1.8 The real numbers  (Page 5/13)

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$\begin{array}{ccccccc}\hfill \frac{7}{4}=1\frac{3}{4}\hfill & & & \hfill -\phantom{\rule{0.2em}{0ex}}\frac{9}{2}=-4\frac{1}{2}\hfill & & & \hfill \frac{8}{3}=2\frac{2}{3}\hfill \end{array}$

[link] shows the number line with all the points plotted.

Locate and label the following on a number line: $4,\frac{3}{4},-\phantom{\rule{0.2em}{0ex}}\frac{1}{4},-3,\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

Locate and plot the integers, $4,-3.$

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 $-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}.$

Now locate the improper fractions $\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2},\frac{7}{3}.$ It is easier to plot them if we convert them to mixed numbers and then plot them as described above: $\frac{6}{5}=1\frac{1}{5},-\phantom{\rule{0.2em}{0ex}}\frac{5}{2}=-2\frac{1}{2},\frac{7}{3}=2\frac{1}{3}.$

Locate and label the following on a number line: $-1,\frac{1}{3},\frac{6}{5},-\phantom{\rule{0.2em}{0ex}}\frac{7}{4},\frac{9}{2},5,-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}.$

Locate and label the following on a number line: $-2,\frac{2}{3},\frac{7}{5},-\phantom{\rule{0.2em}{0ex}}\frac{7}{4},\frac{7}{2},3,-\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

In [link] , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a<b a is less than b ” when a is to the left of b on the number line
• a>b a is greater than b ” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Order each of the following pairs of numbers, using<or>. It may be helpful to refer [link] .

$-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}___-1$ $-3\frac{1}{2}___-3$ $-\phantom{\rule{0.2em}{0ex}}\frac{3}{4}___-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}$ $-2___-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}$

Be careful when ordering negative numbers.

1. $\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}___-1\hfill \\ -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}-1\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}>-1\hfill \end{array}$

2. $\begin{array}{cccccc}& & & & & -3\frac{1}{2}___-3\hfill \\ -3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}-3\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -3\frac{1}{2}<-3\hfill \end{array}$

3. $\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}___-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\hfill \\ -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{is to the left of}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{3}{4}<-\phantom{\rule{0.2em}{0ex}}\frac{1}{4}\hfill \end{array}$

4. $\begin{array}{cccccc}& & & & & -2___-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\hfill \\ -2\phantom{\rule{0.2em}{0ex}}\text{is to the right of}\phantom{\rule{0.2em}{0ex}}-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\phantom{\rule{0.2em}{0ex}}\text{on the number line.}\hfill & & & & & -2>-\phantom{\rule{0.2em}{0ex}}\frac{8}{3}\hfill \end{array}$

Order each of the following pairs of numbers, using<or>:

$-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}___-1$ $-1\frac{1}{2}___-2$ $-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}___-\phantom{\rule{0.2em}{0ex}}\frac{1}{3}$ $-3___-\phantom{\rule{0.2em}{0ex}}\frac{7}{3}.$

> > < <

Order each of the following pairs of numbers, using<or>:

$-1___-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}$ $-2\frac{1}{4}___-2$ $-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}___-\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$ $-4___-\phantom{\rule{0.2em}{0ex}}\frac{10}{3}.$

< < > <

Locate decimals on the number line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Locate 0.4 on the number line.

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] .

Locate on the number line: 0.6.

Locate on the number line: 0.9.

Locate $-0.74$ on the number line.

The decimal $-0.74$ is equivalent to $-\phantom{\rule{0.2em}{0ex}}\frac{74}{100},$ so it is located between 0 and $-1.$ On a number line, mark off and label the hundredths in the interval between 0 and $-1.$ See [link] .

Locate on the number line: $-0.6.$

Locate on the number line: $-0.7.$

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than$0.04 (four cents). So,

$0.40>0.04$

Again, we can use the number line to order numbers.

• a<b a is less than b ” when a is to the left of b on the number line
• a>b a is greater than b ” when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line? See [link] .

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40>0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

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