Exploring Decimals
Exploring Decimals
Activity 1: FINDING DECIMAL EQUIVALENTS In the course of converting fractions to their decimal equivalents, you should have observed the following: Some fractions convert to terminating decimals; others, to repeating decimals. In some fractions that convert to a repeating decimal, the digits begin repeating immediately; in others, one or more digits appear between the decimal point and the repetend. In this activity, you will investigate the answers to the following questions: Is there a way to predict whether the decimal equivalent of a given fraction terminates or repeats? If the decimal equivalent of a fraction repeats, how many digits will appear between the decimal point and the repetend? In this activity, the term fraction refers to a fraction expressed in simplest form. 1. Complete the table on the following page. Then use the data in the table to answer the following questions. If necessary, check fractions other than those in the table. a. Which prime numbers appear in the prime factorization of the denominators of those fractions whose decimal equivalents terminate?
2.
b.
Does the prime factorization of the denominator of any of the fractions whose decimal equivalents repeat contain only the primes in part a? If so, which fractions?
c.
How can you predict whether the decimal equivalent of a given fraction terminates or repeats?
3.
a.
For the fractions whose decimal equivalents terminate, how is the number of digits in the decimal equivalent of the fraction related to the exponents in the prime factorization of the denominator?
b.
Does the relationship in part a hold for the number of digits between the decimal point and the repetend in those cases where the decimal equivalent of the fraction repeats?
c.
How can you predict the number of digits between the decimal point and the repetend in the decimal equivalent of a fraction?
MATH 3443
Modeling: Real Numbers and Statistics
Page 43
Activity 1: FINDING DECIMAL EQUIVALENTS In the course of converting fractions to their decimal equivalents, you should have observed the following: Some fractions convert to terminating decimals; others, to repeating decimals. In some fractions that convert to a repeating decimal, the digits begin repeating immediately; in others, one or more digits appear between the decimal point and the repetend. In this activity, you will investigate the answers to the following questions: Is there a way to predict whether the decimal equivalent of a given fraction terminates or repeats? If the decimal equivalent of a fraction repeats, how many digits will appear between the decimal point and the repetend? In this activity, the term fraction refers to a fraction expressed in simplest form. 1. Complete the table on the following page. Then use the data in the table to answer the following questions. If necessary, check fractions other than those in the table. a. Which prime numbers appear in the prime factorization of the denominators of those fractions whose decimal equivalents terminate?
2.
b.
Does the prime factorization of the denominator of any of the fractions whose decimal equivalents repeat contain only the primes in part a? If so, which fractions?
c.
How can you predict whether the decimal equivalent of a given fraction terminates or repeats?
3.
a.
For the fractions whose decimal equivalents terminate, how is the number of digits in the decimal equivalent of the fraction related to the exponents in the prime factorization of the denominator?
b.
Does the relationship in part a hold for the number of digits between the decimal point and the repetend in those cases where the decimal equivalent of the fraction repeats?
c.
How can you predict the number of digits between the decimal point and the repetend in the decimal equivalent of a fraction?
MATH 3443
Modeling: Real Numbers and Statistics
Page 43