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.