10.4: Addition of Signed Numbers

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Learning Objectives

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.

$A number line with has marks for the numbers -2 to 7. An arrow is drawn from 0 to 2, and from 2 to 5.$

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.

$A number line with has marks for the numbers -7 to 2. An arrow is drawn from 0 to -2, and from -2 to -5.$

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.

Sample Set A

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 $3 + 7 = 10$.

Sample Set A

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

$(0) + \text{(a positive number)} = \text{(that same positive number)}$.
$(0) + \text{(a negative number)} = \text{(that same negative number)}$.

Definition: The Additive Identity Is Zero

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.

$A number line with has marks for the numbers -5 to 4. An arrow is drawn from 2 to -4, and from 0 to 2.$

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.

Sample Set B

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 $7 + (-2) = 5$.

Sample Set B

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, $3 + (-11) = -8$.

Sample Set B

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

$A square with a plus and minus sign.$

key can be used for finding sums of signed numbers.

Sample Set C

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|.