# Deciding if a Fraction is a Finite or Infinite Repeating Decimal

## Rational and Irrational Numbers

The *rational numbers* are numbers that can be written in
the form
$\,\frac{a}{b}\,$,
where $\,a\,$ and $\,b\,$
are integers, and
$\,b\,$ is nonzero.

Recall that the *integers* are:
$\,\ldots , -3, -2, -1, 0, 1, 2, 3,\, \ldots\,$
That is, the integers are the whole numbers, together with their opposites.

Thus, the rational numbers are ratios of integers.

For example, $\,\frac25\,$ and $\,\frac{-7}{4}\,$ are rational numbers.

Every real number is either rational,
or it isn't.
If it isn't rational,
then it is said to be *irrational*.

## Finite and Infinite Repeating Decimals

By doing a long division, every rational number can be written as a finite decimal or an infinite repeating decimal.

A *finite decimal* is one that stops, like
$\,0.157\,$.
An *infinite repeating decimal*
is one that has a specified sequence of digits that repeat,
like:
$$0.263737373737\ldots = 0.26\overline{37}$$
Notice that in an infinite repeating decimal,
the over-bar indicates the digits that repeat.

## Pronunciation of ‘Finite’ and ‘Infinite’

*Finite* is pronounced **FIGH-night** (FIGH rhymes with ‘eye’; long i).
However, *infinite* is pronounced **IN-fi-nit** (both short i's).

## Which Rational Numbers are Finite Decimals, and Which are Infinite Repeating Decimals?

To answer this question:

- Start by putting the fraction in simplest form.
- Then, factor the denominator into primes.
- If there are only prime factors of $\,2\,$ and $\,5\,$ in the denominator, then the fraction has a finite decimal name.

The following example illustrates the idea:

$$ \cssId{s32}{\frac{9}{60}} \cssId{s33}{\ = \ \frac{3}{20}} \cssId{s34}{\ = \ \frac{3}{2\cdot2\cdot 5}\cdot\frac{5}{5}} \cssId{s35}{\ = \ \frac{15}{100}} \cssId{s36}{\ = \ 0.15} $$If there are only factors of $\,2\,$ and $\,5\,$ in the denominator, then additional factors can be introduced, as needed, so that there are equal numbers of twos and fives. Then, the denominator is a power of $\,10\,$, which is easy to write in decimal form.

When the fraction is in simplest form, then any prime factors other than $\,2\,$ or $\,5\,$ in the denominator will give an infinite repeating decimal. For example:

$$ \begin{gather} \cssId{s44}{\frac{1}{6}} \cssId{s45}{= \frac{1}{2\cdot 3}} \cssId{s46}{= 0.166666\ldots} \cssId{s47}{= 0.1\overline{6}}\cr \cssId{s48}{\text{(bar over just the $6$)}}\cr\cr \end{gather} $$ $$ \begin{gather} \cssId{s49}{\frac{2}{7} = 0.\overline{285714}}\cr \cssId{s50}{\text{(bar over the digits $285714$)}}\cr\cr \end{gather} $$ $$ \begin{gather} \cssId{s51}{\frac{3}{11} = 0.\overline{27}}\cr \cssId{s52}{\text{(bar over the digits $27$)}} \end{gather} $$## Examples

Consider the given fraction. In decimal form, determine if the given fraction is a finite decimal, or an infinite repeating decimal.

## Practice

Do not use your calculator for these problems. Feel free, however, to use pencil and paper.