# 2.3 Properties of the real numbers

 Page 1 / 2
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

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

For example, 3 and 11 are real numbers; $3+11=14$ and $3\cdot 11=33,$ 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.

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.)

## The commutative properties

Let $a$ and $b$ represent real numbers.

## The commutative properties

$\begin{array}{cc}\begin{array}{l}\text{COMMUTATIVE}\text{\hspace{0.17em}}\text{PROPERTY}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{OF}\text{\hspace{0.17em}}\text{ADDITION}\end{array}& \begin{array}{l}\text{COMMUTATIVE}\text{\hspace{0.17em}}\text{PROPERTY}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{OF}\text{\hspace{0.17em}}\text{MULTIPLICATION}\end{array}\\ a+b=b+a& a\cdot b=b\cdot a\end{array}$

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.

$\begin{array}{cc}3+4=4+3& \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}7.\end{array}$

$\begin{array}{cc}5+x=x+5& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{sum}\text{.}\end{array}$

$\begin{array}{cc}4\cdot 8=8\cdot 4& \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}\text{32}\text{.}\end{array}$

$\begin{array}{cc}y7=7y& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$

$\begin{array}{cc}5\left(a+1\right)=\left(a+1\right)5& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$

$\begin{array}{cc}\left(x+4\right)\left(y+2\right)=\left(y+2\right)\left(x+4\right)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$

## Practice set a

Fill in the $\left(\begin{array}{cc}& \end{array}\right)$ with the proper number or letter so as to make the statement true. Use the commutative properties.

$6+5=\left(\begin{array}{cc}& \end{array}\right)+6$

$5$

$m+12=12+\left(\begin{array}{cc}& \end{array}\right)$

$m$

$9\cdot 7=\left(\begin{array}{cc}& \end{array}\right)\cdot 9$

$7$

$6a=a\left(\begin{array}{cc}& \end{array}\right)$

$6$

$4\left(k-5\right)=\left(\begin{array}{cc}& \end{array}\right)4$

$\left(k-5\right)$

$\left(9a-1\right)\left(\begin{array}{cc}& \end{array}\right)=\left(2b+7\right)\left(9a-1\right)$

$\left(2b+7\right)$

## The associative properties

Let $a,b,$ and $c$ represent real numbers.

## The associative properties

$\begin{array}{cc}\begin{array}{l}\text{ASSOCIATIVE}\text{\hspace{0.17em}}\text{PROPERTY}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{OF}\text{\hspace{0.17em}}\text{ADDITION}\end{array}& \begin{array}{l}\text{ASSOCIATIVE}\text{\hspace{0.17em}}\text{PROPERTY}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{OF}\text{\hspace{0.17em}}\text{MULTIPLICATION}\end{array}\\ \left(a+b\right)+c=a+\left(b+c\right)& \left(ab\right)c=a\left(bc\right)\end{array}$

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.

$\begin{array}{llll}\left(2+6\right)+1\hfill & =\hfill & 2+\left(6+1\right)\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}8+1\hfill & =\hfill & 2+7\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9\hfill & =\hfill & 9\hfill & \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}\text{9}\text{.}\hfill \end{array}$

$\begin{array}{cc}\left(3+x\right)+17=3+\left(x+17\right)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{sum}\text{.}\end{array}$

$\begin{array}{llll}\left(2\cdot 3\right)\cdot 5\hfill & =\hfill & 2\cdot \left(3\cdot 5\right)\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\cdot 5\hfill & =\hfill & 2\cdot 15\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}30\hfill & =\hfill & 30\hfill & \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}30.\hfill \end{array}$

what are the products of Nano chemistry?
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
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
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.