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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 able to add numbers with like signs and unlike signs, understand addition with zero.

Overview

  • Addition of Numbers with Like Signs
  • Addition with Zero
  • Addition of Numbers with Unlike Signs

Addition of numbers with like signs

Let us add the two positive numbers 2 and 3. We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 3 is positive, we move 3 more units to the right.
We are now located at 5.
Thus, 2 + 3 = 5 .

A number line with arrows on each end, labeled from negative two to eight in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards five.

Summarizing, we have

( 2 positive units ) + ( 3 positive units ) = ( 5 positive units )

Now let us add the two negative numbers 2 and 3 . We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is negative, we move 2 units to the left.
Since 3 is negative, we move 3 more units to the left.
We are now located at 5 .

Thus, ( 2 ) + ( 3 ) = 5 .

A number line with arrows on each end, labeled from negative seven to three in increments of one. There is a curved arrow starting from zero, and pointing towards negative two. There is another curved arrow starting from negative two, and pointing towards negative five

Summarizing, we have

( 2 negative units ) + ( 3 negative units ) = ( 5 negative units )

These two examples suggest that

( positive number ) + ( positive number ) = ( positive number ) ( negative number ) + ( negative number ) = ( negative number )

Adding numbers with the same sign

To add two real numbers that have the same sign, add the absolute values of the numbers and associate the common sign with the sum.

Sample set a

Find the sums.

3 + 7

Add these absolute values . | 3 | = 3 | 7 | = 7 } 3 + 7 = 10 The common sign is "+ ."

3 + 7 = + 10 or 3 + 7 = 10

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( 4 ) + ( 9 )

Add these absolute values . | 4 | = 4 | 9 | = 9 } 4 + 9 = 13 The common sign is " ."

( 4 ) + ( 9 ) = 13

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Practice set a

Find the sums.

( 4 ) + ( 8 )

12

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( 36 ) + ( 9 )

45

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14 + ( 20 )

34

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2 3 + ( 5 3 )

7 3

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2.8 + ( 4.6 )

7.4

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Addition with zero

Notice that

Addition with 0

( 0 ) + ( a positive number ) = ( that same positive number ) ( 0 ) + ( a negative number ) = ( that same negative number)

The additive identity is 0

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity .

Addition of numbers with unlike signs

Now let us perform the addition 2 + ( 6 ) . These two numbers have unlike signs. This type of addition can also be illustrated using the number line.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 6 is negative, we move, from the 2, 6 units to the left.
We are now located at 4 .

A number line with arrows on each end, labeled from negative five to five in increments of one. There is a curved arrow starting from zero, and pointing towards two. There is another curved arrow starting from two, and pointing towards negative four.

A rule for adding two numbers that have unlike signs is suggested by noting that if the signs are disregarded, 4 can be obtained from 2 and 6 by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and 6 . Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.

Adding numbers with unlike signs

To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate the sign of the number with the larger absolute value with this difference.

Sample set b

Find the following sums.

7 + ( 2 )

| 7 | = 7 Larger absolute value . Sign is " + " . | 2 | = 2 Smaller absolute value .

Subtract absolute values: 7 2 = 5. Attach the proper sign: " + " .

7 + ( 2 ) = + 5 or 7 + ( 2 ) = 5

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3 + ( 11 )

| 3 | = 3 Smaller absolute value . | 11 | = 11 Larger absolute value . Sign is " " .

Subtract absolute values: 11 3 = 8. Attach the proper sign: " " .

3 + ( 11 ) = 8

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The morning temperature on a winter's day in Lake Tahoe was 12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?

We need to find 12 + 25 .

| 12 | = 12 Smaller absolute value . | 25 | = 25 Larger absolute value . Sign is "+" .

Subtract absolute values: 25 12 = 13. Attach the proper sign: " + " .

12 + 25 = 13

Thus, the afternoon temperature is 13 degrees.

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Use a calculator. Add 147 + 84 .                                                      Display Reads

Type 147 147 Press + / 147 Press + 147 Type 84 84 Press = 63

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Practice set b

Find the sums.

1345.6 + ( 6648.1 )

7993.7

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Exercises

Find the sums for the the following problems.

( 3 ) + ( 12 )

15

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( 4 ) + ( 8 )

12

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( 16 ) + ( 8 )

24

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( 3 ) + ( 12 )

15

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9 + ( 6 )

15

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16 + ( 9 )

25

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5 + ( 12 ) + ( 4 )

21

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1221 + ( 44 )

1265

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47.03 + ( 22.71 )

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1.998 + ( 4.086 )

6.084

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[ ( 3 ) + ( 4 ) ] + [ ( 6 ) + ( 1 ) ]

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[ ( 2 ) + ( 8 ) ] + [ ( 3 ) + ( 7 ) ]

20

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[ ( 3 ) + ( 8 ) ] + [ ( 6 ) + ( 12 ) ]

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[ ( 8 ) + ( 6 ) ] + [ ( 2 ) + ( 1 ) ]

17

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[ 4 + ( 12 ) ] + [ 12 + ( 3 ) ]

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[ 5 + ( 16 ) ] + [ 4 + ( 11 ) ]

18

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[ 2 + ( 4 ) ] + [ 17 + ( 19 ) ]

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[ 10 + ( 6 ) ] + [ 12 + ( 2 ) ]

14

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14 + [ ( 3 ) + 5 ]

16

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[ 2 + ( 7 ) ] + ( 11 )

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[ 14 + ( 8 ) ] + ( 2 )

4

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In order for a small business to break even on a project, it must have sales of $ 21 , 000 . If the amount of sales was $ 15 , 000 , how much money did this company fall short?

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Suppose a person has $ 56.00 in his checking account. He deposits $ 100.00 into his checking account by using the automatic teller machine. He then writes a check for $ 84.50 . If an error causes the deposit not to be listed into this person's account, what is this person's checking balance?

$ 28.50

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A person borrows $ 7.00 on Monday and then $ 12.00 on Tuesday. How much has this person borrowed?

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A person borrows $ 11.00 on Monday and then pays back $ 8.00 on Tuesday. How much does this person owe?

$ 3.00

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Exercises for review

( [link] ) Simplify 4 ( 7 2 6 2 3 ) 2 2 .

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( [link] ) Simplify 35 a 6 b 2 c 5 7 b 2 c 4 .

5 a 6 c

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( [link] ) Simplify ( 12 a 8 b 5 4 a 5 b 2 ) 3 .

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( [link] ) Determine the value of | 8 | .

8

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( [link] ) Determine the value of ( | 2 | + | 4 | 2 ) + | 5 | 2 .

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Questions & Answers

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
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Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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what does nano mean?
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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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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
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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
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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.
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Do you know which machine is used to that process?
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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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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