# 10.5 Subtraction of signed numbers

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to subtract signed numbers. By the end of the module students should understand the definition of subtraction, be able to subtract signed numbers and be able to use a calculator to subtract signed numbers.

## Section overview

• Definition of Subtraction
• The Process of Subtraction
• Calculators

## Definition of subtraction

We know from experience with arithmetic that the subtraction $5-2$ produces 3, that is $5-2=\text{3}$ . 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 $\text{5}+\left(-2\right)$ . 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 $a+\left(-b\right)$ , 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.

## Sample set a

Perform the indicated subtractions.

$5-3=5+\left(-3\right)=2$

$4-9=\text{4}+\left(-9\right)=-5$

$-4-6=-4+\left(-6\right)=-\text{10}$

$-3-\left(-\text{12}\right)=-3+\text{12}=9$

$0-\left(-15\right)=\text{0}+\text{15}=\text{15}$

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?

We need to find the difference between 26 and -7.

$26-\left(-7\right)=\text{26}+\text{7}=\text{33}$

Thus, the expected temperature drop is 33°F.

$\begin{array}{ccc}-6-\left(-5\right)-10& =& -6+5+\left(-10\right)\hfill \\ & =& \left(-6+5\right)+\left(-10\right)\hfill \\ & =& -1+\left(-10\right)\hfill \\ & =& -11\hfill \end{array}$

## Practice set a

Perform the indicated subtractions.

$9-6$

3

$6-9$

-3

$0-7$

-7

$1-\text{14}$

-13

$-8-\text{12}$

-20

$-\text{21}-6$

-27

$-6-\left(-4\right)$

-2

$8-\left(-\text{10}\right)$

18

$1-\left(-\text{12}\right)$

13

$\text{86}-\left(-\text{32}\right)$

118

$0-\text{16}$

-16

$0-\left(-\text{16}\right)$

16

$0-\left(8\right)$

-8

$5-\left(-5\right)$

10

$\text{24}-\left[-\left(-\text{24}\right)\right]$

0

## Calculators

Calculators can be used for subtraction of signed numbers. The most efficient calculators are those with a key.

## Sample set b

Use a calculator to find each difference.

$3,\text{187}-8,\text{719}$

 Display Reads Type 3187 3187 Press - 3187 Type 8719 8719 Press = -5532

Thus, $3,\text{187}-8,\text{719}=-5,\text{532}$ .

$-\text{156}-\left(-\text{211}\right)$

Method A:

 Display Reads Type 156 156 Press -156 Type - -156 Press 211 211 Type -211 Press = 55

Thus, $-\text{156}-\left(-\text{211}\right)=\text{55}$ .

Method B:

We manually change the subtraction to an addition and change the sign of the number to be subtracted.

$-\text{156}-\left(-\text{211}\right)$ becomes $-\text{156}+\text{211}$

 Display Reads Type 156 156 Press -156 Press + -156 Type 211 211 Press = 55

## Practice set b

Use a calculator to find each difference.

$\text{44}-\text{315}$

-271

$\text{12}\text{.}\text{756}-\text{15}\text{.}\text{003}$

-2.247

$-\text{31}\text{.}\text{89}-\text{44}\text{.}\text{17}$

-76.06

$-0\text{.}\text{797}-\left(-0\text{.}\text{615}\right)$

-0.182

## Exercises

For the following 18 problems, perform each subtraction. Use a calculator to check each result.

$8-3$

5

$\text{12}-7$

$5-6$

-1

$\text{14}-\text{30}$

$-6-8$

-14

$-1-\text{12}$

$-5-\left(-3\right)$

-2

$-\text{11}-\left(-8\right)$

$0-6$

-6

$0-\text{15}$

$0-\left(-7\right)$

7

$0-\left(-\text{10}\right)$

$\text{67}-\text{38}$

29

$\text{142}-\text{85}$

$\text{816}-\text{1140}$

-324

$\text{105}-\text{421}$

$-\text{550}-\left(-\text{121}\right)$

-429

$-\text{15}\text{.}\text{016}-\left(4\text{.}\text{001}\right)$

For the following 4 problems, perform the indicated operations.

$-\text{26}+7-\text{52}$

-71

$-\text{15}-\text{21}-\left(-2\right)$

$-\text{104}-\left(-\text{216}\right)-\left(-\text{52}\right)$

164

$-0\text{.}\text{012}-\left(-0\text{.}\text{111}\right)-\left(0\text{.}\text{035}\right)$

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?

-11

The low temperature today in Denver was $-4°$ F and the high was $-\text{42}°$ F. What is the temperature difference?

## Exercises for review

( [link] ) Convert $\text{16}\text{.}\text{02}\frac{1}{5}$ to a decimal.

16.022

( [link] ) Find 4.01 of 6.2.

( [link] ) Convert $\frac{5}{\text{16}}$ to a percent.

31.25%

( [link] ) Use the distributive property to compute the product: $\text{15}\cdot \text{82}$ .

( [link] ) Find the sum: $\text{16}+\left(-\text{21}\right)$ .

-5

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