# 2.3 Properties of the real numbers

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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

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.

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

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

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

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