# 2.3 Properties of the real numbers  (Page 2/2)

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$\begin{array}{cc}\left(9y\right)4=9\left(y4\right)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$

## Practice set b

Fill in the $\left(\begin{array}{cc}& \end{array}\right)$ to make each statement true. Use the associative properties.

$\left(9+2\right)+5=9+\left(\begin{array}{cc}& \end{array}\right)$

$2+5$

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

$x+5$

$\left(11a\right)6=11\left(\begin{array}{cc}& \end{array}\right)$

$a\cdot 6$

$\left[\left(7m-2\right)\left(m+3\right)\right]\left(m+4\right)=\left(7m-2\right)\left[\left(\begin{array}{cc}& \end{array}\right)\left(\begin{array}{cc}& \end{array}\right)\right]$

$\left(m+3\right)\left(m+4\right)$

## Sample set c

Simplify (rearrange into a simpler form): $5x6b8ac4$ .

According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.

$\begin{array}{ll}5\cdot 6\cdot 8\cdot 4\cdot x\cdot b\cdot a\cdot c\hfill & \hfill \\ 960xbac\hfill & \text{Multiply}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numbers}\text{.}\hfill \\ 960abcx\hfill & \text{By}\text{\hspace{0.17em}}\text{convention,}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{will,}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{possible,}\text{\hspace{0.17em}}\text{write}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}\text{letters}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{alphabetical}\text{\hspace{0.17em}}\text{order}\text{.}\hfill \end{array}$

## Practice set c

Simplify each of the following quantities.

$3a7y9d$

$189ady$

$6b8acz4\cdot 5$

$960abcz$

$4p6qr3\left(a+b\right)$

$72pqr\left(a+b\right)$

## The distributive properties

When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.

$4+4+4=3\cdot 4$

Notice that there are three 4’s, that is, 4 appears 3 times . Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.

When a number $a$ is added repeatedly $n$ times, we have
$\underset{a\text{\hspace{0.17em}}\text{appears}\text{\hspace{0.17em}}n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}$
Then, using multiplication as a description for repeated addition, we can replace
$\begin{array}{ccc}\underset{n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}& \text{with}& na\end{array}$

For example:

$x+x+x+x$ can be written as $4x$ since $x$ is repeatedly added 4 times.

$x+x+x+x=4x$

$r+r$ can be written as $2r$ since $r$ is repeatedly added 2 times.

$r+r=2r$

The distributive property involves both multiplication and addition. Let’s rewrite $4\left(a+b\right).$ We proceed by reading $4\left(a+b\right)$ as a multiplication: 4 times the quantity $\left(a+b\right)$ . This directs us to write

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & \left(a+b\right)+\left(a+b\right)+\left(a+b\right)+\left(a+b\right)\hfill \\ \hfill & =\hfill & a+b+a+b+a+b+a+b\hfill \end{array}$

Now we use the commutative property of addition to collect all the $a\text{'}s$ together and all the $b\text{'}s$ together.

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & \underset{4a\text{'}s}{\underbrace{a+a+a+a}}+\underset{4b\text{'}s}{\underbrace{b+b+b+b}}\hfill \end{array}$

Now, using multiplication as a description for repeated addition, we have

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & 4a+4b\hfill \end{array}$

We have distributed the 4 over the sum to both $a$ and $b$ .

## The distributive property

$\begin{array}{cc}a\left(b+c\right)=a\cdot b+a\cdot c& \left(b+c\right)\end{array}a=a\cdot b+a\cdot c$

The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.

## Sample set d

Use the distributive property to rewrite each of the following quantities.

## Practice set d

What property of real numbers justifies
$a\left(b+c\right)=\left(b+c\right)a?$

the commutative property of multiplication

Use the distributive property to rewrite each of the following quantities.

$3\left(2+1\right)$

$6+3$

$\left(x+6\right)7$

$7x+42$

$4\left(a+y\right)$

$4a+4y$

$\left(9+2\right)a$

$9a+2a$

$a\left(x+5\right)$

$ax+5a$

$1\left(x+y\right)$

$x+y$

## The identity properties

The number 0 is called the additive identity since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, $6+0=6$ .

## Multiplicative identity

The number 1 is called the multiplicative identity since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example $6\cdot 1=6$ .

We summarize the identity properties as follows.

$\begin{array}{cc}\begin{array}{l}\text{ADDITIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \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{PROPERTY}\end{array}& \begin{array}{l}\text{MULTIPLICATIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \text{​}\text{​}\text{​}\text{​}\text{​}\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}}\text{\hspace{0.17em}}\text{PROPERTY}\end{array}\\ \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,\hspace{0.17em}then}& \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,}\text{\hspace{0.17em}}\text{then}\\ a+0=a\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{and}\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}}0+a=a& a\cdot 1=a\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{and}\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}}1\cdot a=a\end{array}$

## The inverse properties

When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to $-3$ the result is 0, that is, $3+\left(-3\right)=0$ . The numbers 3 and $-3$ are additive inverses of each other.

## Multiplicative inverses

When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when 6 and $\frac{1}{6}$ are multiplied together, the result is 1, that is, $6\cdot \frac{1}{6}=1$ . The numbers 6 and $\frac{1}{6}$ are multiplicative inverses of each other.

We summarize the inverse properties as follows.

## The inverse properties

1. If $a$ is any real number, then there is a unique real number $-a$ , such that
$\begin{array}{ccc}a+\left(-a\right)=0& \text{and}& -a+a=0\end{array}$
The numbers $a$ and $-a$ are called additive inverses of each other.
2. If $a$ is any nonzero real number, then there is a unique real number $\frac{1}{a}$ such that
$\begin{array}{ccc}a\cdot \frac{1}{a}=1& \text{and}& \frac{1}{a}\end{array}\cdot a=1$
The numbers $a$ and $\frac{1}{a}$ are called multiplicative inverses of each other.

## Expanding quantities

When we perform operations such as $6\left(a+3\right)=6a+18$ , we say we are expanding the quantity $6\left(a+3\right)$ .

## Exercises

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.

$x+3$

$3+x$

$5+y$

$10x$

$10x$

$18z$

$r6$

$6r$

$ax$

$xc$

$cx$

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

$6\left(s+1\right)$

$\left(s+1\right)6$

$\left(8+a\right)\left(x+6\right)$

$\left(x+16\right)\left(a+7\right)$

$\left(a+7\right)\left(x+16\right)$

$\left(x+y\right)\left(x-y\right)$

$0.06m$

$m\left(0.06\right)$

$5\left(6h+1\right)$

$\left(6h+1\right)5$

$m\left(a+2b\right)$

$k\left(10a-b\right)$

$\left(10a-b\right)k$

$\left(21c\right)\left(0.008\right)$

$\left(-16\right)\left(4\right)$

$\left(4\right)\left(-16\right)$

$\left(5\right)\left(b-6\right)$

$\square \text{\hspace{0.17em}}\cdot ○$

$○\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\square$

Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.

$9x2y$

$18xy$

$5a6b$

$2a3b4c$

$24abc$

$5x10y5z$

$1u3r2z5m1n$

$30mnruz$

$6d4e1f2\left(g+2h\right)$

$\left(\frac{1}{2}\right)d\left(\frac{1}{4}\right)e\left(\frac{1}{2}\right)a$

$\frac{1}{16}ade$

$3\left(a+6\right)2\left(a-9\right)6b$

$1\left(x+2y\right)\left(6+z\right)9\left(3x+5y\right)$

$9\left(x+2y\right)\left(6+z\right)\left(3x+5y\right)$

For the following problems, use the distributive property to expand the quantities.

$2\left(y+9\right)$

$b\left(r+5\right)$

$br+5b$

$m\left(u+a\right)$

$k\left(j+1\right)$

$jk+k$

$x\left(2y+5\right)$

$z\left(x+9w\right)$

$xz+9wz$

$\left(1+\text{\hspace{0.17em}}d\right)e$

$\left(8+\text{\hspace{0.17em}}2f\right)g$

$8g+2fg$

$c\left(2a+\text{\hspace{0.17em}}10b\right)$

$15x\left(2y+\text{\hspace{0.17em}}3z\right)$

$30xy+45xz$

$8y\left(12a+b\right)$

$z\left(x+y+m\right)$

$xz+yz+mz$

$\left(a+6\right)\left(x+y\right)$

$\left(x+10\right)\left(a+b+c\right)$

$ax+bx+cx+10a+10b+10c$

$1\left(x+y\right)$

$1\left(a+16\right)$

$a+16$

$0.48\left(0.34a+0.61\right)$

$21.5\left(16.2a+3.8b+0.7c\right)$

$348.3a+81.7b+15.05c$

$2{z}_{t}\left({L}_{m}+8k\right)$

$2{L}_{m}{z}_{t}+16k{z}_{t}$

## Exercises for review

( [link] ) Find the value of $4\cdot 2+5\left(2\cdot 4-6÷3\right)-2\cdot 5$ .

( [link] ) Is the statement $3\left(5\cdot 3-3\cdot 5\right)+6\cdot 2-3\cdot 4<0$ true or false?

false

( [link] ) Draw a number line that extends from $-2$ to 2 and place points at all integers between and including $-2$ and 3.

( [link] ) Replace the $\ast$ with the appropriate relation symbol $\left(<,>\right).-7\ast -3$ .

$<$

( [link] ) What whole numbers can replace $x$ so that the statement $-2\le x<2$ is true?

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