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 \(3\).

4/3

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.

REPEATING DECIMAL

A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.

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.

\(1.333 \ldots=1 . \overline{3}\)	\(3\) is the repeating digit
\(4.1666 \ldots=4.1 \overline{6}\)	\(6\) is the repeating digit
\(4.161616 \ldots=4 . \overline{16}\)	\(16\) is the repeating block
\(0.271271271 \ldots=0 . \overline{271}\)	\(271\) is the repeating block

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
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