Find the fractional equivalent for 0.142857142857142857....
Problem H2
Shigeto and Consuela were computing the decimal expansion of 1/19. Since Shigeto used scratch paper, he had only a little room to write his answer. He continued writing the digits on the next line, and in the end, his answer looked like this:
Shigeto noticed a pattern in these numbers. Describe his pattern.
Look at the columns of digits. What do you notice? Close Tip
Problem H3
Consuela did her computation on a narrow notepad. Her answer looked like this:
After looking at Shigeto's pattern, Consuela tried to find a pattern in her answer. What observations can you make about Consuela's pattern?
Look at the columns of digits. What do you notice? Close Tip
Problem H4
David started to compute the decimal expansion of 1/47. He got tired after computing this much of the expansion:
Shigeto had no trouble finishing the expansion using his pattern. How about you? Can you finish the expansion and explain your answer?
What are the possible periods for the decimal expansion of 1/47? Can you predict the actual period based on how many digits there are in the decimal expansion so far? How would you arrange those digits so that the columns add up to 9? Close Tip
Problem H5
Does the length of the period of your expansion make sense? Explain why or why not.
Problem H6
Consuela looked at David's work and knew immediately that her method would not be helpful. Explain why not.
Problem H7
Mr. Teague asked the class to compute a decimal expansion with period 42. Unfortunately, his dog spilled paint on Shigeto's and Consuela's answers. Use the visible information in Shigeto's and Consuela's answers and the patterns you have seen to find the complete decimal expansion:
Problem H8
Find the fraction with this particular decimal expansion.
Problem H9
Is it possible to represent the number 1 as a repeating decimal?
Think about the decimal expansion for 1/3 = 0.333333.... What would 2/3 be? What fraction would be closest or equal to 1? Close Tip
Problem H10
Is it possible to predict the period of 1/14 if you know the period of 1/7 (i.e., six)?
Problem H11
Is it possible to provide a convincing argument to prove that the decimal expansion of 1/n has a period that is less than n?
