Repeating Decimals

So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let's see what happens when we convert the fraction \(\frac{4}{3}\) to a decimal. First, notice that \(\frac{4}{3}\) is an improper fraction. Its value is greater than \(1\). The equivalent decimal will also be greater than \(1\).

We divide \(4\) by \(3\).

No matter how many more zeros we write, there will always be a remainder of \(1\), and the threes in the quotient will go on forever. The number \(1.333..\) is called a repeating decimal. Remember that the "\(...\)" means that the pattern repeats.

How do you know how many 'repeats' to write? Instead of writing \(1.333\)... we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal \(1.333 \ldots\) is written \(1. \overline{3}\). The line above the \(3\) tells you that the \(3\) repeats endlessly. So \(1.333..=1. \overline{3}\)

For other decimals, two or more digits might repeat. Table 5.5 shows some more examples of repeating decimals.

Table 5.5

It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.

Source: Rice University, https://openstax.org/books/prealgebra/pages/5-3-decimals-and-fractions
This work is licensed under a Creative Commons Attribution 4.0 License.

EXAMPLE 5.30

Write \(\frac{43}{22}\) as a decimal.

EXAMPLE 5.31

Simplify: \(\frac{7}{8}+6.4\).

TRY IT 5.59

Write as a decimal: \(\frac{27}{11}\).

TRY IT 5.60

Write as a decimal: \(\frac{51}{22}\).

EXAMPLE 5.30

Divide \(43\) by \(22\).

Notice that the differences of \(120\) and \(100\) repeat, so there is a repeat in the digits of the quotient; \(54\) will repeat endlessly. The first decimal place in the quotient, \(9\), is not part of the pattern. So,

\[\frac{43}{22}=1.9 \overline{54}\]

EXAMPLE 5.31

		\(\frac{7}{8}+6.4\)
Change \(\frac{7}{8}\) to a decimal.		\(0.875+6.4\)
Add.		\(7.275\)

TRY IT 5.59

\(2 . \overline{45}\)

TRY IT 5.60

\(2.3 \overline{18}\)