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)