1393 Words6 Pages

Recurring Decimals
Infinite yet rational, recurring decimals are a different breed of numbers. Mathematicians, in turn, have been fascinated by these special numbers for over two thousand years. The Hindu-Arabic base 10 system we use today was inspired by the Chinese method of decimals which was actually around 10000 years old. Decimals may have been around for a very long time, but what about recurring decimals? In fact the ancient Greeks were one of the first to deal with recurring decimals. The Greek mathematician Zeno had a paradox in which the answer was a finite number that was a sum of an infinite sequence. The answer to his problem was a recurring decimal, and it definitely would not be the last time recurring decimals played*…show more content…*

What about the number “n”? Can the period of the number 1/n be found also? These questions are what I hope to answer in my project. Before utilizing mathematical resources, I decided to approach my problem from scratch and just experiment with numbers on a lazy Sunday morning. The patterns I encountered when working with these numbers were stunning; to keep things simple, I started with small numbers as denominators. Soon enough, I realized that the primes 2 and 5 do not produce any recurring decimals. 3 only provides one repeating digit, and yet 9 does too. The number 7 produces 6 repeating digits, but the number 11 only produces two digits. What is going on here? Well, it appears that some primes p have p-1 repeating digits, but not all of them. However, all primes except 2 and 5 seem to have a period that is (p-1)/n long, where n is some integer. What about the squares of primes? I would like to give the example of 3; although 9 (3x3) has the same number of repeating digits as 3 (1), 27 actually has three repeating digits. Even more stunning, 81 (3^4) has nine repeating digits. It appears as if as the number is raised by a power of 3, then the number of repeating digits is also raised to some power. The number 3 is actually an exception to the general rule that I will mention, and the repeating digits of 3 to the nth power is just 3^(n-2). A conventional example is 7; 7 has 6 repeating digits but 49 (7x7) has 42 (6x7)

What about the number “n”? Can the period of the number 1/n be found also? These questions are what I hope to answer in my project. Before utilizing mathematical resources, I decided to approach my problem from scratch and just experiment with numbers on a lazy Sunday morning. The patterns I encountered when working with these numbers were stunning; to keep things simple, I started with small numbers as denominators. Soon enough, I realized that the primes 2 and 5 do not produce any recurring decimals. 3 only provides one repeating digit, and yet 9 does too. The number 7 produces 6 repeating digits, but the number 11 only produces two digits. What is going on here? Well, it appears that some primes p have p-1 repeating digits, but not all of them. However, all primes except 2 and 5 seem to have a period that is (p-1)/n long, where n is some integer. What about the squares of primes? I would like to give the example of 3; although 9 (3x3) has the same number of repeating digits as 3 (1), 27 actually has three repeating digits. Even more stunning, 81 (3^4) has nine repeating digits. It appears as if as the number is raised by a power of 3, then the number of repeating digits is also raised to some power. The number 3 is actually an exception to the general rule that I will mention, and the repeating digits of 3 to the nth power is just 3^(n-2). A conventional example is 7; 7 has 6 repeating digits but 49 (7x7) has 42 (6x7)

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