We have discussed that terminating and repeating decimals represent rational numbers.

Let's look at how we can plot terminating decimals.

Solution

\( 1.7 = \text{one and seven tenths} \)

How do we plot a repeating decimal?

\( 0.\overline{7} = 0.777\ldots \)

Since a repeating decimal never ends, we are not able to plot them exactly on the number line using place values like we did for terminating decimals.

For now, we will compare repeating decimals to terminating decimals that we can plot on the number line.

This comparison will allow us to plot the approximate location of any repeating decimal.

Solution

Take a moment to consider the following questions before we continue.

1. How could we use the hundredths digit to be more accurate?
2. Should we continue in the same pattern and use the thousandths digit to be even more accurate?

We need to acknowledge that no matter how many digits we choose to use when plotting, our plot of \( 0.\overline{7} \) on the number line is always going to be an approximation.

The decimal representation cannot be plotted accurately on the number line.

The fractional form can easily be used to plot  the exact location of the number.