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.
Lamb, Robert. "How are Fibonacci numbers expressed in nature?" HowStuffWorks. Discovery Communications, 24 June 2008. Web. 28 Jan. 2010. .
Fractions have been a around long enough for me to understand that I do not like them, but they play a significant part in simplifying, for some, division of goods or time. There is no one person who can be credited with the invention of fractions, but their use has been traced back as early as 1000 BC, in Egypt--using the formula to trade tangibles, currency, and build pyramids.
The experience of the APEC Youth Science festival was incredible. It has had an enormous impact on me in many ways, changing the way I look at the world and connecting me with people and events far beyond my formerly limited experience. I am extremely glad to have had this opportunity. It was a wonderful experience on multiple levels. It challenged me and expanded me intellectually and socially. I feel that this experience has had an immense impact on me.
I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every math student will recognize these periods of frustration and helplessness.
Wigner, Eugene P. 1960. The Unreasonable Effectiveness of Mathematics. Communications on Pure and Applied Mathematics 13: 1-14.
The Fibonacci sequence is often defined as {F_n }_(n=1)^∞ containing the numbers 1, 1, 2, 3, 5, 8, 13, 21, and so on [1]. A formula for determining the numbers contained in the set is given by F_1=1 and F_2=1, with the recursive formula F_n=F_(n-1)+F_(n-2). In other words, the formula doesn't start until n=3, and computing elements of the set just involves adding the two previous numbers together to get the next. Using the Fibonacci sequence, it is true that the infinite sequence r_n,
The recursive sequence of numbers that bear his name is a discovery for which Fibonacci is popularly known. While it brought him little recognition during the course of his life, is was nearly 100 years later when the majority of the mathematicians recognized and appreciated his invention. This fascinating and unique study of recursive numbers possess all sorts of intriguing properties that can be discovered by applying different mathematical procedures to a set of numbers in a given sequence. The recursive / Fibonacci numbers are present in everyday life and they are manifested in the everyday life in which we live. The formed patterns perplex and astonish the minds in real world perspectives. The recursive sequences are beautiful to study and much of their beauty falls in nature. They highlight the mathematical complexity and the incredible order of the world that we live in and this gives a clear view of the algorithm that God used to create some of these organisms and plants. Such patterns seem not have been evolved by accident but rather, they seem to have evolved by the work of God who created both heaven and
The question that Leonardo Fibonacci solved with Fibonacci Numbers was about rabbit reproduction. The question was asking, “Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age...
...atics in six countries, Mathematics Teaching in the 21st Century, Center for Research in Mathematics and Science Education, Michigan State University.
Towers, J., Martin, L., & Pirie, S. (2000). Growing mathematical understanding: Layered observations. In M.L. Fernandez (Ed.), Proceedings of the Annual Meetings of North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ, 225-230.
A somewhat underused strategy for teaching mathematics is that of guided discovery. With this strategy, the student arrives at an understanding of a new mathematical concept on his or her own. An activity is given in which "students sequentially uncover layers of mathematical information one step at a time and learn new mathematics" (Gerver & Sgroi, 2003). This way, instead of simply being told the procedure for solving a problem, the student can develop the steps mainly on his own with only a little guidance from the teacher.
The Fibonacci Sequence was discovered by Leonardo Fibonacci. It is a sequence of numbers in which the next number is the sum of the two numbers previous to it. There is a direct correlation between this sequence of numbers and the Golden Ratio. If you were to take two adjacent numbers in the Fibonacci sequence and divide them, their ratio would be very close to the Golden Ratio. Also, as the numbers get higher, their ratios get closer to the golden ratio itself. For example, the ratio of 5 to 3 is 1.666… But the ratio of 21 to 12 is 1.615… Getting even higher, the ratio of 233 to 144 is
The golden ration can occur anywhere. The golden proportion is the ratio of the shorter length to the longer length which equals the ratio of the longer length to the sum of both lengths.
...re encompassing way, it becomes very clear that everything that we do or encounter in life can be in some way associated with math. Whether it be writing a paper, debating a controversial topic, playing Temple Run, buying Christmas presents, checking final grades on PeopleSoft, packing to go home, or cutting paper snowflakes to decorate the house, many of our daily activities encompass math. What has surprised me the most is that I do not feel that I have been seeking out these relationships between math and other areas of my life, rather the connections just seem more visible to me now that I have a greater appreciation and understanding for the subject. Math is necessary. Math is powerful. Math is important. Math is influential. Math is surprising. Math is found in unexpected places. Math is found in my worldview. Math is everywhere. Math is Beautiful.
Abstractions from nature are one the important element in mathematics. Mathematics is a universal subject that has connections to many different areas including nature. [IMAGE] [IMAGE] Bibliography: 1. http://users.powernet.co.uk/bearsoft/Maths.html 2. http://weblife.bangor.ac.uk/cyfrif/eng/resources/spirals.htm 3.