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.

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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