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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.)
Let $a$ and $b$ represent real numbers.
The commutative properties tell us that two numbers can be added or multiplied in any order without affecting the result.
The following are examples of the commutative properties.
$\begin{array}{cc}3+4=4+3& \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}7.\end{array}$
$\begin{array}{cc}5+x=x+5& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{sum}\text{.}\end{array}$
$\begin{array}{cc}4\cdot 8=8\cdot 4& \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}\text{32}\text{.}\end{array}$
$\begin{array}{cc}y7=7y& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$
$\begin{array}{cc}5(a+1)=(a+1)5& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$
$\begin{array}{cc}(x+4)(y+2)=(y+2)(x+4)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$
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.
$9\cdot 7=\left(\begin{array}{cc}& \end{array}\right)\cdot 9$
$7$
$4(k-5)=\left(\begin{array}{cc}& \end{array}\right)4$
$(k-5)$
$(9a-1)(\begin{array}{cc}& \end{array})=\left(2b+7\right)(9a-1)$
$(2b+7)$
Let $a,b,$ and $c$ represent real numbers.
The associative properties tell us that we may group together the quantities as we please without affecting the result.
The following examples show how the associative properties can be used.
$$\begin{array}{llll}(2+6)+1\hfill & =\hfill & 2+(6+1)\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}8+1\hfill & =\hfill & 2+7\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9\hfill & =\hfill & 9\hfill & \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}\text{9}\text{.}\hfill \end{array}$$
$\begin{array}{cc}(3+x)+17=3+(x+17)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{sum}\text{.}\end{array}$
$$\begin{array}{llll}(2\cdot 3)\cdot 5\hfill & =\hfill & 2\cdot (3\cdot 5)\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}6\cdot 5\hfill & =\hfill & 2\cdot 15\hfill & \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}30\hfill & =\hfill & 30\hfill & \text{Both}\text{\hspace{0.17em}}\text{equal}\text{\hspace{0.17em}}30.\hfill \end{array}$$
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