Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Monthly Update sign up
Mailing List signup
Search
MENU
Learning Math Home
Number and Operations Session 7, Part A: Fractions to Decimals
 
session7 Part A Part B Part C Homework
 
Glossary
number Site Map
Session 7 Materials:
Notes
Solutions
Video

Session 7, Part A:
Fractions to Decimals

In This Part: Terminating Decimals | Repeating Decimals | Repeating Decimal Rings

All the fractions we've looked at so far were terminating decimals, and their denominators were all powers of 2 and/or 5. The fractions in this section have other factors in their denominators, and as a result they will not have terminating decimal representations.

As you can see in the division problem below, the decimal expansion of 1/3 does not fit the pattern we've observed so far in this session:

Since the remainder of this division problem is never 0, this decimal does not end, and the digit 3 repeats infinitely. For decimals of this type, we can examine the period of the decimal, or the number of digits that appear before the digit string begins repeating itself. In the decimal expansion of 1/3, only the digit 3 repeats, and so the period is one.

To indicate that 3 is a repeating digit, we write a bar over it, like this:

The fraction 1/7 converts to 0.142857142857.... In this case, the repeating part is 142857, and its period is six. We write it like this:

The repetend is the digit or group of digits that repeats infinitely in a repeating decimal. For example, in the repeating decimal 0.3333..., the repetend is 3 and, as we've just seen, the period is one; in 0.142857142857..., the repetend is 142857, and the period is six.

Problem A9

  

Investigate the periods of decimal expansions by completing the table below for unit fractions with prime denominators less than 20. (If you're using a calculator, make sure that it gives you all the digits, including the ones that repeat. If your calculator won't do this, use long division.)

Fraction

Denominator

Period

Decimal Representation

1/2

2

terminating

0.5

1/3

3

1

0.333...

1/5

5

terminating

0.2

1/7

7

6

0.142857...

1/11

11

1/13

13

1/17

17

1/19

19

show answers

Fraction

Denominator

Period

Decimal Representation

1/2

2

terminating

0.5

1/3

3

1

0.333...

1/5

5

terminating

0.2

1/7

7

6

0.142857...

1/11

11

2

0.090909...

1/13

13

6

0.076923076923...

1/17

17

16

0.05882352941176470588...

1/19

19

18

0.05263157894736842105...


hide answers


Take it Further

Problem A10

Solution

Notice that the period for 1/7 is six, which is one less than the denominator. Why can't the period for this fraction be any greater than six?


Think about the remainders when you use long division to divide 1 by 7. When would a decimal terminate? When would it begin to repeat? What would happen if you saw the same remainder more than once?   Close Tip
 

 

Problem A11

Solution  

Do the decimal expansions for the denominators 17 and 19 follow the same period pattern as 7?


 

Problem A12

Solution  

Describe the behavior of the periods for the fractions 1/11 and 1/13.


Take it Further

Problem A13

Solution

Complete the table for the next six prime numbers:

Fraction

Denomin-
ator

Period

Decimal Representation

1/23

23

 0.0434782608695652173913...

1/29

29

 0.0344827586206896551724137931...

1/31

31

 0.032258064516129...

1/37

37

 

1/41

41

 

1/43

43

 0.023255813953488372093...

show answers


Fraction

Denomin-
ator

Period

Decimal Representation

1/23

23

22

 0.0434782608695652173913...

1/29

29

28

 0.0344827586206896551724137931...

1/31

31

15

 0.032258064516129...

1/37

37

3

 0.027027...

1/41

41

5

 0. 0243902439...

1/43

43

21

 0.023255813953488372093...


hide answers


 

Problem A14

Solution

Discuss the periods of the decimal representations of these prime numbers.


 

Problem A15

Solution

Predict, without computing, the period of the decimal representation of 1/47.


 


video thumbnail
 

Video Segment
In this video segment, the participants analyze the remainders of unit fractions whose denominators are prime numbers. They notice some interesting patterns in the relationship between the fractions denominators and the number of repeating digits in their decimal representation.

Do you notice any other patterns?

If you are using a VCR, you can find this segment on the session video approximately 11 minutes and 2 seconds after the Annenberg Media logo.

 

Next > Part A (Continued): Repeating Decimal Rings

Learning Math Home | Number Home | Glossary | Map | ©

Session 7: Index | Notes | Solutions | Video

Home | Catalog | About Us | Search | Contact Us | Site Map

  • Follow The Annenberg Learner on Facebook

© Annenberg Foundation 2013. All rights reserved. Privacy Policy