Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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

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...

 Problem A10 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 Do the decimal expansions for the denominators 17 and 19 follow the same period pattern as 7?

 Problem A12 Describe the behavior of the periods for the fractions 1/11 and 1/13.

Problem A13

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...

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...

 Problem A14 Discuss the periods of the decimal representations of these prime numbers.

 Problem A15 Predict, without computing, the period of the decimal representation of 1/47.

 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.

 Session 7: Index | Notes | Solutions | Video