# Fractions and Decimals Part A: Fractions to Decimals (65 minutes)

In This Part: Terminating Decimals
In Part A of this session, you’ll examine the process of converting fractions to decimals, which will help you better understand the relationship between the two. You will be able to predict the number of decimal places in terminating decimals and the number of repeating digits in non-terminating decimals. You will also begin to understand which types of fractions terminate and which repeat, and why all rational numbers must fit into one of these categories. See Note 1 below.

Problem A1
A unit fraction is a fraction that has 1 as its numerator. The table below lists the decimal representations for the unit fractions 1/2, 1/4, and 1/8:

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal
Representa-
tion

 1/2 2 21 1 0.5 1/4 4 22 2 0.25 1/8 8 23 3 0.125

Make a conjecture about the number of places in the decimal representation for 1/16.

Notice that the denominator 4 can be written as 22, and the decimal representation has two decimal places; 8 can be written as 23, and the decimal representation has three decimal places.

Problem A2
How do these decimal representations relate to the powers of five? If you know that 54is 625, does that help you find the decimal representation for 1/16 (i.e., 1/24)?

Remember that the decimal system is based on powers of 10. Why would this be important in finding decimal representations for fractional powers of two?

Problem A3
Complete the table for unit fractions with denominators that are powers of two. (Use a calculator, if you like, for the larger denominators.) Click on the Show Answers button and a completed version of the table will appear below. You can record exponential values in the form of 2^n.

See if you can use the powers-of-five trick you just learned.

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/2 2 21 1 0.5 1/4 4 22 2 0.25 1/8 8 23 3 0.125 1/16 16 1/32 32 1/64 64 1/1,024 1,024 1/2n 2n 2n

Problem A4
Explain how you arrived at the decimal expression for 1/2n.

Think about the relationship of this type of fraction to the powers of five.

Video Segment
Why are powers of 10 important when converting fractions to decimals? Watch this segment to see how Professor Findell and the participants reasoned about this question.

You can find this segment on the session video approximately 2 minutes and 30 seconds after the Annenberg Media logo.

Problem A5
Complete the table below to see whether unit fractions with denominators that are powers of 5 show a similar pattern to those that are powers of 2. Click on the Show Answers button and a completed version of the table will appear below. You can record exponential values in the form of 5^n.

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/5 5 51 1 0.2 1/25 25 52 2 0.04 1/125 125 53 3 0.008 1/625 625 1/3,125 3,125 1/15,625 15,625 1/5n 5n 5n

Problem A6
Now complete the table below to see what happens when we combine powers of 2 and 5. You can record exponential values in the form of 2^n or 5^n.

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/10 10 21 • 51 1/20 20 22 • 51 1/50 50 21 • 52 1/200 200 1/500 500 1/4,000 4,000 1/(2n • 5m) 2n • 5m 2n • 5m

Problem A7
a. Do the decimal conversions of fractions with denominators whose factors are only 2s and/or 5s always terminate?
b. Explain why or why not.

Problem A8

##### Fractions to Decimals is adapted from Findell, Carol, and Masunaga, David. No More Fractions — PERIOD! Student Math Notes, Volume 3. pp.119-121. © 2000 by the National Council of Teachers of Mathematics. Used with permission. All rights reserved.

In This Part: Repeating Decimals

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

TAKE IT FURTHER
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?

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.

TAKE IT FURTHER
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…

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.

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?

You can find this segment on the session video approximately 11 minutes and 2 seconds after the Annenberg Media logo.

In This Part: Repeating Decimal Rings
Here’s another interesting phenomenon of repeating decimals.

We’ve explored repeating patterns for decimal expansions of such fractions as 1/7 (or other fractions with prime denominators larger than 7). What happens when the numerator is larger than 1? If you know the decimal representation of 1/7, is there an easy way to find the decimal representation of, say, 2/7?

One way would be to multiply the digits of the repeating part by 2. When we display the repeating parts in one or two rings, some interesting patterns emerge.

Use the following Interactive Activity to explore the repeating parts of decimal expansions for such fractions as 1/7, 2/7, …, and 6/7 and 1/13, 2/13, 3/13, …, and 12/13:

For a non-interactive version of this activity, follow the instructions and complete Problem A16.

Problem A16
a.
Arrange the digits for one period of the repeating decimal expansion for 1/7 in a circle. Now find the decimal expansions for 2/7, 3/7, …, and 6/7. How are 1/7 and 6/7 related? How are 2/7 and 5/7 related? How about 3/7 and 4/7?
b.
You might try thinking that 7 has one ring and that the size of the ring is six. Try the same idea with the number 13. How would you describe 13?
c.
Explore this idea with other prime numbers.

### Notes

Note 1
Most people don’t think of decimals as fractions. Decimals are fractions, but we don’t write the denominators of these fractions since they are powers of 10. Decimal numbers greater than 1 should really be called decimal fractions, because the word “decimal” actually refers only to the part to the right of the decimal point.

### Solutions

Problem A1
It appears that the number of decimal places equals the power of 2. Therefore, 1/16 should have four decimal places. Checking the value by long division or using a calculator confirms this, since 1/16 = 0.0625.

Problem A2
Since 1/2 = 5/10, 1/2n = 5n/10n. Here, 10n dictates the number of decimal places, and 5n dictates the actual digits in the decimal. Since 54 = 625, 1/24 = 0.0625 (four decimal places).

Problem A3

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/2 2 21 1 0.5 1/4 4 22 2 0.25 1/8 8 23 3 0.125 1/16 16 24 4 0.0625 1/32 32 25 5 0.03125 1/64 64 26 6 0.015625 1/1,024 1,024 210 10 0.0009765625 1/2n 2n 2n n 0.5n (with enough leading zeroes to give n decimal places)

Notice that the decimal will include the power of 5, with some leading zeros. For example, 55 is 3,125, so “3125” shows up in the decimal, with enough leading zeros for it to comprise five digits: .03125. Similarly, 56 is 15,625, so the decimal is .015625 (six digits).

Problem A4
We know that 1/2n= 5n/10n, so for all unit fractions with denominators that are the nth power of 2, the decimal will consist of the digits of 5with enough leading zeros to give n decimal places.

Problem A5

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/5 5 51 1 0.2 1/25 25 52 2 0.04 1/125 125 53 3 0.008 1/625 625 54 4 0.0016 1/3,125 3,125 55 5 0.00032 1/15,625 15,625 56 6 0.000064 1/5n 5n 5n n 0.2n (with enough leading zeroes to give n decimal places)

Notice that the decimal will include the power of 2, with some leading zeros. For example, 25 is 32, so the decimal is .00032 (five digits).

Problem A6

Fraction

Denominator

Prime Factorization

Number of Decimal
Places

Decimal Representa-
tion

 1/10 10 21 • 51 1 0.1 1/20 20 22 • 51 2 0.05 1/50 50 21 • 52 2 0.02 1/200 200 23 • 52 3 0.005 1/500 500 22 • 53 3 0.002 1/4,000 4,000 25 • 53 5 0.00025 1/(2n • 5m) 2n • 5m 2n • 5m max(n,m) For m > n: (10-m)(2m-n) or for n > m: (10-n)(5n-m)
“Max (n,m)” means the larger of m or n — in other words, the greater exponent between the power of 2 and the power of 5 in the second column.

All of the fractions we’ve looked at so far convert to terminating decimals; that is, their decimal equivalents have a finite number of decimal places. Another way to describe this is that if you used long division to convert the fraction to a decimal, eventually your remainder would be 0.

Problem A7
a. Yes, they will all terminate.
b. A decimal terminates whenever it can be written as n/10k for some integer n and k. Then n will be the decimal, and there will be k decimal places. Since any number whose factors are 2s and 5s must be a factor of 10k for some k, the decimal must terminate. Specifically, k will be the larger number between the powers of 2 and 5 in the denominator. (See the table in Problem A6 for some examples.)

Problem A8
Write and reflect. Answers will vary.

Problem A9

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
Since the decimal cannot terminate (because the denominator contains factors other than powers of two and/or five), the remainder 0 is not possible. That means that there are only six possible remainders when we divide by 7: 1 through 6. When any remainder is repeated, the decimal will repeat from that point. If, after six remainders, you have not already repeated a remainder, the next remainder must repeat one of the previous remainders, because you only had six to choose from. Therefore, there can be no more than six possible remainders before the remainder begins repeating itself.

Problem A11
Yes, the period is one less than the denominator — it can never be more. For example, when dividing by 19, there are 18 possible remainders.

Problem A12
The period for each of these is not one less than the denominator, but it is a factor of the number that is one less than the denominator. For example, 12 is one less than 13; 1/13 has a period of six, and 6 is a factor of 12.

Solution A13

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
In all cases, the period is a factor of one less than the prime number in the denominator. For example, 1/41 has a period of five, and 5 is a factor of 40 (i.e., 41 – 1).

Problem A15
Judging from the pattern, we might expect the period to be a factor of 46. The possible factors are 1, 2, 23, and 46. The actual period is 46.

Problem A16
a. 1/7 = , 6/7 = . Sliding the expansion of 1/7 by three digits yields the expansion of 6/7. All of the expansions of 2/7 through 6/7 can be built this way, by sliding the expansion of 1/7 by one through five digits. If the digits are written in a circle, the first digit of 6/7 will be directly opposite the first digit of 1/7. Similarly, the first digit of 5/7 will be opposite the first digit of 2/7, and the first digit of 4/7 will be opposite the first digit of 3/7. In every case, the two fractions add up to 7/7, or 1.

b. Thirteen has two rings:

 • 1/13 =  can be used to generate 10/13, 9/13, 12/13, 3/13, and 4/13. • 2/13 =  can be used to generate the others: 7/13, 5/13, 11/13, 6/13, and 8/13.

c. Answers will vary, but if you try this with enough prime numbers, you should find that the size of a ring is the same as the period of the expansion, and this determines the number of rings. For 41, each ring has five numbers, and there are eight rings (since there are 40 possible fractions from 1/41 to 40/41).