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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