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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses absolute value. By the end of the module students should understand the geometric and algebraic definitions of absolute value.

Section overview

  • Geometric Definition of Absolute Value
  • Algebraic Definition of Absolute Value

Geometric definition of absolute value

Absolute value-geometric approach

Geometric definition of absolute value:
The absolute value of a number a size 12{a} {} , denoted a size 12{ \lline a \rline } {} , is the distance from a to 0 on the number line.

Absolute value answers the question of "how far," and not "which way." The phrase "how far" implies "length" and length is always a nonnegative quantity . Thus, the absolute value of a number is a nonnegative number.

Sample set a

Determine each value.

4 = 4 size 12{ lline 4 rline =4} {}

A number line with hash marks from 0 to 6, with zero to 4 marked as 4 units in length.

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4 = 4 size 12{ lline - 4 rline =4} {}

A number line with hash marks from -6 to 0, with -4 to 0 marked as 4 units in length.

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0 = 0 size 12{ lline 0 rline =0} {}

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5 = 5 size 12{ - lline 5 rline = - 5} {} . The quantity on the left side of the equal sign is read as "negative the absolute value of 5." The absolute value of 5 is 5. Hence, negative the absolute value of 5 is -5.

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3 = 3 size 12{ - lline - 3 rline = - 3} {} . The quantity on the left side of the equal sign is read as "negative the absolute value of -3." The absolute value of -3 is 3. Hence, negative the absolute value of -3 is - ( 3 ) = - 3 .

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Practice set a

By reasoning geometrically, determine each absolute value.

7 size 12{ lline 7 rline } {}

7

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3 size 12{ lline - 3 rline } {}

3

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12 size 12{ lline "12" rline } {}

12

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0 size 12{ lline 0 rline } {}

0

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9 size 12{ - lline 9 rline } {}

-9

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6 size 12{ - lline - 6 rline } {}

-6

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Algebraic definition of absolute value

From the problems in [link] , we can suggest the following algebraic defini­tion of absolute value. Note that the definition has two parts.

Absolute value—algebraic approach

Algebraic definition of absolute value
The absolute value of a number a is
| a | = a , if  a 0 - a , if < 0

The algebraic definition takes into account the fact that the number a size 12{a} {} could be either positive or zero a 0 size 12{ left (a>= 0 right )} {} or negative a < 0 size 12{ left (a<0 right )} {} .

  1. If the number a size 12{a} {} is positive or zero a 0 size 12{ left (a>= 0 right )} {} , the upper part of the definition applies. The upper part of the definition tells us that if the number enclosed in the absolute value bars is a nonnegative number, the absolute value of the number is the number itself.
  2. The lower part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.
The definition says that the vertical absolute value lines may be elimi­nated only if we know whether the number inside is positive or negative.

Sample set b

Use the algebraic definition of absolute value to find the following values.

8 size 12{ lline 8 rline } {} . The number enclosed within the absolute value bars is a nonnegative number, so the upper part of the definition applies. This part says that the absolute value of 8 is 8 itself.

8 = 8 size 12{ lline 8 rline =8} {}

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3 size 12{ lline - 3 rline } {} . The number enclosed within absolute value bars is a negative number, so the lower part of the definition applies. This part says that the absolute value of -3 is the opposite of -3, which is 3 size 12{ - left ( - 3 right )} {} . By the definition of absolute value and the double-negative property,

3 = 3 = 3 size 12{ lline - 3 rline = - left ( - 3 right )=3} {}

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Practice set b

Use the algebraic definition of absolute value to find the following values.

7 size 12{ lline 7 rline } {}

7

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9 size 12{ lline 9 rline } {}

9

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12 size 12{ lline - "12" rline } {}

12

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5 size 12{ lline - 5 rline } {}

5

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8 size 12{ - lline 8 rline } {}

-8

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1 size 12{ - lline 1 rline } {}

-1

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52 size 12{ - lline - "52" rline } {}

-52

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31 size 12{ - lline - 31 rline } {}

-31

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Exercises

Determine each of the values.

5 size 12{ lline 5 rline } {}

5

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3 size 12{ lline 3 rline } {}

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6 size 12{ lline 6 rline } {}

6

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9 size 12{ lline -9 rline } {}

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1 size 12{ lline -1 rline } {}

1

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4 size 12{ lline -4 rline } {}

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3 size 12{- lline 3 rline } {}

-3

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7 size 12{- lline 7 rline } {}

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- 14 size 12{- lline –14 rline } {}

-14

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0 size 12{ lline 0 rline } {}

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26 size 12{ lline -"26" rline } {}

26

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26 size 12{- lline -"26" rline } {}

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4 size 12{- left (- lline 4 rline right )} {}

4

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2 size 12{- left (- lline 2 rline right )} {}

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6 size 12{- left (- lline -6 rline right )} {}

6

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42 size 12{- left (- lline -"42" rline right )} {}

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5 2 size 12{ lline 5 rline - lline -2 rline } {}

3

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2 3 size 12{ lline -2 rline rSup { size 8{3} } } {}

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2 3 size 12{ lline - left (2 cdot 3 right ) rline } {}

6

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2 9 size 12{ lline -2 rline - lline -9 rline } {}

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6 + 4 2 size 12{ left ( lline -6 rline + lline 4 rline right ) rSup { size 8{2} } } {}

100

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1 1 3 size 12{ left ( lline -1 rline - lline 1 rline right ) rSup { size 8{3} } } {}

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4 + 6 2 2 3 size 12{ left ( lline 4 rline + lline -6 rline right ) rSup { size 8{2} } - left ( lline -2 rline right ) rSup { size 8{3} } } {}

92

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{ 4 + 3 3 } 2 size 12{- left lbrace left none - left [- lline -4 rline + lline -3 rline right ] rSup { size 8{3} } right rbrace right none rSup { size 8{2} } } {}

-1

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A Mission Control Officer at Cape Canaveral makes the statement “lift-off, T minus 50 seconds.” How long is it before lift-off?

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Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $2,400,000. Use absolute value notation to describe this company’s debt.

$ 2, 400 , 000 size 12{-$ lline -2,"400","000" rline } {}

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A particular machine is set correctly if upon action its meter reads 0. One particular machine has a meter reading of - 1.6 upon action. How far is this machine off its correct setting?

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Exercises for review

( [link] ) Find the sum: 9 70 + 5 21 + 8 15 size 12{ { {9} over {"70"} } + { {5} over {"21"} } + { {8} over {"15"} } } {} .

9 10 size 12{ { {9} over {"10"} } } {}

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( [link] ) Find the value of 3 10 + 4 12 19 20 size 12{ { { { {3} over {"10"} } + { {4} over {"12"} } } over { { {"19"} over {"20"} } } } } {} .

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( [link] ) Convert 3 . 2 3 5 size 12{3 "." 2 { {3} over {5} } } {} to a fraction.

3 13 50 or 163 50 size 12{3 { {"13"} over {"50"} } " or " { {"163"} over {"50"} } } {}

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( [link] ) The ratio of acid to water in a solution is 3 8 size 12{ { {3} over {8} } } {} . How many mL of acid are there in a solution that contain 112 mL of water?

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( [link] ) Find the value of 6 ( 8 ) size 12{-6- \( -8 \) } {} .

2

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Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
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?
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?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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Anassong
How can I make nanorobot?
Lily
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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
7hours 36 min - 4hours 50 min
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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