# 3.1 Signed numbers

 Page 1 / 1
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: be familiar with positive and negative numbers and with the concept of opposites.

## Overview

• Positive and Negative Numbers
• Opposites

## Positive and negative numbers

When we studied the number line in Section [link] we noted that

Each point on the number line corresponds to a real number, and each real number is located at a unique point on the number line.

## Positive and negative numbers

Each real number has a sign inherently associated with it. A real number is said to be a positive number if it is located to the right of 0 on the number line. It is a negative number if it is located to the left of 0 on the number line.

## The notation of signed numbers

A number is denoted as positive if it is directly preceded by a $"+"$ sign or no sign at all.
A number is denoted as negative if it is directly preceded by a $"-"$ sign.

The $"+"$ and $"-"$ signs now have two meanings:

$+$ can denote the operation of addition or a positive number.
$-$ can denote the operation of subtraction or a negative number.

## Read the $"-"$ Sign as "negative"

To avoid any confusion between "sign" and "operation," it is preferable to read the sign of a number as "positive" or "negative."

## Sample set a

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

$4+\left(-2\right)$ should be read as "four plus negative two" rather than "four plus minus two."

$-6+\left(-3\right)$ should be read as "negative six plus negative three" rather than "minus six plusminus three."

$-15-\left(-6\right)$ should be read as "negative fifteen minus negative six" rather than "minus fifteenminus minus six."

$-5+7$ should be read as "negative five plus seven" rather than "minus five plus seven."

$0-2$ should be read as "zero minus two."

## Practice set a

Write each expression in words.

$4+10$

four plus ten

$7+\left(-4\right)$

seven plus negative four

$-9+2$

negative nine plus two

$-16-\left(+8\right)$

negative sixteen minus positive eight

$-1-\left(-9\right)$

negative one minus negative nine

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

zero plus negative seven

## Opposites

On the number line, each real number 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 $a$ is any real number, then $-a$ is its opposite. Notice that the letter $a$ is a variable. Thus, $"a"$ need not be positive, and $"-a"$ need not be negative.

If $a$ is a real number, $-a$ is opposite $a$ on the number line and $a$ is opposite $-a$ on the number line.

$-\left(-a\right)$ is opposite $-a$ on the number line. This implies that $-\left(-a\right)=a$ .

This property of opposites suggests the double-negative property for real numbers.

## The double-negative property

If $a$ is a real number, then
$-\left(-a\right)=a$

## Sample set b

If $a=3$ , then $-a=-3$ and $-\left(-a\right)=-\left(-3\right)=3$ .

If $a=-4$ , then $-a=-\left(-4\right)=4$ and $-\left(-a\right)=a=-4$ .

## Practice set b

Find the opposite of each real number.

8

$-8$

17

$-17$

$-6$

6

$-15$

15

$-\left(-1\right)$

$-1$ , since $-\left(-1\right)=1$

$-\left[-\left(-7\right)\right]$

7

Suppose that $a$ is a positive number. What type of number is $-a$ ?

If $a$ is positive, $-a$ is negative.

Suppose that $a$ is a negative number. What type of number is $-a$ ?

If $a$ is negative, $-a$ is positive.

Suppose we do not know the sign of the number $m$ . Can we say that $-m$ is positive, negative, or that we do notknow ?

We must say that we do not know.

## Exercises

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 ____________________ .

For the following problems, how should the real numbers be read ? (Write in words.)

$-5$

a negative five

$-3$

12

twelve

10

$-\left(-4\right)$

negative negative four

$-\left(-1\right)$

For the following problems, write the expressions in words.

$5+7$

five plus seven

$2+6$

$11+\left(-2\right)$

eleven plus negative two

$1+\left(-5\right)$

$6-\left(-8\right)$

six minus negative eight

$0-\left(-15\right)$

Rewrite the following problems in a simpler form.

$-\left(-8\right)$

$-\left(-8\right)=8$

$-\left(-5\right)$

$-\left(-2\right)$

2

$-\left(-9\right)$

$-\left(-1\right)$

1

$-\left(-4\right)$

$-\left[-\left(-3\right)\right]$

$-3$

$-\left[-\left(-10\right)\right]$

$-\left[-\left(-6\right)\right]$

$-6$

$-\left[-\left(-15\right)\right]$

$-\left\{-\left[-\left(-26\right)\right]\right\}$

26

$-\left\{-\left[-\left(-11\right)\right]\right\}$

$-\left\{-\left[-\left(-31\right)\right]\right\}$

31

$-\left\{-\left[-\left(-14\right)\right]\right\}$

$-\left[-\left(12\right)\right]$

12

$-\left[-\left(2\right)\right]$

$-\left[-\left(17\right)\right]$

17

$-\left[-\left(42\right)\right]$

$5-\left(-2\right)$

$5-\left(-2\right)=5+2=7$

$6-\left(-14\right)$

$10-\left(-6\right)$

16

$18-\left(-12\right)$

$31-\left(-1\right)$

32

$54-\left(-18\right)$

$6-\left(-3\right)-\left(-4\right)$

13

$2-\left(-1\right)-\left(-8\right)$

$15-\left(-6\right)-\left(-5\right)$

26

$24-\left(-8\right)-\left(-13\right)$

## Exercises for review

( [link] ) There is only one real number for which ${\left(5a\right)}^{2}=5{a}^{2}$ . What is the number?

0

( [link] ) Simplify $\left(3xy\right)\left(2{x}^{2}{y}^{3}\right)\left(4{x}^{2}{y}^{4}\right)$ .

( [link] ) Simplify ${x}^{n+3}\cdot {x}^{5}$ .

${x}^{n+8}$

( [link] ) Simplify ${\left({a}^{3}{b}^{2}{c}^{4}\right)}^{4}$ .

( [link] ) Simplify ${\left(\frac{4{a}^{2}b}{3x{y}^{3}}\right)}^{2}$ .

$\frac{16{a}^{4}{b}^{2}}{9{x}^{2}{y}^{6}}$

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.