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Mathematics LibreTexts

6.6: Terminating or Repeating?

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You’ve seen that when you write a fraction as a decimal, sometimes the decimal terminates, like:

12=0.5and33100=0.033.

However, some fractions have decimal representations that go on forever in a repeating pattern, like:

13=0.33333and67=0.857142857142857142857142

It’s not totally obvious, but it is true: Those are the only two things that can happen when you write a fraction as a decimal.

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

0.1010010001000010000001andπ=3.14159265358979

But these numbers can never be written as a nice fraction ab where a and bare whole numbers. They are called irrational numbers. The reason for this name: Fractions like ab are also called ratios. Irrational numbers cannot be expressed as a ratio of two whole numbers.

For now, we’ll think about the question: Which fractions have decimal representations that terminate, and which fractions have 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 1n for some whole number n.

Think / Pair / Share
  • Which of the following fractions have infinitely long decimal representations and which do not? 1213141516171819110.
  • Try some more examples on your own. Do you have a conjecture?

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

________________________________.

Problem 7

Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 2. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)

Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction 12n?

Fraction Denominator Decimal
12 2 0.5
14 22 0.25
18 23 0.125
116    
132    
164    
1128    
1256  
Problem 8

Complete the table below which shows the decimal expansion of unit fractions where the denominator is a power of 5. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)

Try even more examples until you can make a conjecture: What is the decimal representation of the unit fraction 15n?

Fraction Denominator Decimal
15 5 0.2
125 52 0.04
1125 53  
1625    
13125    
115625  

Marcus noticed a pattern in the table from Problem 7, but was having trouble explaining exactly what he noticed. Here’s what he said to his group:

I remembered that when we wrote fractions as decimals before, we tried to make the denominator into a power of ten. So we can do this: 12=1255=510=0.5.14=142525=25100=0.25.18=18125125=1251000=0.125.

When we only have 2’s, we can always turn them into 10’s by adding enough 5’s.

Think / Pair / Share
  • Write out several more examples of what Marcus discovered.
  • If Marcus had the unit fraction 12n, what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?
  • Now think about unit fractions with powers of 5 in the denominator. If Marcus had the unit fraction 15n, what would be his first step to turn it into a decimal? What would the decimal expansion look like and why?

Marcus had a really good insight, but he didn’t explain it very well. He doesn’t really mean that we “turn 2’s into 10’s.” And he’s not doing any addition, so talking about “adding enough 5’s” is pretty confusing.

Problem 9
  1. Complete the statement below by filling in the numerator of the fraction.

    The unit fraction 12n has a decimal representation that terminates. The representation will have n decimal digits, and will be equivalent to the fraction ?10n.

  2. Write a better version of Marcus’s explanation to justify why this fact is true.
Problem 10

Write a statement about the decimal representations of unit fractions ?5n and justify that your statement is correct. (Use the statement in Problem 9 as a model.)

Problem 11

Each of the fractions listed below has a terminating decimal representation. Explain how you could know this for sure, without actually calculating the decimal representation. 1101201501200150014000.

The Period of a Repeating Decimal

If the denominator of a fraction can be factored into just 2’s and 5’s, you can always form an equivalent fraction where the denominator is a power of ten.

For example, if we start with the fraction

12a5b,

we can form an equivalent fraction

12a5b=12a5b2b5a2b5a=2b5a2a+b5a+b=2b5a10a+b.

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

What about fractions where the denominator has other prime factors besides 2’s and 5’s? Certainly we can’t turn the denominator into a power of 10, because powers of 10 have just 2’s and 5’s as their prime factors. So in this case the decimal expansion will go on forever. But why will it have a repeating pattern? And is there anything else interesting we 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

13=0.33333=0.ˉ3.

The repeating part is just the single digit 3, so the period of this repeating decimal is one.

Similarly, we know that

67=0.857142857142857142857142=0.¯857142.

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

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

Problem 12

Complete the table below which shows the decimal expansion of unit fractions where the denominator has prime factors besides 2 and 5. (You may want to use a calculator to compute the decimal representations. The point is to look for and then explain a pattern, rather than to compute by hand.)

Try even more examples until you can make a conjecture: What can you say about the period of the fraction 1n when n has prime factors besides 2 and 5?

Fraction Denominator Decimal
13 0.1ˉ6 1
16 0.¯142857 6
17    
19    
111    
112    
113    
114    

Imagine you are doing the “Dots & Boxes” division to compute the decimal representation of a unit fraction like 16. You start with a single dot in the ones box:

unitfracs-768x143.png

To find the decimal expansion, you “unexplode” dots, form groups of six, see how many dots are left, and repeat.

Draw your own pictures to follow along this explanation:

Picture 1: When you unexplode the first dot, you get 10 dots in the 110 box, which gives one group of six with remainder of 4.

Picture 2: When you unexplode those four dots, you get 40 dots in the 1100 box, which gives six group of six with remainder of 4.

Picture 3: Unexplode those 4 dots to get 40 in the next box to the right.

Picture 4: Make six groups of 6 dots with remainder 4.

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

  • unexplode 4 dots to get 40 in the next box to the right,
  • make six groups of 6 dots with remainder 4,
  • unexplode 4 dots to get 40 in the next box to the right,
  • make six groups of 6 dots with remainder 4,
  • and so on forever…

On Your Own

Work on the following exercises on your own or with a partner.

  1. Use “Dots & Boxes” division to compute the decimal representation of 111. Explain how you know for sure the process will repeat forever.
  2. Use “Dots & Boxes” division to compute the decimal representation of 112. Explain how you know for sure the process will repeat forever.
  3. What are the possible remainders you can get when you use division to compute the fraction 17? How can you be sure the process will eventually repeat?
  4. What are the possible remainders you can get when you use division to compute the fraction 19? How can you be sure the process will eventually repeat?
Problem 13

Suppose that n is a whole number, and it has some prime factors besides 2’s and 5’s. Write a convincing argument that:

  1. The decimal representation of 1n will go on forever (it will not terminate).
  2. The decimal representation of 1n will be an infinite repeating decimal.
  3. The period of the decimal representation of 1n will be less than n.
Problem 14
  1. Find the “decimal” expansion for 12 in the following bases. Be sure to show your work: two,three,four,five,six,seven.
  2. Make a conjecture: If I write the decimal expansion of 12 in base b, when will that expansion be finite and when will it be an infinite repeating decimal expansion?
  3. Can you prove your conjecture is true?

This page titled 6.6: Terminating or Repeating? is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michelle Manes via source content that was edited to the style and standards of the LibreTexts platform.

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