1.8 The real numbers  (Page 5/13)

Locate and plot the integers, $4,\mathrm{-3}.$

Locate the proper fraction $\frac{3}{4}$ first. The fraction $\frac{3}{4}$ is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot $\frac{3}{4}.$ Similarly plot $-\frac{1}{4}.$

Now locate the improper fractions $\frac{6}{5},-\frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\frac{5}{2}=\mathrm{-2}\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

Be careful when ordering negative numbers.

1. ⓐ
   $\begin{array}{cccccc}& & & & & -\frac{2}{3}\_\_\_\mathrm{-1}\hfill \\ -\frac{2}{3}\text{is to the right of}\mathrm{-1}\text{on the number line.}\hfill & & & & & -\frac{2}{3}>\mathrm{-1}\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccccc}& & & & & \mathrm{-3}\frac{1}{2}\_\_\_\mathrm{-3}\hfill \\ \mathrm{-3}\frac{1}{2}\text{is to the left of}\mathrm{-3}\text{on the number line.}\hfill & & & & & \mathrm{-3}\frac{1}{2}<\mathrm{-3}\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccccc}& & & & & -\frac{3}{4}\_\_\_-\frac{1}{4}\hfill \\ -\frac{3}{4}\text{is to the left of}-\frac{1}{4}\text{on the number line.}\hfill & & & & & -\frac{3}{4}<-\frac{1}{4}\hfill \end{array}$
   
   
4. ⓓ
   $\begin{array}{cccccc}& & & & & \mathrm{-2}\_\_\_-\frac{8}{3}\hfill \\ \mathrm{-2}\text{is to the right of}-\frac{8}{3}\text{on the number line.}\hfill & & & & & \mathrm{-2}>-\frac{8}{3}\hfill \end{array}$

A proper fraction has value less than one. The decimal number 0.4 is equivalent to $\frac{4}{10},$ a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See [link] .

The decimal $\mathrm{-0.74}$ is equivalent to $-\frac{74}{100},$ so it is located between 0 and $\mathrm{-1}.$ On a number line, mark off and label the hundredths in the interval between 0 and $\mathrm{-1}.$ See [link] .