﻿ Deciding if a Fraction is a Finite or Infinite Repeating Decimal
DECIDING IF A FRACTION IS A FINITE OR INFINITE REPEATING DECIMAL
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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?

• 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:

$\displaystyle\frac{9}{60} \ = \ \frac{3}{20} \ = \ \frac{3}{2\cdot2\cdot 5}\cdot\frac{5}{5} \ = \ \frac{15}{100} \ = \ 0.15$

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:

$\displaystyle\frac{1}{6} = \frac{1}{2\cdot 3} = 0.166666\ldots = 0.1\overline{6}$     (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: $\displaystyle\frac25$
Fraction: $\displaystyle\frac57$
Master the ideas from this section

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: