The rational numbers are numbers that can be written in
the form
$\displaystyle\,\frac{a}{b}\,$,
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.
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.
Finite is pronounced FIGH-night (FIGH rhymes with ‘eye’; long i).
However, infinite is pronounced IN-fi-nit (both short i's).
To answer this question:
The following example illustrates the idea:
$\displaystyle
\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:
$\displaystyle
\cssId{s44}{\frac{1}{6}}
\cssId{s45}{= \frac{1}{2\cdot 3}}
\cssId{s46}{= 0.166666\ldots}
\cssId{s47}{= 0.1\overline{6}}
$
(bar over just the $6$)
$\displaystyle\frac{2}{7}
= 0.\overline{285714}
$
(bar over the digits $285714$)
$\displaystyle\frac{3}{11}
= 0.\overline{27}
$
(bar over the digits $27$)
Consider the given fraction.
In decimal form, determine if the given fraction is a finite decimal, or an infinite repeating decimal.
DO NOT USE YOUR CALCULATOR FOR THESE PROBLEMS.
Feel free, however, to use pencil and paper.