0 1 2 3 4 5 6 7 8 9 10 20 30 40 50 +(–30) –50 –40 –30 –20 –10 0 Since the second term takes you farther away from zero in the same direction (positively or negatively) as the first addend, you can simply add the two distances from zero (their absolute values) (Step 3) and attach the sign they share (Step 4). Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 537 MODELS Model 1 Evaluate: –8 + (–25) + (–4) + (–10) Step 1 The four addends are –8, –25, –4, and –10. All are negative. −8 = 8 Step 2 Step 3 −25 = 25 −4 = 4 −10 = 10 8 25 4 10 Step 4 47 Answer: –47 absolute value of the answer Model 2 Simplify: 0.17 + 2.8 + 6.42 Step 1 0.17, 2.8, and 6.42 are all positive. Step 2 Absolute values are 0.17, 2.8, 6.42 0.17 2.80 6.42 Step 3 9.39 Model 3 Add: Step 4 Answer: +9.39 or simply 9.39 absolute value of the answer Special Case: Adding Signed Fractions − Rewrite: 3 ⎛⎜ 2 ⎞⎟ + ⎜− ⎟ 4 ⎜⎝ 3 ⎟⎠ To add signed fractions, first rewrite the fractions with a common denominator. 3 3 ⎛⎜ 2 4 ⎞⎟ × + ⎜− × ⎟ 4 3 ⎜⎝ 3 4 ⎟⎠ ⎛ 8⎞ 9 =− + ⎜⎜− ⎟⎟⎟ 12 ⎜⎝ 12 ⎠ − −9 −8 + 12 12 −9 + −8 = 12 = Attach the sign of each fraction to its numerator and use the appropriate methodology to add the numerators. continued on the next page Chapter 6 — Signed Numbers, Exponents, and Order of Operations 538 Apply the methodology to add the terms in the numerator: Steps 1 & 2 both are negative −9 = 9 −8 = 8 absolute value of the answer Step 3 9 + 8 = 17 Step 4 numerator sum is negative: –17 Answer : −17 17 5 =− = −1 12 12 12 METHODOLOGY Adding Two Numbers with Opposite Signs ► ► Example 1: Evaluate 15 + (–32) Example 2: Evaluate –28 + 31 Try It! Steps in the Methodology Example 1 Identify the two terms. +15 and –32 Example 2 Step 1 Identify terms. Step 2 Determine absolute values. Determine the absolute value of each term. Step 3 Subtract the smaller absolute value from the larger absolute value. Subtract absolute values. Step 4 Present the answer. In this step, compute only with absolute values, not signs. To present your answer, attach the sign of the number with the larger absolute value. ???? Why do you do Steps 3 and 4? 15 = 15 −32 = 32 32 −15 17 THINK −32 > 15 –17 ???? Why do you do Steps 3 and 4? It may be helpful to visualize a few examples of the addition process by using number lines. In the examples on the next page, the pattern for adding numbers with opposite signs becomes clear. Simply stated, each answer’s absolute value is the difference between the absolute values of the original numbers (Step 3) and each has the sign of the original number with the largest absolute value (Step 4). Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 539 Example 1 Start with an overdrawn checking account with a balance of –$20 and deposit a $60 paycheck. In other words, compute –$20 + $60. The first $20 of the check cancels out the –$20 balance and brings you back up to a $0 balance. What’s left of your deposit is your new positive balance, +$40. = –20 + 20 + 40 +40 = –60 –50 –40 –30 –20 –10 Start 0 10 20 30 40 50 Answer } +20 60 } –20 + +60 0 = 60 + 40 40 Since addition is commutative, –20 + 60 is the same as +60 + (–20). Picture 60 + (–20) on a number line: +(–20) } 60 + (–20) –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 Answer 60 = 40 + Start = } = 40 + 20 + (–20) 0 40 Example 2 Start with a +$15 balance in your account and write a check for $35. That is, compute +$15 + (–$35). The first $15 of the check cancels out your $15 balance and brings you down to a $0 balance. What’s left of the check accounts for your new negative balance, –$20. = –35 –30 –25 –20 –15 Answer –10 –5 0 5 10 15 Start 20 } = 15 + (–15) + (–20) + (–15) +(–20) (–35) } 15 + + (–35) 0 –20 = 25 + (–20) Example 3 Visualize: –50 + 40 –50 –50 –40 Start –30 + 40 = –10 + (–40) + 40 –20 –10 0 Answer 10 20 30 40 50 60 = –10 + = } –60 } + 40 –10 0 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 540 MODELS Model 1 Add: –73 + 25 Step 1 Step 2 The two terms are –73 and +25, with opposite signs. −73 = 73 25 = 25 Step 3 73 −25 absolute value of the answer 48 Step 4 -73 > 25 answer will be negative THINK Answer: –48 Model 2 Simplify: 42.25 + (–16.9) Step 1 The two terms are 42.25 and –16.9, with opposite signs. Steps 2 & 3 42.25 −16.90 25.35 Step 4 THINK absolute value of the answer 42.25 > -16.9 answer will be positive Answer: +25.35 or 25.35 Model 3 Evaluate: − 2 1 + 5 3 6 5 −6 + 5 + = 15 15 15 Apply the Methodology to add the terms in the numerator: Rewrite with a common denominator: − Step 1 opposite signs Step 2 −6 = 6, Step 3 6 – 5 =1 Step 4 (Refer to Special Case, Model 3 on page 537.) THINK 5 =5 -6 > 5 numerator sum is negative, - 1 1 Answer: −1 = – − 15 15 Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 541 TECHNIQUES To add more than two signed numbers, use either of the following two techniques. Adding Three or More Signed Numbers Technique #1 Add the first two numbers, using the appropriate Methodology for Adding Signed Numbers. Then add each succeeding number as you work left to right. Technique #2 Find the sum of the positive numbers and the sum of the negative numbers. ??? Then add the two sums. Why can you do this? ???? Why can you do Technique #2? Because of the Commutative Property of Addition, you can rearrange the terms: For example, –8 + 3 + 2 + (–2) + (–10) + 9 = 3 + 2 + 9 + (–8) + (–2) + (–10) The Associative Property of Addition allows you to group the addends as you wish to simplify your computation: = [3 + 2 + 9] + [(–8) + (–2) + (–10)] sum of the positives + sum of the negatives Model Simplify: –8 + 3 + 2 + (–2) + (–10) + 9 Using Technique #1, working left to right: –8 + 3 = –5 –5 + 2 = –3 –3 + (–2) = –5 –5 + (–10) = –15 –15 + 9 = –6 Answer Using Technique #2: –8 + 3 + 2 + (–2) + (–10) + 9 Add the positive numbers: 3 + 2 + 9 = 14 Add the negative numbers: –8 + (–2) + (–10) = –20 Add the sums 14 + (–20) = –6 Answer Chapter 6 — Signed Numbers, Exponents, and Order of Operations 542 Subtracting Signed Numbers Predict the answers to the following examples: Example Your Prediction Translation into an Expression You begin with a negative checkbook balance, –$14, and are charged an overdrawn check fee penalty of $25. What is your new balance? Subtract the fee from the starting balance: –$14 – $25 Your phone card balance is 600 minutes. You make a call lasting 20 minutes, and there is an additional 10-minute charge for using a pay phone for the call. What is your new balance? Subtract the used minutes and the extra charge: 600 min. – 20 min. – 10 min. OR Answer –$39 Add the negative penalty fee to the starting negative balance: –$14 + (–$25) OR Think of the minutes used and the extra pay phone charge as negative numbers and add to the starting balance: 570 mins. 600 min + (–20 min.) + (–10 min.) You owe $75 on your charge account, and return an item you had previously purchased for $25. Represent your new account balance as a signed number. Remove (subtract) the previous purchase from your current debt: –$75 – (–$25) You owe $75 on your charge account, and return an item you had previously purchased for $95. Represent your new account balance as a signed number. Remove (subtract) the previous purchase from your current debt: –$75 – (–$95) OR Add the refund to your debt: –$75 + $25 OR Add the refund to your debt: –$75 + $95 Notice the pattern in the examples—subtracting a number is the same as adding its opposite. This the basis for the next methodology. –$50 +$20 Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 543 METHODOLOGY Subtracting Two Signed Numbers ► ► Example 1: Evaluate –54 – (+14) Example 2: Evaluate 16 – (–7) Steps in the Methodology Step 1 Copy the problem. Step 2 Identify the second term. Step 3 Change to add the opposite. Write the problem exactly as given. Identify the second term—the number you are subtracting from the first Change the operation sign to addition, and change the sign of the second term. ??? Try It! Example 1 –54 – (+14) THINK subtracting +14 –54 + (–14) Why do you do this? Step 4 Add appropriately. For the expression in Step 3, determine whether you are adding two numbers with the same sign or two numbers with opposite signs and follow the appropriate Methodology for Adding Signed Numbers. THINK –54 and –14 are both negative Add their absolute values. 54 +14 68 Attach a negative sign. –68 Step 5 Present the answer. Present your answer. –68 Example 2 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 544 ??? Why do you do Step 3? It may be helpful to visualize the subtraction process on a number line. Intuitively, it makes sense to move in the negative direction when you subtract, as in the following two examples of subtracting a positive number. minus 20 or +(–20) Example: 40 – 20 40 – 20 = 40 + (–20) –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 = +20 minus 30 or +(–30) Example: –10 – 30 –10 – 30 = –10 + (–30) –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 = –40 However, when you subtract a negative number (that is, when you subtract “the opposite of a positive number), you reverse the movement of the subtraction to the positive direction. Example: 40 – (–20) minus –20 or +(+20) 40 – (–20) = 40 + (+20) –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 = +60 minus –30 or +(+30) Example: –10 – (–30) –10 – (–30) = –10 + (+30) –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 = +20 As the examples in the table proceeding the methodology and above demonstrate, subtracting a number is the same as adding its opposite. There is another important reason for changing subtraction to addition of the opposite. Subtraction is not commutative. For example, 7 – 4 ≠ 4 – 7. However, once you make the proper changes and rewrite the problem as an addition problem, you can apply the Properties of Addition— the Commutative and Associative Properties—to simplify your calculation. This is especially useful when you add and subtract more than two terms within the same expression (see Models 1 and 2 on pages 546 and 547). Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 545 MODELS Model 1 Evaluate: –20 – (–9) subtraction sign Step 1 Step 2 Step 3 Step 4 –20 – (–9) subtracting negative 9 THINK –20 + (+9) Solve this addition problem. Addends are –20 and +9, opposite signs THINK Subtract the absolute values. 20 –9 11 Step 5 absolute value of the answer −20 > 9 , so attach a negative sign Answer: –11 Model 2 Simplify: –23 – 75 subtraction: “–23 minus 75” Step 1 –23 – 75 Step 2 THINK Step 3 Step 4 subtracting +75 –23 + (–75) THINK Solve this addition problem. same sign, both negative Add the absolute values. 23 +75 Step 5 Answer: –98 98 Attach the common sign, negative. Model 3 Subtract: 8.25 – 19.73 Steps 1, 2 & 3 8.25 – 19.73 = 8.25 + (–19.73) Step 4 opposite signs THINK 19.73 −8.25 11.48 Step 5 Answer: –11.48 Solve this addition problem. −19.73 > 8.25 , so attach a negative sign Chapter 6 — Signed Numbers, Exponents, and Order of Operations 546 Model 4 Evaluate: − 11 ⎛⎜ 1 ⎞⎟ − ⎜− ⎟ 15 ⎜⎝ 3 ⎟⎠ Steps 1, 2 & 3 Step 4 − 11 ⎛⎜ 1 ⎞⎟ 11 ⎛ 1 ⎞ − ⎜− ⎟⎟ = − +⎜⎜+ ⎟⎟⎟ 15 ⎜⎝ 3 ⎠ 15 ⎜⎝ 3 ⎠ First rewrite with a common denominator: 11 ⎛⎜ 1 ⎞⎟ 11 ⎛⎜ 5 ⎞⎟ −11 + 5 + ⎜+ ⎟ = − + ⎜+ ⎟⎟ = − 15 ⎝⎜ 3 ⎠ 15 ⎜⎝ 15 ⎟⎠ 15 THINK opposite signs 11 – 5 = 6 −11 > +5 = Step 5 −6 15 Reduce: −6 ÷ 3 −2 = 15 ÷ 3 5 attach negative sign numerator = –6 −2 2 =− Answer 5 5 TECHNIQUE Use the following technique when the expression contains both addition and subtraction signs. Adding and Subtracting Signed Numbers in the Same Expression Technique Change each subtraction in the expression to addition of the opposite (do not change additions) and apply a Technique for adding three or more numbers. MODELS Model 1 Simplify: –14 – (–6) + (–2) – 1 subtraction signs Change each subtraction: –14 + (+6) + (–2) + (–1) Apply addition Technique –14 + 6 = –8 and work left to right –8 + (–2) = –10 –10 + (–1) = –11 Answer Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 547 Model 2 Evaluate: –11 + 15 – 2 + (–21) – 7 – (–28) + 25 – 6 subtraction signs = –11 + 15 + (–2) + (–21) + (–7) + (+28) + 25 + (–6) Add the positives: 15 + 28 + 25 = +68 Add the negatives: –11 + (–2) + (–21) + (–7) + (–6) = –47 Add the two sums: +68 + (–47) = +21 or 21 Answer Multiplying and Dividing Signed Numbers Predict the answers to the following examples: Example For 5 weeks you must lose 3 pounds per week. What will be your weight change for the five weeks? Your Prediction Translation into an Expression 5 weeks × (–3) lbs./week 5 × (–3) Your doctor wants you to lose 20 pounds. You have 5 weeks before your next appointment. What should be your average weight change per week? A small plane descended 23 feet per second until it descended 1150 feet. How many seconds did this descent take? –20 lbs. 5 weeks –1150 ft. –23 ft./sec (a loss of 15 pounds) –4 lbs. per week –1150 ÷ (–23) OR 50 seconds This is the same as 1150 ft. ÷ 23 ft./sec Consider time past as negative: –4 weeks × (–2) lbs./week –4 × (–2) OR THINK The temperature has dropped 5 degrees every hour for the last 6 hours. How much higher was the temperature 6 hours ago? –15 lbs. –20 ÷ 5 THINK For the past 4 weeks you lost 2 pounds per week. How much more did you weigh 4 weeks ago? Answer This is the same as 4 weeks × 2 lbs./week +8 pounds, that is, 8 pounds more 4 weeks ago Consider time past as negative: –6 hours × (–5) degrees/hour –6 × (–5) OR THINK This is the same as 6 hours × 5 degrees/hour +30 degrees (30 degrees higher 6 hours ago) Chapter 6 — Signed Numbers, Exponents, and Order of Operations 548 METHODOLOGY The Methodology for Multiplying or Dividing Signed Numbers is based upon whether the signs of the numbers are the same or different. Multiplying or Dividing Two Signed Numbers ► ► Example 1: –42 × 6 Example 2: –162 ÷ (–9) Try It! Steps in the Methodology Step 1 Determine the sign of the answer. Determine sign of answer. • If the two numbers have opposite signs, the answer will be negative. • If the two numbers have the same sign, the answer will be positive. ??? Why do you do this? Step 2 Determine absolute value. Step 3 Multiply or divide absolute values. Determine the absolute value of each term. Present the answer. –42 × 6 opposite signs The answer will be negative. −42 = 42 6 =6 Calculate the product (for multiplication) or quotient (for division) of the absolute values of the numbers. In this step, compute only with absolute values, not signs. Step 4 Example 1 To present your answer, attach the correct sign (as determined in Step 1) to the product or quotient. 42 ×6 252 –252 Example 2 Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 549 ??? Why do you do Step 1? For Multiplication: Multiplication is repeated addition. When you multiply a negative number, say (–5), by a positive number, say 7, you can think of it as adding (–5) to itself seven times. 7 × (–5) = (–5) + (–5) + (–5) + (–5) + (–5) + (–5) + (–5), which equals –35 by the Addition Methodolgy. Now consider a negative number, say (–7), times a negative number, say (–5): (–7) × (–5) Think of this computation as being “the opposite of (or the negative of)” 7 times (–5). The opposite of 7 × (–5) is the opposite of (–35) which is +35. For Division: This is the inverse operation to multiplication. –35 ÷ 7 = –5 because –5 × 7 = –35 35 ÷ (–7) = –5 because –5 × (–7) = +35 35 ÷ 7 = +5 because 5 × 7 = +35 –35 ÷ (–7) = +5 because 5 × (–7) = –35 MODELS Model 1 Evaluate: –8 (–12) THINK “negative eight times negative twelve” Step 1 The factors have the same sign. The answer will be positive. Step 2 −8 = 8 Step 3 8 × 12 = 96 Step 4 Answer: +96 or 96 −12 = 12 absolute value of the answer Chapter 6 — Signed Numbers, Exponents, and Order of Operations 550 Model 2 Simplify: −4.82 0.2 absolute value of the answer Step 1 opposite signs; answer will be negative Step 2 −4.82 = 4.82 Step 4 0.2 = 0.2 Step 3 2 4.1 0.2 4.8 2 −4 08 −8 ) 02 −2 0 Answer: –24.1 Model 3 ⎛ 2⎞ ⎛ 1⎞ Evaluate: ⎜⎜− ⎟⎟⎟ • ⎜⎜− ⎟⎟⎟ ⎜⎝ 3 ⎠ ⎜⎝ 8 ⎠ Step 1 factors have the same sign; answer will be positive Step 2 − Step 4 Answer : + 2 2 = 3 3 − 1 1 1 = 8 8 1 or 12 Step 3 1 12 2 1 1 ×4 = 3 12 8 absolute value of the answer TECHNIQUE Use the following technique when multiplying more than two signed factors. Multiplying Three or More Signed Factors Technique Multiply the first two factors, then multipy by each succeeding number as you work left to right. Shortcut Determining the sign of the product first (see page 551, Models 1 & 2) Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers MODELS Model 1 Simplify: ⎛ 1⎞ 4 × (−2) × ⎜⎜− ⎟⎟⎟ × 3 × (−2) ⎜⎝ 2 ⎠ Work left to right, keeping track of the sign for each operation. 4 × (−2) opposite signs = −8 ⎛ 1⎞ −8 × ⎜⎜− ⎟⎟⎟ ⎝⎜ 2 ⎠ 4 same signs =− + 4×3 same signs = +12 +12 × (−2) Shortcut 8 ⎛⎜ 1 × ⎜− 1 ⎜⎜⎝ 1 2 ⎞⎟ ⎟⎟ = + 4 = +4 ⎟⎟ 1 ⎠ opposite signs = –24 Answer Determining the Sign of the Product First Determine the sign of the answer first by counting the negative factors. • An even number of negative factors yields a positive product. • An odd number of negative factors yields a negative product. ⎛ 1⎞ 4 × (– 2) × ⎜⎜– ⎟⎟⎟ × 3 × (– 2) ⎜⎝ 2 ⎠ three negative factors; the answer will be negative Then simply multiply the absolute values of the factors and attach the sign. 1 4 2 1 3 2 24 Answer: –24 × ×1 × × = 1 1 1 2 1 1 Model 2 Evaluate: –5 × 4 × 2 × (–0.5) × 2 Use shortcut: two negative factors; the answer will be positive 5 × 4 × 2 × 0.5 × 2 = 20 × 2 × 0.5 × 2 = 40 × 0.5 × 2 = 20 × 2 = 40 Answer: +40 or 40 551 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 552 Validation for Adding, Subtracting, Multiplying, and Dividing Signed Numbers After simplifying an expression, you can validate your answer. Start with your answer and use the opposite operation or operations to work back to the first term in the original expression, always keeping in mind that the opposite operation has its own unique set of steps for signed numbers. Following are validation models for each of the basic operations. Addition: Validate addition by subtracting. Example 1 –19 + (–24) = –43 Validation: –43 – (–24) = –43 + (+24) = –19 9 Example 2 – Validation: Example 3 15 + (–32) = –17 Validation: –17 – (–32) = –17 + (+32) = 15 9 5 3 ⎛⎜ 2 ⎞⎟ + ⎜− ⎟⎟ = −1 12 4 ⎜⎝ 3 ⎠ 5 ⎛⎜ 2 ⎞⎟ − ⎜− ⎟ 12 ⎜⎝ 3 ⎟⎠ 5 ⎛ 8⎞ = −1 − ⎜⎜− ⎟⎟⎟ 12 ⎜⎝ 12 ⎠ 17 ⎛ 8 ⎞ −17 + 8 = − + ⎜⎜+ ⎟⎟⎟ = 12 12 ⎜⎝ 12 ⎠ −1 = Example 4 –8 + 3 + 2 + (–2) + (–10) + 9 = –6 Validation: Work backwards and subtract all terms but the first. –6 – 9 – (–10) – (–2) – 2 – 3 = –6 + (–9) + (+10) + (+2) + (–2) + (–3) = [–6 + (–9) + (–2) + (–3)] + [(+10) + (+2)] = –20 + (+12) = –8 9 −9 −3 3 = =− 9 12 4 4 Subtraction: Validate addition by adding. Example 5 –54 – 14 = –68 Validation: –68 + 14 = –54 9 Example 6 68 −14 Validation: 54 negative 8.25 – 19.73 = –11.48 –11.48 + 19.73 = +8.25 9 19.73 −11.48 8.25 positive Addition and Subtraction: Validate by using successive opposite operations to work back to the first term. Example 7 –11 + 15 – 2 + (–21) – 7 – (–28) + 25 – 6 = 21 Validation: 21 + 6 – 25 + (–28) + 7 – (–21) + 2 – 15 = 21 + 6 + (–25) + (–28) + 7 + (+21) + 2 + (–15) = [21 + 6 + (+7) + (+21) + 2] + [(–25) + (–28) + (–15)] = 57 + (–68) = –11 9 Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 553 Multiplication: Validate multiplication by dividing. Example 8 Validation: –42 × 6 = –252 42 negative 6 252 −24 –252 ÷ 6 ) = –42 9 Example 9 –5 × 4 × 2 × (–0.5) × 2 = 40 Validation: 40 ÷ 2 = 20 20 ÷ (–0.5)= –40 –40 ÷ 2 = –20 12 −12 –20 ÷ 4 = –5 9 0 40. negative 0.5 20.0 −20 ) 0 −0 0 Division: Validate division by multiplying. Example 10 –4.82 = –24.1 0.2 Validation: –24.1 × 0.2 = –4.82 9 24.1 ×0.2 4.82 negative ADDRESSING COMMON ERRORS Issue Misunderstanding the addition process when the terms have opposite signs Incorrect Process Evaluate: 6.2 + (–41.9) 4 .9 41 +6.2 48.1 Answer: –48.1 41.9 –6.2 35.7 Answer: w r: 3 35.7 . Resolution Visualize the process (as in “Why Do You Do Steps 3 and 4?” in the methodologies for addition). To add two numbers with different signs, always find the difference in their absolute values to determine the absolute value of the answer. The sign of the answer will always be the sign of the term farther from zero on the number line. Correct Process Evaluate: 6.2 + (–41.9) The terms are +6.2 and –41.9 −41.9 > 6.2 The answer will be negative. 41.9 −6.2 35.7 Validation –35.7 – (–41.9) = –35.7+(+41.9) 41.9 −35.7 6.2 +41.9 > −35.7 = +6.2 9 35.7 is the absolute value of the answer. Answer: –35.7 Not changing the sign(s) of the term(s) when converting subtraction to addition Simplify: 1 – (–3) 6 − 15 6 + 15 + (– (–3) = 21 + (–3) (–3)= )= +18 Answer: +18 It takes two changes for each conversion of a subtraction operation to an addition operation—a change of the operator sign as well as a change to the sign of the second term. Simplify: 6 − 15 – (–3) 6+(–15)+(+3)= –9 + (+3)= –6 Answer: –6 –6 + (–3)= –9 –9 + 15 = +6 9 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 554 Issue Interpreting a multiplication problem as a subtraction problem Incorrect Process Evaluate: 6) –5 ((–6) –5 – 6 = –5 + (–6)) = –11 Answer: –11 Incorrectly determining the sign of the product of several factors Evaluate: –7•(–6)•3•(–1)•4 7•6•3•1•4 = 42 2•3•1•4 = 126 • 4 = 504 Answer: +504 Incorrectly ordering negative numbers List from smallest to largest: –6, 3, 8, –8, –¾ Answer: wer: –¾,–6, – –8 –8,, 3, 8 Resolution Recognize the various ways of representing multiplication. To multiply two signed numbers, a and b: a × b a × (b) a•b a • (b) a (b) (a) (b) To be sure that you have correctly applied the shortcut for determining the sign for a product of several factors, count the negative factors again. (The alternative is to compute the problem left to right, being attentive to the correct sign for each succeeding product). Use a number line to visualize the order of a set of signed numbers. The negative number with the largest absolute value is the farthest to the left on the number line. Correct Process Validation Evaluate: –5 (–6) +30 ÷ (–6) = –5 × (–6) = +30 = –5 9 Answer: +30 Evaluate: –7•(–6)•3•(–1)•4 –504 ÷ 4 = –126 There are three –126÷(–1)=+126 negative factors +126 ÷ 3 = 42 (–7, –6, and –1), so the answer will 42 ÷ (–6) = –7 9 be negative, –504. Alternately, –7 • (–6) = +42 42 • 3 = 126 126 • (–1)= –126 –126 • 4 = –504 List from smallest to largest: –6, 3, 8, –8, –¾ 0 –8 –6 –¾ Answer: –8, –6, –¾, 3, 8 PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with signed numbers the meaning of absolute values how to order a set of signed numbers how to add numbers with the same sign how to add two numbers with opposite signs how to convert from subtraction of signed numbers to addition how to multiply and divide signed numbers how, in general, to validate signed number computations 3 8 Section 6.1 ACTIVITY Introduction to Negative Numbers and Computing with Signed Numbers PERFORMANCE CRITERIA • Ordering a set of signed numbers from least to greatest • Adding and subtracting signed numbers – correct absolute value of the answer – correct sign of the answer • Multiplying and dividing signed numbers – correct absolute value of the answer – correct sign of the answer CRITICAL THINKING QUESTIONS 1. What makes a number negative? 2. What is the absolute value of a number? 3. What is the result of adding any number to its opposite? 4. How do you determine the sign of the answer to an addition problem? 555 556 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 5. What does it mean to convert a subtraction problem into an addition problem? 6. How do you determine the sign of the answer to a multiplication or division problem? 7. In an addition problem with more than two numbers, why can you add all the positive numbers and all the negative numbers first and then find the sum of those two numbers? 8. In a multiplication problem with more than two factors, why does an even number of negative factors produce a positive answer and an odd number produce a negative answer? Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers 557 9. When adding signed fractions, where should you attach their signs for ease of computation? 10. What will be your strategy to assure that your answer to a signed number problem is correct? TIPS FOR SUCCESS • Use a number line to visualize the order of a set of numbers. • Use a number line to help visualize addition and subtraction. • When computing with signed whole numbers and decimals, it is helpful to think in terms of dollars and cents. • After copying a subtraction expression, write out the problem on the next line with the addition sign and the opposite sign of the second term. Then compute the answer. • Once you have properly converted a subtraction problem to an addition problem, do not confuse yourself by looking back at the subtraction problem. Concentrate on solving the addition problem, applying the appropriate methodology for addition. Chapter 6 — Signed Numbers, Exponents, and Order of Operations 558 DEMONSTRATE YOUR UNDERSTANDING Evaluate each of the following (a) through (j) by doing the calculation “in your head.” Answer Answer a) –9 + 12 ______ f) 80 + (–90) ______ b) (–12) + (–3) ______ g) 25 ÷ (–5) ______ c) –16 + 8 ______ h) 7 (–8) ______ d) 36 +10 ______ i) (–2) (–9) ______ e) 15 + (–10) ______ j) –12 ÷ 3 ______ MENTAL MATH 1. Order the following sets of numbers from smallest to largest. a) 2, –1.5, –3, –7, 0.5, 4 b) –2, 8, ¾, 0, –6, –¼, 3 2. Evaluate each of the following: Worked Solution a) –49 + (–18) b) –22.4 + 48.7 c) 20 – 32 Validation (optional) Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers Worked Solution d) –37 – (–14) e) –24 + 5 – 8 f) 33 + (–23) – 17 – (–2) g) –26 – 14 + (–13) + 12 h) 3.72 + (–1.4) Validation (optional) 559 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 560 Worked Solution i) (–0.025) – 1.23 2 1 − 5 3 j) − k) 2 ⎛⎜ 7 ⎞⎟ + ⎜− ⎟ 5 ⎜⎝ 8 ⎟⎠ l) m) –3 (–8) –162 ÷ 9 Validation (optional) Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers Worked Solution n) –55 ÷ 2.5 o) 12 × (–1) × (–3) p) –2 • 4 • (–5) q) 0.2 (–1.3) r) − 3 ⎛⎜ 5 ⎞⎟ × ⎜− ⎟ 4 ⎜⎝ 9 ⎟⎠ Validation (optional) 561 Chapter 6 — Signed Numbers, Exponents, and Order of Operations 562 Worked Solution s) Validation (optional) 12 ⎛⎜ 3 ⎞⎟ ÷ ⎜− ⎟ 35 ⎜⎝ 7 ⎟⎠ t) –5 (–4) (2) (0) (–10) u) 2 × (−2) × 6 × (−4) 3 IDENTIFY AND CORRECT THE ERRORS Identify the error(s) in the following worked solutions. If the worked solution is correct, write “Correct” in the second column. If the worked solution is incorrect, solve the problem correctly in the third column. Worked Solution What is Wrong Here? 1) 36 – 48 Identify the Errors Did not rewrite as an addition problem. 36 + (–48) Answer should be negative. Correct Process 36 − 48 = 36 + (−48) = −12 Answer: –12 48 −36 12 −48 > 36 Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers Worked Solution What is Wrong Here? 2) 2 ⎛⎜ 1 ⎞⎟ + ⎜− ⎟ 3 ⎜⎝ 2 ⎟⎠ 3) –14 (–6) 4) –47.9 + (–1.1) 5) −5.4 −0.2 Identify the Errors 563 Correct Process Chapter 6 — Signed Numbers, Exponents, and Order of Operations 564 Worked Solution What is Wrong Here? 6) Identify the Errors Correct Process –1 (–2) (3) (–5) 7) List in order from smallest to largest: − 1 1 , − 1, 3, − 5, 2 5 TEAM EXERCISE In some applications involving signed numbers, you may be given a previous number and a current number and must calculate the change, including the direction of the change, positive or negative. To do this, you must subtract the previous number from the current number. In each of the following, calculate the change requested, including its sign. Situation Five hours ago the temperature was 53°F. Currently, it is 65°F. What was the change from the previous temperature until now? Expression 65º current 53º previous current – previous = _____ – ______ 0º Answer Section 6.1 — Introduction to Negative Numbers and Computing with Signed Numbers Situation The current temperature is 2°F. Ten hours ago, it was 6 degrees below zero (–6°F). What is the change from the previous temperature until now? Expression Answer 2º current current – previous 0º = _____ – ______ –6º previous The temperature now is 65°F. Seven hours ago it was 95°F. What is the change in temperature from the previous to the present temperature? 95º previous current – previous = _____ – ______ 65º current 0º This year’s end-ofthe-year balance is $50,000. Last year’s end-of-theyear balance was negative $30,000. What is the change from last year to this year? This year’s end-ofthe-year balance is –$30,000. Last year’s end-of-theyear balance was +$50,000. What is the change from last year to this year? current – previous = _____ – ______ –30,000 0 50,000 current previous current – previous = _____ – ______ –30,000 current 0 565 50,000 previous Chapter 6 — Signed Numbers, Exponents, and Order of Operations 566 ADDITIONAL EXERCISES Evaluate each of the following expressions. 1. –14 + 32 2. 24 – (–11) 3. –18.9 + 15 4. − 7 1 + 12 4 5. –5 – 20 6. 6.34 + (–10.2) 7. –18 + (–95) 8. –29 + 18 9. –15 – (–9) 10. 24.3 – 34.1 11. –22 + 7 – (–6) – 9 12. 2 + (–4) – 6 – 1 + 8 13. –8 (–12) ⎛ 3 ⎞⎛ 5 ⎞ 14. ⎜⎜⎜− ⎟⎟⎟⎜⎜⎜ ⎟⎟⎟ ⎝ 7 ⎠⎝ 9 ⎠ 15. 2.1 −0.03 16. 15 (–4) 17. –4.82 ÷ 0.2 18. –6 • 2 • (–3) • (–10) 19. –5.7 – 12.3 + 18 20. –84 ÷ (–3)

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