How Do You Know if a Fraction Is Terminating or Repeating

Identify Value and Decimals

Terminating or Repeating?

You lot've seen that when y'all write a fraction every bit a decimal, sometimes the decimal terminates, like:

$\frac 1 2 = 0.5 \quad \text{ and } \quad \frac {33} {1000} = 0.033.$

All the same, some fractions have decimal representations that become on forever in a repeating design, like:

$\frac 1 3 = 0.33333\dots \quad \text{ and } \quad \frac {6} {7} = 0.857142857142857142857142\dots.$

Information technology's non totally obvious, but it is truthful: Those are the but two things that can happen when you lot write a fraction as a decimal.

Of course, you can imagine (just never write down) a decimal that goes on forever but doesn't repeat itself, for example:

$0.101001000100001000001\dots \quad \text{ and } \quad \pi = 3.14159265358979\dots.$

Simply these numbers can never be written as a dainty fraction $\frac a b$ where $a$ and $b$ are whole numbers. They are chosen irrational numbers. The reason for this proper name: Fractions like $\frac a b$ are likewise called ratios. Irrational numbers cannot be expressed as a ratio of 2 whole numbers.

For now, we'll think near the question: Which fractions have decimal representations that finish, and which fractions take decimal representations that repeat forever? We'll focus just on unit fractions.

Definition

A unit fraction is a fraction that has 1 in the numerator. It looks like $\frac 1 n$ for some whole number $n$ .

Retrieve / Pair / Share

Which of the following fractions accept infinitely long decimal representations and which do not?
$\frac 12 \qquad \frac 13 \qquad \frac 14 \qquad \frac15 \qquad \frac16 \qquad \frac 17 \qquad \frac 18 \qquad \frac 19 \qquad \frac 1{10}.$

Try some more examples on your own. Do you have a theorize?

A fraction $\frac 1 b$ has an infinitely long decimal expansion if:

________________________________.

Marcus noticed a pattern in the tabular array from Problem vii, but was having trouble explaining exactly what he noticed. Here's what he said to his group:

I remembered that when nosotros wrote fractions as decimals before, we tried to brand the denominator into a power of 10. Then nosotros can do this:

$\frac12 = \frac12 \cdot \frac55 = \frac{5}{10} = 0.5.$

$\frac14 = \frac14 \cdot \frac{25}{25} = \frac{25}{100} = 0.25.$

$\frac18 = \frac18 \cdot \frac{125}{125} = \frac{125}{1000} = 0.125.$

When we merely accept 2'southward, we tin can always turn them into 10's by adding plenty 5'south.

Think / Pair / Share

Marcus had a really good insight, but he didn't explain it very well. He doesn't really mean that nosotros "turn ii'due south into 10's." And he's not doing any addition, so talking about "adding plenty 5's" is pretty disruptive.

Problem 9

Complete the statement below past filling in the numerator of the fraction.

The unit fraction $\frac 1{2^n}$ has a decimal representation that terminates. The representation volition take $n$ decimal digits, and volition be equivalent to the fraction

$\frac{?}{10^n}.$
Write a improve version of Marcus's explanation to justify why this fact is true.

Problem x

Write a statement about the decimal representations of unit fractions $\frac 1{5^n}$ and justify that your statement is right. (Use the statement in Trouble 9 as a model.)

Trouble 11

Each of the fractions listed beneath has a terminating decimal representation. Explain how you could know this for sure, without actually computing the decimal representation.

$\frac 1{10} \qquad\quad \frac 1{20} \qquad\quad \frac 1{50} \qquad\quad \frac 1{200} \qquad\quad \frac 1{500} \qquad\quad \frac 1{4000}.$

The Period of a Repeating Decimal

If the denominator of a fraction tin can be factored into just 2'southward and 5'south, yous tin always form an equivalent fraction where the denominator is a power of 10.

For example, if we commencement with the fraction

$\frac 1 {2^a 5^b},$

we can course an equivalent fraction

$\frac 1 {2^a 5^b} \ = \ \frac 1 {2^a 5^b} \cdot \frac {2^b 5^a} {2^b 5^a} \ = \ \frac {2^b 5^a} {2^{a+b} 5^{a+b}} \ = \ \frac {2^b 5^a} {10^{a+b}} .$

The denominator of this fraction is a power of x, then the decimal expansion is finite with (at most) $a+b$ places.

What almost fractions where the denominator has other prime factors also 2's and v'southward? Certainly we can't turn the denominator into a power of 10, considering powers of 10 have only 2'due south and 5'southward every bit their prime factors. So in this case the decimal expansion volition go on forever. Merely why will it have a repeating pattern? And is at that place anything else interesting nosotros can say in this case?

Definition

The period of a repeating decimal is the smallest number of digits that repeat.

For example, we saw that

$\frac 13 \ = \ 0.33333\dots = 0.\overline{3}.$

The repeating part is just the single digit iii, and then the period of this repeating decimal is one.

Similarly, we know that

$\frac 67 \ =\ 0.857142857142857142857142\dots \ = \ 0.\overline{857142}.\phantom{857142857142857142}.$

The smallest repeating office is the digits $857142$ , so the period of this repeating decimal is 6.

You can call up of it this way: the period is the length of the string of digits nether the vinculum (the horizontal bar that indicates the repeating digits).

Imagine you are doing the "Dots & Boxes" partition to compute the decimal representation of a unit fraction like $\frac 1 6$ . You start with a single dot in the ones box:

To detect the decimal expansion, you "unexplode" dots, class groups of six, meet how many dots are left, and repeat.

Draw your own pictures to follow along this explanation:

Picture 1: When you unexplode the get-go dot, you become 10 dots in the $\frac 1{10}$ box, which gives one group of vi with remainder of four.

Picture show 2: When you unexplode those four dots, yous get 40 dots in the $\frac 1{100}$ box, which gives six group of six with remainder of 4.

Picture three: Unexplode those 4 dots to go 40 in the adjacent box to the correct.

Picture 4: Make 6 groups of vi dots with balance four.

Since the remainder repeated (we got a rest of 4 over again), we can come across that the process will now repeat forever:

unexplode 4 dots to get 40 in the next box to the right,
make half dozen groups of 6 dots with residual 4,
unexplode 4 dots to go xl in the adjacent box to the right,
make six groups of six dots with remainder four,
and so on forever…

On Your Ain

Work on the post-obit exercises on your own or with a partner.

Use "Dots & Boxes" division to compute the decimal representation of $\frac 1{11}$ . Explain how you know for sure the process volition repeat forever.
Use "Dots & Boxes" division to compute the decimal representation of $\frac 1{12}$ . Explain how you know for sure the process will echo forever.
What are the possible remainders you can get when yous utilise division to compute the fraction $\frac 1 7$ ? How can you lot be sure the procedure will somewhen repeat?
What are the possible remainders you tin get when you use division to compute the fraction $\frac 1 9$ ? How can you exist sure the procedure will somewhen repeat?

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Source: http://pressbooks-dev.oer.hawaii.edu/math111/chapter/terminating-or-repeating/

How Do You Know if a Fraction Is Terminating or Repeating

Terminating or Repeating?

Definition

Retrieve / Pair / Share

Think / Pair / Share

Problem 9

Problem x

Trouble 11

The Period of a Repeating Decimal

Definition

On Your Ain

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