# 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

#### Questions & Answers

How does cognition and memory links with perception?
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Catalina
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Ruth
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Ruth
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Sujeet Reply
how to read human behaviour
Sujeet
Describe frontal lobe syndrome
Bridget Reply
Frontal lobe syndrome is an impairment of the frontal lobe that occurs due to disease or head trauma.
Zafar
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average
Angela
I'M A VERY CURIOUS PERSON WHO LOVES TO LEARN ABOUT DIFFERENT TYPES
Sheri Reply
I AM A PERSON WHO IS NOT CHOOSING TO LET MY MENTAL HEALTH AFFECT ME AND I AM AN
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rilek
Indeed....different types means what exactly and how does these different types transpire into our lives? there are many different ways of allowing ourselves to accommodate and continiously play out the fact we have mental illnesses!!!
Elizabeth
social learning theory
CONFERENCE Reply
how we know the people are telling lie
Vasu
human brain
SERAJ
if you want to learn how to detect the lie you have to observe their behaviour...
Aayan
that's common things..
MOHD
anything specially, tht we can say either person is telling yes or no..?
MOHD
Yes
rochell
I believe there are many factors that play into being a human lie detector.
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agii
Ex. Talking too fast while explaining a story might indicate that the person is indeed hiding something. Other than that, people, usually with histrionic personality disorder, also tend to exaggerate stories.
agii
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dorfrlyn
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dorfrlyn
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dorfrlyn
some people have the habit of speaking too fast, so in that case how can we identify whether the person is lying or not?
Labdhee
pay attention on facial expressions
El
When people talk too fast to the point that it's incomprehensible, it could mean that that person is trying to have you move one from one topic to another to prevent you from asking questions.
agii
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Zafar
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Ashley
detecting lies is easier than many think. the dynamics of listening, body language, facial expression, story line, deductive reasoning and most of all analytical skills all play a role in determining if someone is lying. micro expressions can be very hard to discern requiring close analysis!
Kyth
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Andrew Reply
intresting I mean
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Sheri
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Aisha Reply
depression
Irfan
yes. it's depression.
Paul
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Tahmetrea
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Diana
not accepting the fact or situation or the reality. the adolescents can't face rejections, as soon as they are rejected , they commit suicide.
Labdhee
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Jody
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Thuy
well, I've a problem. I feel very conscious when I'm in a crowd. when I'm partying with friends, I feels like everyone is looking at me. when I look at them, each one of them have their eyes on me. I don't know how I should deal with this? can someone help me with that ?
puneet
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Thuy
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Thuy
Bullying that leads to anxiety, shame and depression, sadly.
Rachael
Where do I go to find research or tips for integrating sex and mental heath?
Felicia Reply
integrating a sexual routine and a mental illness that has kept o e from intimacy for a prolonged time? can u be a Lil more specific
lindsay
I have experience with both
lindsay
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MOHD
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Abu
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MOHD
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MOHD
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MOHD
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Abu
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Wendy
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Wendy
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Mohammad
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Felicia
I can help on these subjects
Abiral
@Abiral what information do you have?
Felicia
Hello
Irfan
Spending quiet time alone, with nature can improve mental health. Offer your opinions?
tsav Reply
mmmm but me i cn not spend time alone
Gift
Different strokes for different folks 🤷🏽‍♂️
tsav
what do u mean
Gift
Some people prefer a quiet life to a busy, bright city life. Some lead very stressful lives. Those people often don't pay attention to how they feel apart from the obvious fact that they're tired and stressed
tsav
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Twisted
You're absolutely right
tsav
what is with human being brain that we don't know?
imran Reply
I guess there are lots of things we don't know, but perhaps your referring to that thing called Consciousness
GoldenWizard
Anything that happens, question it!
tsav
in addition we may ask the doctor's
Gift
the science of many feelings and emotions can be hard to describe. our brains light up at some things but we can't always understand why
Twisted
But yet we must persist
tsav
Using relevant examples discuss challenges encounter by the beginning teachers during the learning and teaching process
LEILA Reply
how do I know when someone is using psychology on me?
Danielle Reply
If yes, it would always bring a solution.
Professor
Psychology cannot be used *on* something. You can use your *knowledge* of psychology to help tackle a situation. But I think I know what you're asking: pay attention to someone when you're talking, if they are looking at you up and down, they're analysing you
tsav
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ibraa Reply
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