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 a role in mathematics. Famous mathematicians such as Euler, Gauss, and Fermat all have contributed their own discoveries about the nature of these numbers. Fittingly, recurring decimals fall under the elegant category of number theory in mathematics, called the “queen of mathematical studies” by Gauss. We have learned much about modular arithmetic and its useful applications; my investigation will revolve around the relationship between modular arithmetic and recurring decimals. Euler’s totient function, which is just a generalized form of Fermat’s Little Theorem, appears to parallel the period of recurring decimals, the number of digits the decimal expansion goes before repeating once more. The totient function can be defined as the following: it is the number of positive integers
2
less than or equal to a number “n” that are relatively prime to “n”. For example, the number 7 has a totient of 6 because 1,2,3,4,5, and 6 are the only numbers that satisfy the conditions. If you punch in “1/7” in...
... middle of paper ...
...about recurring decimals and doing research in
5
general. Working on an original research project was definitely something new to me, and I truly value all the things I was able to learn the last few weeks at COSMOS.
Works Cited
Ball, Keith. Strange Curves, Counting Rabbits, and Other Mathematical Explorations. Princeton: Princeton University Press, 2003.
Burger, Edward B. and Michael Starbird. The Heart of Mathematics: An Invitation to Effective Thinking. United States: Key College Publishing, 2000.
Fractions Calculator. Dr. R Knott. 14 Aug. 2000. Acumedia. 29 July 2005.
Weisstein, Eric W. "Decimal Expansion." From MathWorld--A Wolfram Web Resource.
Wikipedia. “Recurring Decimals”. Wikipedia 2005. Wikipedia. 27 July 2005.

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