Session 7, Part A:
Fractions to Decimals (65 minutes)

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

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. Note 1

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.    Close Tip 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 54 is 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?   Close Tip 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.    Close Tip 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

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 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.   Close Tip 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. If you are using a VCR, 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

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

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

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

 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 Mathemtics. Used with permission. All rights reserved.
 Session 7: Index | Notes | Solutions | Video