10.4: Addition of Signed Numbers
- be able to add numbers with like signs and with unlike signs
- be able to use the calculator for addition of signed numbers
Addition of Numbers with Like Signs
The addition of the two positive numbers 2 and 3 is performed on the number line as follows.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since 3 is positive, move 3 more units to the right.
We are now located at 5.
Thus, \(2 + 3 = 5\).
Summarizing, we have
\(\text{(2 positive units)} + \text{(3 positive units)} = \text{(5 positive units)}\)
The addition of the two negative numbers -2 and -3 is performed on the number line as follows.
Begin at 0, the origin.
Since -2 is negative, move 2 units to the left.
Since -3 is negative, move 3 more units to the left.
We are now located at -5.
Thus, \((-2) + (-3) = -5\)
Summarizing, we have
\(\text{(2 negative units)} + \text{(3 negative units)} = \text{(5 negative units)}\)
Observing these two examples, we can suggest these relationships:
\(\text{(postitive number)} + \text{(positive number)} = \text{(positive number)}\)
\(\text{(negative number)} + \text{(negative number)} = \text{(negative number)}\)
Adding Numbers with the Same
Sign
Addition of numbers with like sign:
To add two real numbers that have the
same
sign, add the absolute values of the numbers and associate with the sum the common sign.
Find the sums.
3 + 7
Solution
\(\begin{array} {l} {|3| = 3} \\ {|7| = 7} \end{array} \big \}\) Add these absolute values.
3 + 7 = 10
The common sign is “+.”
Thus, \(3 + 7 = +10\), or \(3 + 7 = 10\).
Find the sums.
(-4) + (-9)
Solution
\(\begin{array} {l} {|-4| = 4} \\ {|-9| = 9} \end{array} \big \}\) Add these absolute values.
4 + 9 = 13
The common sign is “-.”
Thus, \((-4) + (-9) = -13\).
Practice Set A
Find the sums.
8 + 6
- Answer
-
14
Practice Set A
41 + 11
- Answer
-
52
Practice Set A
(-4) + (-8)
- Answer
-
-12
Practice Set A
(-36) + (-9)
- Answer
-
-45
Practice Set A
-14 + (-20)
- Answer
-
-34
Practice Set A
\(-\dfrac{2}{3} + (-\dfrac{5}{3})\)
- Answer
-
\(-\dfrac{7}{3}\)
Practice Set A
-2.8 + (-4.6)
- Answer
-
-7.4
Practice Set A
0 + (-16)
- Answer
-
-16
Addition With Zero
Addition with Zero
Notice that
\((0) + \text{(a positive number)} = \text{(that same positive number)}\).
\((0) + \text{(a negative number)} = \text{(that same negative number)}\).
Since adding zero to a real number leaves that number unchanged, zero is called the additive identity .
Addition of Numbers with Unlike Signs
The addition \(2 + (-6)\),
two numbers with unlike signs , can also be illustrated using the number line.
Begin at 0, the origin.
Since 2 is positive, move 2 units to the right.
Since -6 is negative, move, from 2, 6 units to the left.
We are now located at -4.
We can suggest a rule for adding two numbers that have unlike signs by noting that if the signs are disregarded, 4 can be obtained by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and -6. Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.
Adding Numbers with Unlike Signs
Addition of numbers with unlike signs: To add two real numbers that have
unlike signs
, subtract the smaller absolute value from the larger absolute value and associate with this difference the sign of the number with the larger absolute value.
Find the following sums.
7 + (-2)
Solution
\(\underbrace{|7| = 7}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\) \(\underbrace{|-2| = 2}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\)
Subtract absolute values: 7 - 2 = 5.
Attach the proper sign: "+."
Thus, \(7 + (-2) = +5\) or \(7 + (-2) = 5\).
3 + (-11)
Solution
\(\underbrace{|3| = 3}_{\begin{array} {c} {\text{Smaller absolute}} \\ {\text{value}}\end{array}}\) \(\underbrace{|-11| = 11}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is negative.}}\end{array}}\)
Subtract absolute values: 11 - 3 = 8.
Attach the proper sign: "-."
Thus, \(3 + (-11) = -8\).
The morning temperature on a winter's day in Lake Tahoe was -12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?
Solution
We need to find \(-12 + 25\).
\(\underbrace{|-12| = 12}_{\begin{array} {c} {\text{Smaller absolute}} \\ {\text{value}}\end{array}}\) \(\underbrace{|25| = 25}_{\begin{array} {c} {\text{Larger absolute}} \\ {\text{value. Sign is positive.}}\end{array}}\)
Subtract absolute values: 25 - 12 = 16.
Attach the proper sign: "+."
Thus, \(-12 + 25 = 13\).
Practice Set B
Find the sums.
4 + (-3)
- Answer
-
1
Practice Set B
-3 + 5
- Answer
-
2
Practice Set B
15 + (-18)
- Answer
-
-3
Practice Set B
0 + (-6)
- Answer
-
-6
Practice Set B
-26 + 12
- Answer
-
-14
Practice Set B
35 + (-78)
- Answer
-
-43
Practice Set B
15 + (-10)
- Answer
-
5
Practice Set B
1.5 + (-2)
- Answer
-
-0.5
Practice Set B
-8 + 0
- Answer
-
-8
Practice Set B
0 + (0.57)
- Answer
-
0.57
Practice Set B
-879 + 454
- Answer
-
-425
Calculators
Calculators having the
key can be used for finding sums of signed numbers.
Use a calculator to find the sum of -147 and 84.
| Display Reads | |||
| Type | 147 | 147 | |
| Press | +/- | -147 | This key changes the sign of a number. It is different than -. |
| Press | + | -147 | |
| Type | 84 | 84 | |
| Press | = | -63 |
Practice Set C
Use a calculator to find each sum.
673 + (-721)
- Answer
-
-48
Practice Set C
-8,261 + 2,206
- Answer
-
-6,085
Practice Set C
-1,345.6 + (-6,648.1)
- Answer
-
-7,993.7
Exercises
Find the sums in the following 27 problems. If possible, use a calculator to check each result.
Exercise \(\PageIndex{1}\)
4 + 12
- Answer
-
16
Exercise \(\PageIndex{2}\)
8 + 6
Exercise \(\PageIndex{3}\)
(-3) + (-12)
- Answer
-
-15
Exercise \(\PageIndex{4}\)
(-6) + (-20)
Exercise \(\PageIndex{5}\)
10 + (-2)
- Answer
-
8
Exercise \(\PageIndex{6}\)
8 + (-15)
Exercise \(\PageIndex{7}\)
-16 + (-9)
- Answer
-
-25
Exercise \(\PageIndex{8}\)
-22 + (-1)
Exercise \(\PageIndex{9}\)
0 + (-12)
- Answer
-
-12
Exercise \(\PageIndex{10}\)
0 + (-4)
Exercise \(\PageIndex{11}\)
0 + (24)
- Answer
-
24
Exercise \(\PageIndex{12}\)
-6 + 1 + (-7)
Exercise \(\PageIndex{13}\)
-5 + (-12) + (-4)
- Answer
-
-21
Exercise \(\PageIndex{14}\)
-5 + 5
Exercise \(\PageIndex{15}\)
-7 + 7
- Answer
-
0
Exercise \(\PageIndex{16}\)
-14 + 14
Exercise \(\PageIndex{17}\)
4 + (-4)
- Answer
-
0
Exercise \(\PageIndex{18}\)
9 + (-9)
Exercise \(\PageIndex{19}\)
84 + (-61)
- Answer
-
23
Exercise \(\PageIndex{20}\)
13 + (-56)
Exercise \(\PageIndex{21}\)
452 + (-124)
- Answer
-
328
Exercise \(\PageIndex{22}\)
636 + (-989)
Exercise \(\PageIndex{23}\)
1,811 + (-935)
- Answer
-
876
Exercise \(\PageIndex{24}\)
-373 + (-14)
Exercise \(\PageIndex{25}\)
-1,211 + (-44)
- Answer
-
-1,255
Exercise \(\PageIndex{26}\)
-47.03 + (-22.71)
Exercise \(\PageIndex{27}\)
-1.998 + (-4.086)
- Answer
-
-6.084
Exercise \(\PageIndex{28}\)
In order for a small business to break even on a project, it must have sales of $21,000. If the amount of sales was $15,000, by how much money did this company fall short?
Exercise \(\PageIndex{29}\)
Suppose a person has $56 in his checking account. He deposits $100 into his checking account by using the automatic teller machine. He then writes a check for $84.50. If an error causes the deposit not to be listed into this person’s account, what is this person’s checking balance?
- Answer
-
-$28.50
Exercise \(\PageIndex{30}\)
A person borrows $7 on Monday and then $12 on Tuesday. How much has this person borrowed?
Exercise \(\PageIndex{31}\)
A person borrows $11 on Monday and then pays back $8 on Tuesday. How much does this person owe?
- Answer
-
$3.00
Exercises for Review
Exercise \(\PageIndex{32}\)
Find the reciprocal of \(8 \dfrac{5}{6}\).
Exercise \(\PageIndex{33}\)
Find the value of \(\dfrac{5}{12} + \dfrac{7}{18} - \dfrac{1}{3}\).
- Answer
-
\(\dfrac{17}{36}\)
Exercise \(\PageIndex{34}\)
Round 0.01628 to the nearest tenth.
Exercise \(\PageIndex{35}\)
Convert 62% to a fraction.
- Answer
-
\(\dfrac{62}{100} = \dfrac{31}{50}\)
Exercise \(\PageIndex{36}\)
Find the value of |-12|.