<< Chapter < Page Chapter >> Page >
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.

A number line with arrows on each end, labeled from negative six to six in increments of one. There are two closed circles at negative two and four, respectively.

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."

Got questions? Get instant answers now!

4 + ( 2 ) should be read as "four plus negative two" rather than "four plus minus two."

Got questions? Get instant answers now!

6 + ( 3 ) should be read as "negative six plus negative three" rather than "minus six plusminus three."

Got questions? Get instant answers now!

15 ( 6 ) should be read as "negative fifteen minus negative six" rather than "minus fifteenminus minus six."

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Practice set a

Write each expression in words.

7 + ( 4 )

seven plus negative four

Got questions? Get instant answers now!

9 + 2

negative nine plus two

Got questions? Get instant answers now!

16 ( + 8 )

negative sixteen minus positive eight

Got questions? Get instant answers now!

1 ( 9 )

negative one minus negative nine

Got questions? Get instant answers now!

0 + ( 7 )

zero plus negative seven

Got questions? Get instant answers now!

Opposites

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.

Two number lines with arrows on each end. The first number line has three labels, zero at the center, negative a to the left of zero and a to the right of zero. Negative a and a are equidistant from zero. The second line has three labels, zero at the center, a to the left of zero and negative a to the right of zero. The points a and negative a are equidistant from zero.

( a ) is opposite a on the number line. This implies that ( a ) = a .

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

The double-negative property

If a is a real number, then
( a ) = a

Sample set b

If a = 3 , then a = 3 and ( a ) = ( 3 ) = 3 .

A number line with arrows on each end, labeled from negative three to three in increments of three. Negative three is labeled as negative a, and three is labeled as a. There is an additional label for three as the opposite of negative a.

Got questions? Get instant answers now!

If a = 4 , then a = ( 4 ) = 4 and ( a ) = a = 4 .

A number line with arrows on each end, labeled from negative four to four in increments of three. Negative four is labeled as a, and four is labeled as negative a. There is an additional label for negative four as the opposite of negative a.

Got questions? Get instant answers now!

Practice set b

Find the opposite of each real number.

( 1 )

1 , since ( 1 ) = 1

Got questions? Get instant answers now!

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

If a is positive, a is negative.

Got questions? Get instant answers now!

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

If a is negative, a is positive.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Exercises

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

a plus sign or no sign at all

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

( 4 )

negative negative four

Got questions? Get instant answers now!

For the following problems, write the expressions in words.

11 + ( 2 )

eleven plus negative two

Got questions? Get instant answers now!

6 ( 8 )

six minus negative eight

Got questions? Get instant answers now!

Rewrite the following problems in a simpler form.

( 8 )

( 8 ) = 8

Got questions? Get instant answers now!

[ ( 3 ) ]

3

Got questions? Get instant answers now!

[ ( 6 ) ]

6

Got questions? Get instant answers now!

{ [ ( 26 ) ] }

26

Got questions? Get instant answers now!

{ [ ( 11 ) ] }

Got questions? Get instant answers now!

{ [ ( 31 ) ] }

31

Got questions? Get instant answers now!

{ [ ( 14 ) ] }

Got questions? Get instant answers now!

5 ( 2 )

5 ( 2 ) = 5 + 2 = 7

Got questions? Get instant answers now!

6 ( 3 ) ( 4 )

13

Got questions? Get instant answers now!

2 ( 1 ) ( 8 )

Got questions? Get instant answers now!

15 ( 6 ) ( 5 )

26

Got questions? Get instant answers now!

24 ( 8 ) ( 13 )

Got questions? Get instant answers now!

Exercises for review

( [link] ) There is only one real number for which ( 5 a ) 2 = 5 a 2 . What is the number?

0

Got questions? Get instant answers now!

( [link] ) Simplify ( 3 x y ) ( 2 x 2 y 3 ) ( 4 x 2 y 4 ) .

Got questions? Get instant answers now!

( [link] ) Simplify x n + 3 x 5 .

x n + 8

Got questions? Get instant answers now!

( [link] ) Simplify ( a 3 b 2 c 4 ) 4 .

Got questions? Get instant answers now!

( [link] ) Simplify ( 4 a 2 b 3 x y 3 ) 2 .

16 a 4 b 2 9 x 2 y 6

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask