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DECIDING IF A FRACTION IS A FINITE OR INFINITE REPEATING DECIMAL
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The concepts for this exercise are summarized below.
For a complete discussion, read the text.
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RATIONAL and IRRATIONAL NUMBERS
The rational numbers are numbers that can be written in
the form
ab
,
where a and
b are integers, and
b is nonzero.
Thus, the rational numbers are ratios of integers.
For example, 25
and -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...=0.26
37¯
.
Notice that in an infinite repeating decimal, the over-bar indicates the digits that repeat.
WHICH RATIONAL NUMBERS ARE FINITE DECIMALS,
and WHICH ARE INFINITE REPEATING DECIMALS?
To answer this question,
start by putting the fraction in simplest form,
and 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:
960
=320
=3
2⋅2⋅5
⋅5
5=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 2s and 5s .
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:
16
=12
⋅3=0
.166666...=0.16
¯
27
=0.285714
¯
311
=0.27
¯
EXAMPLES:
Determine if the given fraction is a finite decimal, or an infinite repeating decimal.
Fraction: 2/5
Answer: FINITE
Fraction: 5/7
Answer: INFINITE REPEATING
DO NOT USE YOUR CALCULATOR FOR THESE PROBLEMS.
Feel free, however, to use pencil and paper.