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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand the closure, commutative, associative, and distributive properties, understand the identity and inverse properties.

Overview

  • The Closure Properties
  • The Commutative Properties
  • The Associative Properties
  • The Distributive Properties
  • The Identity Properties
  • The Inverse Properties

Property

A property of a collection of objects is a characteristic that describes the collection. We shall now examine some of the properties of the collection of real numbers. The properties we will examine are expressed in terms of addition and multiplication.

The closure properties

The closure properties

If a and b are real numbers, then a + b is a unique real number, and a b is a unique real number.

For example, 3 and 11 are real numbers; 3 + 11 = 14 and 3 11 = 33 , and both 14 and 33 are real numbers. Although this property seems obvious, some collections are not closed under certain operations. For example,

The real numbers are not closed under division since, although 5 and 0 are real numbers, 5 / 0 and 0 / 0 are not real numbers.

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The natural numbers are not closed under subtraction since, although 8 is a natural number, 8 8 is not. ( 8 8 = 0 and 0 is not a natural number.)

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The commutative properties

Let a and b represent real numbers.

The commutative properties

COMMUTATIVE PROPERTY OF ADDITION COMMUTATIVE PROPERTY OF MULTIPLICATION a + b = b + a a b = b a

The commutative properties tell us that two numbers can be added or multiplied in any order without affecting the result.

Sample set a

The following are examples of the commutative properties.

3 + 4 = 4 + 3 Both equal 7.

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5 + x = x + 5 Both represent the same sum .

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4 8 = 8 4 Both equal 32 .

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y 7 = 7 y Both represent the same product .

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5 ( a + 1 ) = ( a + 1 ) 5 Both represent the same product .

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( x + 4 ) ( y + 2 ) = ( y + 2 ) ( x + 4 ) Both represent the same product .

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Practice set a

Fill in the ( ) with the proper number or letter so as to make the statement true. Use the commutative properties.

4 ( k 5 ) = ( ) 4

( k 5 )

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( 9 a 1 ) ( ) = ( 2 b + 7 ) ( 9 a 1 )

( 2 b + 7 )

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The associative properties

Let a , b , and c represent real numbers.

The associative properties

ASSOCIATIVE PROPERTY OF ADDITION ASSOCIATIVE PROPERTY OF MULTIPLICATION ( a + b ) + c = a + ( b + c ) ( a b ) c = a ( b c )

The associative properties tell us that we may group together the quantities as we please without affecting the result.

Sample set b

The following examples show how the associative properties can be used.

( 2 + 6 ) + 1 = 2 + ( 6 + 1 ) 8 + 1 = 2 + 7 9 = 9 Both equal 9 .

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( 3 + x ) + 17 = 3 + ( x + 17 ) Both represent the same sum .

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( 2 3 ) 5 = 2 ( 3 5 ) 6 5 = 2 15 30 = 30 Both equal 30.

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Questions & Answers

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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
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s. Reply
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Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
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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?
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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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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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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
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what is the function of carbon nanotubes?
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what is nanomaterials​ and their applications of sensors.
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what is nano technology
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what is system testing?
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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anybody can imagine what will be happen after 100 years from now in nano tech world
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after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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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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