2.3 Properties of the real numbers

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.)

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

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

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

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

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

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

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

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

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