DECIDING IF A FRACTION IS A FINITE OR INFINITE REPEATING DECIMAL

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
 
RATIONAL and IRRATIONAL NUMBERS

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.

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:

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 $27$)

EXAMPLES:

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

Fraction: $\displaystyle\frac25$
Answer: FINITE DECIMAL
Fraction: $\displaystyle\frac57$
Answer: INFINITE REPEATING DECIMAL
Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Deciding if Numbers are Equal or Approximately Equal

 
 

DO NOT USE YOUR CALCULATOR FOR THESE PROBLEMS.
Feel free, however, to use pencil and paper.

Consider this fraction:
In decimal form, this number is a:


    
(an even number, please)