<< Chapter < Page Chapter >> Page >
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
  • Reading Signed 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.

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.

Reading signed numbers

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."

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set a

Write each expression in words.

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

six plus one

Got questions? Get instant answers now!

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

two plus negative eight

Got questions? Get instant answers now!

- 7 + 5 size 12{"–7"+5} {}

negative seven plus five

Got questions? Get instant answers now!

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

negative ten minus three

Got questions? Get instant answers now!

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

negative one minus negative eight

Got questions? Get instant answers now!

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

zero plus negative eleven

Got questions? Get instant answers now!

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 a size 12{a} {} is any real number, then a size 12{ - a} {} is its opposite.

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

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

Two number lines. One number line with hash marks from left to right, -a, 0, and a. This number line is titled a positive.  A second number line with hash marks from left to right, a, 0, and -a. This number line is titled a negative.

The double-negative property

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

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

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

Sample set b

Find the opposite of each number.

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

A number line with hash marks from left to right, -2, 0, and 2. Below the -2 is -a, and below the 2 is a, or -(-a).

Got questions? Get instant answers now!

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

A number line with hash marks from left to right, -4, 0, and 4. Below the -4 is a, or -(-a), and below the 2 is -a.

Got questions? Get instant answers now!

Practice set b

Find the opposite of each number.

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

a size 12{ - a} {} is negative

Got questions? Get instant answers now!

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

a size 12{ - a} {} is positive

Got questions? Get instant answers now!

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

We must say that we do not know.

Got questions? Get instant answers now!

Exercises

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

+ (or no sign)

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

7 size 12{-7} {}

negative seven

Got questions? Get instant answers now!

15 size 12{"15"} {}

fifteen

Got questions? Get instant answers now!

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

negative negative one, or opposite negative one

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

five plus three

Got questions? Get instant answers now!

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

fifteen plus negative three

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

negative seven minus negative two

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

-8

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Exercises for review

( [link] ) Find the quotient; 8 ÷ 27 size 12{8÷"27"} {} .

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

Got questions? Get instant answers now!

( [link] ) Solve the proportion: 5 9 = 60 x size 12{ { {5} over {9} } = { {"60"} over {x} } } {}

Got questions? Get instant answers now!

( [link] ) Use the method of rounding to estimate the sum: 5829 + 8767 size 12{"5829"+"8767"} {}

6, 000 + 9, 000 = 15 , 000   ( 5, 829 + 8, 767 = 14 , 59 6 )   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"} {}

Got questions? Get instant answers now!

( [link] ) Use a unit fraction to convert 4 yd to feet.

Got questions? Get instant answers now!

( [link] ) Convert 25 cm to hm.

0.0025 hm

Got questions? Get instant answers now!

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
7hours 36 min - 4hours 50 min
Tanis Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of mathematics' conversation and receive update notifications?

Ask