<< Chapter < Page Chapter >> Page >
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.

Got questions? Get instant answers now!

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.)

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

4 8 = 8 4 Both equal 32 .

Got questions? Get instant answers now!

y 7 = 7 y Both represent the same product .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

( 2 b + 7 )

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
Eleven fifteenths of two more than a number is eight.
Leny Reply
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask