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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."

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

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

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

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

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

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Practice set a

Write each expression in words.

7 + ( 4 )

seven plus negative four

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9 + 2

negative nine plus two

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16 ( + 8 )

negative sixteen minus positive eight

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1 ( 9 )

negative one minus negative nine

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0 + ( 7 )

zero plus negative seven

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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.

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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.

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Practice set b

Find the opposite of each real number.

( 1 )

1 , since ( 1 ) = 1

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

If a is positive, a is negative.

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

If a is negative, a is positive.

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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.

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Exercises

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

a plus sign or no sign at all

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

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

( 4 )

negative negative four

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For the following problems, write the expressions in words.

11 + ( 2 )

eleven plus negative two

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6 ( 8 )

six minus negative eight

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Rewrite the following problems in a simpler form.

( 8 )

( 8 ) = 8

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[ ( 3 ) ]

3

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[ ( 6 ) ]

6

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{ [ ( 26 ) ] }

26

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{ [ ( 11 ) ] }

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{ [ ( 31 ) ] }

31

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{ [ ( 14 ) ] }

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5 ( 2 )

5 ( 2 ) = 5 + 2 = 7

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6 ( 3 ) ( 4 )

13

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2 ( 1 ) ( 8 )

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15 ( 6 ) ( 5 )

26

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24 ( 8 ) ( 13 )

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Exercises for review

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

0

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

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

x n + 8

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

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

16 a 4 b 2 9 x 2 y 6

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Questions & Answers

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Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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?
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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absolutely yes
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characteristics of micro business
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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
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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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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