10.5: Subtraction of Signed Numbers
- Page ID
- 48896
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- understand the definition of subtraction
- be able to subtract signed numbers
- be able to use a calculator to subtract signed numbers
Definition of Subtraction
We know from experience with arithmetic that the subtraction 5 - 2 produces 3, that is 5 - 2 = 3. We can suggest a rule for subtracting signed numbers by illustrating this process on the number line.
Begin at 0, the origin.
Since 5 is positive, move 5 units to the right.
Then, move 2 units to the left to get to 6. (This reminds us of addition with a negative number.)
From this illustration we can see that 5 - 2 is the same as 5 + (-2). This leads us directly to the definition of subtraction.
Definition of Subtraction
If \(a\) and \(b\) are real numbers, \(a - b\) is the same as \(a + (-b)\), where \(-b\) is the opposite of \(b\).
The Process of Subtraction
From this definition, we suggest the following rule for subtracting signed numbers.
Subtraction of Signed Numbers
To perform the subtraction \(a - b\), add the opposite of \(b\) to \(a\), that is, change the sign of \(b\) and add.
Perform the indicated subtractions.
5 - 3 = 5 + (-3) = 2
4 - 9 = 4 + (-9) = -5
-4 - 6 = -4 + (-6) = -10
-3 - (-12) = -3 + 12 = 9
-3 - (-12) = -3 + 12 = 9
The high temperature today in Lake Tahoe was 26°F. The low temperature tonight is expected to be -7°F. How many degrees is the temperature expected to drop?
Solution
We need to find the difference between 26 and -7.
26 - (-7) = 26 + 7 = 33
Thus, the expected temperature drop is 33°F.
\(\begin{array} {rcl} {-6 - (-5) - 10} & = & {-6 + 5 + (-10)} \\ {} & = & {(-6 + 5) + (-10)} \\ {} & = & {-1 + (-10)} \\ {} & = & {-11} \end{array}\)
Practice Set A
Perform the indicated subtractions.
9 − 6
- Answer
-
3
Practice Set A
6 - 9
- Answer
-
-3
Practice Set A
0 - 7
- Answer
-
-7
Practice Set A
1 - 14
- Answer
-
-13
Practice Set A
-8 - 12
- Answer
-
-20
Practice Set A
-21 - 6
- Answer
-
-27
Practice Set A
-6 - (-4)
- Answer
-
-10
Practice Set A
8 - (-10)
- Answer
-
18
Practice Set A
1 - (-12)
- Answer
-
13
Practice Set A
86 - (-32)
- Answer
-
118
Practice Set A
0 - 16
- Answer
-
-16
Practice Set A
0 - (-16)
- Answer
-
16
Practice Set A
0 - (8)
- Answer
-
-8
Practice Set A
5 - (-5)
- Answer
-
10
Practice Set A
24 - [-(-24)]
- Answer
-
0
Calculators
Calculators can be used for subtraction of signed numbers. The most efficient calculators are those with a
key.
Use a calculator to find each difference.
3,187 - 8,719
Solution
| Display Reads | ||
| Type | 3187 | 3187 |
| Press | - | 3187 |
| Type | 8719 | 8719 |
| Press | = | -5532 |
Thus, 3,187 - 8,719 - -5,532
-156 - (-211)
Solution
Method A:
| Display Reads | ||
| Type | 156 | 156 |
| Press |  |
-156 |
| Type | - | -156 |
| Press | 211 | 211 |
| Type |  |
-211 |
| Press | = | 55 |
Thus, -156 - (-211) = 55.
Method B:
We manually change the subtraction to an addition and change the sign of the number to be subtracted. \(-156 - (-211)\) because -156 + 211
| Display Reads | ||
| Type | 156 | 156 |
| Press | |
-156 |
| Press | + | -156 |
| Type | 211 | 211 |
| Press | = | 55 |
Practice Set B
Use a calculator to find each difference.
44 - 315
- Answer
-
-271
Practice Set B
12.756 - 15.003
- Answer
-
-2.247
Practice Set B
-31.89 - 44.17
- Answer
-
-76.06
Practice Set B
-0.797 - (-0.615)
- Answer
-
-0.182
Exercises
For the following 18 problems, perform each subtraction. Use a calculator to check each result.
Exercise \(\PageIndex{1}\)
8 - 3
- Answer
-
5
Exercise \(\PageIndex{2}\)
12 - 7
Exercise \(\PageIndex{3}\)
5 - 6
- Answer
-
-1
Exercise \(\PageIndex{4}\)
14 - 30
Exercise \(\PageIndex{5}\)
-6 - 8
- Answer
-
-14
Exercise \(\PageIndex{6}\)
- 1 - 12
Exercise \(\PageIndex{7}\)
-5 - (-3)
- Answer
-
-2
Exercise \(\PageIndex{8}\)
-11 - (-8)
Exercise \(\PageIndex{9}\)
0 - 6
- Answer
-
-6
Exercise \(\PageIndex{10}\)
0 - 15
Exercise \(\PageIndex{11}\)
0 - (-7)
- Answer
-
7
Exercise \(\PageIndex{12}\)
0 - (-10)
Exercise \(\PageIndex{13}\)
67 - 38
- Answer
-
29
Exercise \(\PageIndex{14}\)
142 - 85
Exercise \(\PageIndex{15}\)
816 - 1140
- Answer
-
-324
Exercise \(\PageIndex{16}\)
105 - 421
Exercise \(\PageIndex{17}\)
-550 - (-121)
- Answer
-
-429
Exercise \(\PageIndex{18}\)
-15.016 - (4.001)
For the following 4 problems, perform the indicated operations.
Exercise \(\PageIndex{19}\)
-26 + 7 - 52
- Answer
-
-71
Exercise \(\PageIndex{20}\)
-15 - 21 - (-2)
Exercise \(\PageIndex{21}\)
-104 - (-216) - (-52)
- Answer
-
164
Exercise \(\PageIndex{22}\)
-0.012 - (-0.111) - (0.035)
Exercise \(\PageIndex{23}\)
When a particular machine is operating properly, its meter will read 34. If a broken bearing in the machine causes the meter reading to drop by 45 units, what is the meter reading?
- Answer
-
-11
Exercise \(\PageIndex{24}\)
The low temperature today in Denver was \(-4^{\circ}\)F and the high was \(-42^{\circ}\)F. What is the temperature difference?
Exercises for Review
Exercise \(\PageIndex{25}\)
Convert \(16.02 \dfrac{1}{5}\) to a decimal.
- Answer
-
16.022
Exercise \(\PageIndex{26}\)
Find 4.01 of 6.2.
Exercise \(\PageIndex{27}\)
Convert \(\dfrac{5}{16}\) to a percent.
- Answer
-
31.25%
Exercise \(\PageIndex{28}\)
Use the distributive property to compute the product: \(15 \cdot 82\).
Exercise \(\PageIndex{29}\)
Find the sum: \(16 + (-21)\).
- Answer
-
-5