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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: understand the definition of subtraction, be able to subtract signed numbers.

Overview

  • Definition of Subtraction
  • Subtraction of Signed Numbers

Definition of subtraction

We know from our experience with arithmetic that the subtraction 5 2 produces 3, that is, 5 2 = 3 . Illustrating this process on the number line suggests a rule for subtracting signed numbers.

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

We begin at 0, the origin.
Since 5 is positive, we move 5 units to the right.
Then, we move 2 units to the left to get to 3. (This reminds us of addition with a negative number.)

This illustration suggests that 5 2 is the same as 5 + ( 2 ) .
This leads us directly to the definition of subtraction.

Definition of subtraction

If a and b are real numbers, a b is the same as a + ( b ) , where b is the opposite of b .

Subtraction of signed numbers

The preceding definition suggests the rule for subtracting signed numbers.

Subtraction of signed numbers

To perform the subtraction a b , add the opposite of b to a , that is, change the sign of b and add.

Sample set a

Perform the subtractions.

5 3 = 5 + ( 3 ) = 2

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

4 9 = 4 + ( 9 ) = 5

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

4 6 = 4 + ( 6 ) = 10

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

3 ( 12 ) = 3 + 12 = 9

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

0 ( 15 ) = 0 + 15 = 15

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

The high temperature today in Lake Tahoe was 26 F . The low temperature tonight is expected to be 7 F . How many degrees is the temperature expected to drop?
We need to find the difference between 26 and 7 .

26 ( 7 ) = 26 + 7 = 33

Thus, the expected temperature drop is 33 F .

6 ( 5 ) 10 = 6 + 5 + ( 10 ) = ( 6 + 5 ) + ( 10 ) = 1 + ( 10 ) = 11

Practice set a

Perform the subtractions.

9 6

3

6 9

3

0 7

7

1 14

13

8 12

20

21 6

27

6 ( 4 )

2

8 ( 10 )

18

1 ( 12 )

13

86 ( 32 )

118

0 16

16

0 ( 16 )

16

0 ( 8 )

8

5 ( 5 )

10

24 ( ( 24 ) )

0

Exercises

For the following exercises, perform the indicated operations.

8 3

5

12 7

5 6

1

14 30

2 15

13

5 18

1 7

6

4 11

6 5

11

8 14

1 12

13

4 4

6 8

14

1 12

5 ( 3 )

2

11 ( 8 )

7 ( 12 )

5

2 ( 10 )

4 ( 15 )

11

11 ( 16 )

1 ( 6 )

5

8 ( 14 )

15 ( 10 )

5

11 ( 4 )

16 ( 8 )

8

12 ( 11 )

0 6

6

0 15

0 ( 7 )

7

0 ( 10 )

67 38

29

142 85

816 1140

324

105 421

550 ( 121 )

429

15.016 ( 4.001 )

26 + 7 52

71

15 21 ( 2 )

104 ( 216 ) ( 52 )

164

0.012 ( 0.111 ) ( 0.035 )

[ 5 + ( 6 ) ] [ 2 + ( 4 ) ]

1

[ 2 + ( 8 ) ] [ 5 + ( 7 ) ]

[ 4 + ( 11 ) ] [ 2 + ( 10 ) ]

1

[ 9 + ( 6 ) ] [ 4 + ( 12 ) ]

( 11 8 ) ( 1 6 )

8

( 5 12 ) ( 4 10 )

( 1 10 ) ( 2 15 )

4

( 0 8 ) ( 4 12 )

( 4 + 7 ) ( 2 5 )

6

( 6 + 2 ) ( 5 11 )

[ 8 + ( 5 + 3 ) ] [ 9 ( 3 5 ) ]

27

[ 4 + ( 1 + 6 ) ] [ 7 ( 6 1 ) ]

[ 2 ( 6 + 10 ) ] [ 1 ( 2 11 ) ]

12

[ 5 ( 2 5 ) ] [ 2 ( 1 4 ) ]

When a particular machine is operating properly, its meter will read 34. If a broken bearing in the machine causes the meter reading to drop by 45 units, what is the meter reading?

11

The low temperature today in Denver was 4 F and the high was 42 F . What is the temperature difference?

Exercises for review

( [link] ) Use the distributive property to expand 4 x ( 5 y + 11 ) .

20 x y + 44 x

( [link] ) Simplify 2 ( 3 x 2 y 2 ) 3 ( 2 x 4 y 3 ) 0 27 x 4 y 3 . Assume x 0 , y 0 .

( [link] ) Simplify | ( 4 2 + 2 2 3 2 ) | .

11

( [link] ) Find the sum. 8 + ( 14 ) .

( [link] ) Find the sum. 3 + ( 6 ) .

3

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
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?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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
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On having this app for quite a bit time, Haven't realised there's a chat room in it.
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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