# 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.

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

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

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

#### Questions & Answers

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
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
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?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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. By  By By   By   By 