# Connexions

You are here: Home » Content » Derived copy of Fundamentals of Mathematics » Signed Numbers

• Preface
• Acknowledgements

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Endorsed by (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
• CCQ

This module is included in aLens by: Community College of QatarAs a part of collection: "Fundamentals of Mathematics"

"Used as supplemental materials for developmental math courses."

Click the "CCQ" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

• College Open Textbooks

This module is included inLens: Community College Open Textbook Collaborative
By: CC Open Textbook CollaborativeAs a part of collection: "Fundamentals of Mathematics"

"Reviewer's Comments: 'I would recommend this text for a basic math course for students moving on to elementary algebra. The information in most chapters is useful, very clear, and easily […]"

Click the "College Open Textbooks" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Featured Content

This module is included inLens: Connexions Featured Content
By: ConnexionsAs a part of collection: "Fundamentals of Mathematics"

"Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and […]"

Click the "Featured Content" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

#### Also in these lenses

• UniqU content

This module is included inLens: UniqU's lens
By: UniqU, LLCAs a part of collection: "Fundamentals of Mathematics"

Click the "UniqU content" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Textbook):

Textbook by: Ron Stewart. E-mail the author

# Signed Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses signed numbers. By the end of the module students be able to distinguish between positive and negative real numbers, be able to read signed numbers and understand the origin and use of the double-negative product property.

## Section Overview

• Positive and Negative Numbers
• Opposites
• The Double-Negative Property

## Positive and Negative Numbers

### Positive and Negative Numbers

Each real number other than zero has a sign associated with it. A real number is said to be a positive number if it is to the right of 0 on the number line and negative if it is to the left of 0 on the number line.

### Note: THE NOTATION OF SIGNED NUMBERS:

#### + and -- Notation

A number is denoted as positive if it is directly preceded by a plus sign or no sign at all.
A number is denoted as negative if it is directly preceded by a minus sign.

The plus and minus signs now have two meanings:

The plus sign can denote the operation of addition or a positive number.

The minus sign can denote the operation of subtraction or a negative number.

To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative." When "+" is used as an operation sign, it is read as "plus." When "- -" is used as an operation sign, it is read as "minus."

### Sample Set A

Read each expression so as to avoid confusion between "operation" and "sign."

#### Example 1

- 8 -8 should be read as "negative eight" rather than "minus eight."

#### Example 2

4 +(- 2)4 +(- 2) size 12{"4 "+ $$"–2"$$ } {} should be read as "four plus negative two" rather than "four plus minus two."

#### Example 3

- 6+(- 3)- 6+(- 3) size 12{"–6 "+ $$"–3"$$ } {}should be read as "negative six plus negative three" rather than "minus six plus minus three."

#### Example 4

- 15 - (- 6)- 15 - (- 6) size 12{"–15 – " $$"–6"$$ } {}should be read as "negative fifteen minus negative six" rather than "minus fifteen minus minus six."

#### Example 5

- 5+ 7- 5+ 7 size 12{"–5 "+" 7"} {} should be read as "negative five plus seven" rather than "minus five plus seven."

#### Example 6

0 - 20 - 2 size 12{"0 – 2"} {} should be read as "zero minus two."

### Practice Set A

Write each expression in words.

#### Exercise 1

6 + 16 + 1 size 12{"6 "+" 1"} {}

six plus one

#### Exercise 2

2+(- 8)2+(- 8) size 12{2+ $$"–8"$$ } {}

##### Solution

two plus negative eight

#### Exercise 3

- 7+5- 7+5 size 12{"–7"+5} {}

##### Solution

negative seven plus five

#### Exercise 4

- 10 -(+3)- 10 -(+3) size 12{"–10 – " $$+3$$ } {}

##### Solution

negative ten minus three

#### Exercise 5

- 1 - (- 8)- 1 - (- 8) size 12{"–1 –" $$"–8"$$ } {}

##### Solution

negative one minus negative eight

#### Exercise 6

0 +(- 11)0 +(- 11) size 12{"0 "+ $$"–11"$$ } {}

##### Solution

zero plus negative eleven

## Opposites

### Opposites

On the number line, each real number, other than zero, has an image on the opposite side of 0. For this reason, we say that each real number has an opposite. Opposites are the same distance from zero but have opposite signs.

The opposite of a real number is denoted by placing a negative sign directly in front of the number. Thus, if aa size 12{a} {} is any real number, then aa size 12{ - a} {} is its opposite.

### Note:

The letter "aa size 12{a} {}" is a variable. Thus, "aa size 12{a} {}" need not be positive, and "- a- a size 12{–a} {}" need not be negative.

If aa size 12{a} {} is any real number, aa size 12{ - a} {} is opposite aa size 12{a} {} on the number line.

## The Double-Negative Property

The number aa size 12{a} {} is opposite aa size 12{ - a} {} on the number line. Therefore, (a)(a) size 12{ - $$- a$$ } {} is opposite aa size 12{ - a} {} on the number line. This means that

( a ) = a ( a ) = a size 12{ - $$- a$$ =a} {}

From this property of opposites, we can suggest the double-negative property for real numbers.

### Double-Negative Property: −(−a)=a−(−a)=a size 12{ - $$- a$$ =a} {}

If aa size 12{a} {} is a real number, then
( a ) = a ( a ) = a size 12{ - $$- a$$ =a} {}

### Sample Set B

Find the opposite of each number.

#### Example 7

If a = 2a = 2 size 12{"a "=" 2"} {}, then - a=- 2- a=- 2 size 12{"–a "=" –2"} {}. Also, (a)=(2)=2(a)=(2)=2 size 12{ - $$- a$$ = - $$- 2$$ =2} {}.

#### Example 8

If a =- 4a =- 4 size 12{"a "=" –4"} {}, then - a=-(- 4)= 4- a=-(- 4)= 4 size 12{"–a "="– " $$"–4"$$ =" 4"} {}. Also, -(- a)=a =- 4-(- a)=a =- 4 size 12{– $$"–a"$$ =" a "= – " 4"} {}.

### Practice Set B

Find the opposite of each number.

8

-8

17

-17

-6

6

-15

15

-(-1)

-1

#### Exercise 12

- [ - ( - 7 ) ] -[-(-7)]

7

#### Exercise 13

Suppose aa size 12{a} {} is a positive number. Is aa size 12{ - a} {} positive or negative?

##### Solution

aa size 12{ - a} {} is negative

#### Exercise 14

Suppose aa size 12{a} {} is a negative number. Is aa size 12{ - a} {} positive or negative?

##### Solution

aa size 12{ - a} {} is positive

#### Exercise 15

Suppose we do not know the sign of the number kk size 12{k} {}. Is kk size 12{ - k} {} positive, negative, or do we not know?

##### Solution

We must say that we do not know.

## Exercises

### Exercise 16

A number is denoted as positive if it is directly preceded by


.

+ (or no sign)

### Exercise 17

A number is denoted as negative if it is directly preceded by


.

How should the number in the following 6 problems be read? (Write in words.)

### Exercise 18

77 size 12{-7} {}

negative seven

### Exercise 19

55 size 12{-5} {}

### Exercise 20

1515 size 12{"15"} {}

fifteen

11

### Exercise 22

11 size 12{- left (-1 right )} {}

#### Solution

negative negative one, or opposite negative one

### Exercise 23

55 size 12{- left (-5 right )} {}

For the following 6 problems, write each expression in words.

### Exercise 24

5+35+3 size 12{5+3} {}

five plus three

### Exercise 25

3+83+8 size 12{3+8} {}

### Exercise 26

15+315+3 size 12{"15"+ left (-3 right )} {}

#### Solution

fifteen plus negative three

### Exercise 27

1+91+9 size 12{1+ left (-9 right )} {}

### Exercise 28

7272 size 12{-7- left (-2 right )} {}

#### Solution

negative seven minus negative two

### Exercise 29

012012 size 12{0- left (-"12" right )} {}

For the following 6 problems, rewrite each number in simpler form.

### Exercise 30

22 size 12{- left (-2 right )} {}

2

### Exercise 31

1616 size 12{- left (-"16" right )} {}

### Exercise 32

88 size 12{- left [- left (-8 right ) right ]} {}

-8

### Exercise 33

2020 size 12{- left [- left (-"20" right ) right ]} {}

### Exercise 34

7373 size 12{7- left (-3 right )} {}

#### Solution

7+3=107+3=10 size 12{7+3="10"} {}

### Exercise 35

6464 size 12{6- left (-4 right )} {}

### Exercises for Review

#### Exercise 36

((Reference)) Find the quotient; 8÷278÷27 size 12{8÷"27"} {}.

##### Solution

0.296¯0.296¯ size 12{0 "." {overline {"296"}} } {}

#### Exercise 37

((Reference)) Solve the proportion: 59=60x59=60x size 12{ { {5} over {9} } = { {"60"} over {x} } } {}

#### Exercise 38

((Reference)) Use the method of rounding to estimate the sum: 5829+87675829+8767 size 12{"5829"+"8767"} {}

##### Solution

6,000+9,000=15,000  (5,829+8,767=14,596)  or 5,800+8,800=14,6006,000+9,000=15,000  (5,829+8,767=14,596)  or 5,800+8,800=14,600 size 12{6,"000"+9,"000"="15","000" $$5,"829"+8,"767"="14","59"6$$ " or 5,""800"+8,"800"="14","600"} {}

#### Exercise 39

((Reference)) Use a unit fraction to convert 4 yd to feet.

#### Exercise 40

((Reference)) Convert 25 cm to hm.

0.0025 hm

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks