Read Me - Repeating Decimals Instructions (Word Format) (23.0K)

The Math Investigator is a data collection software program that may be used to collect data for the investigation. You may type answers onto the Word file or copy and paste in data from the Investigator.

Math Investigator 6.1

REPEATING DECIMALS on the Math Investigator prints decimals for rational numbers. If the decimal is repeating, it counts the number of digits in the nonrepeating part (if any) and the number of digits in the repetend.

1. Use the program REPEATING DECIMALS to obtain a printout of the decimals for fractions of the form 1/N, with N = 2 to 50. (Note: By repeatedly using the Edit menu to copy and paste text items, fractions and their decimals can be pasted and printed together.) You may find it helpful to use colored pens to mark the different types of decimals: terminating decimals, repeating decimals, and decimals with more than one digit before the repetend.

a. What type of fractions have terminating decimals? Form a conjecture and test it for other fractions.

b. Form a conjecture about the type of fractions which have repeating decimals with a nonrepeating part. For example, 1/12 = .08333 . . . has two digits before the repetend which is repeating 3s. Is there a pattern for predicting the number of digits in the nonrepeating part of a decimal?

c. If a repeating decimal does not have a nonrepeating part, it is called purely periodic. For example, 1/7 = .142857 . . . is purely periodic. What types of fractions will have decimals that are purely periodic?

d. The number of digits in the repetend of the decimals for some of these fractions is one less than the denominator. For example, the repetend for 1/17 has 16 digits. For what types of numbers n will the repetend for 1/n have n - 1 digits?

2. (Challenge) In writing the computer program for this investigation, it was necessary to tell the computer how many digits to check in order to find the repetend. The number of digits in the repetend of the decimal for 1/n will have at most n - 1 digits. Thus, the repetend for 1/119 will not have more than 118 digits. However, since 119 = 7 x 17, it is natural to wonder if the repetends for 1/7 and 1/17 can be used to determine the repetend for 1/119. In general, can the prime factors of n be used to provide information about the number of digits in the repetend for 1/n? Investigating this question will involve gathering data for many special cases and looking for patterns. We hope you will enjoy this challenge.