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.
Kieren, T., Gordon-Calvert, L., Reid, D. & Simmt, E. (1995). An enactivist research approach to mathematical activity: Understanding, reasoning, and beliefs. Paper presented at the meeting of the Ame rican Educational Research Association, San Francisco.
Restivo, Sal, Jean Paul Van Bendegen, and Roland Fischer. Math Works: Philosophical and Social Studies of Mathematics and Mathematics Education. Albany, New York: State University of New York Press, 1993.
The curriculum implies that teachers will teach students the skills they need for the future. Valley View’s High School math department announces, “Students will learn how to use mathematics to analyze and respond to real-world issues and challenges, as they will be expected to do college and the workplace.” Also, the new integrates math class allows students to distinguish the relationship between algebra and geometry. Although students are not being instructed a mathematical issue in depth, they are rapidly going through all the different topics in an integrated math class. Nowadays, students are too worried to pass the course to acquire a problem-solving mind. Paul Lockhart proclaims the entire problem of high school students saying, “I do not see how it's doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams and dear memories of hating them.” A mathematics class should not be intended to make a student weep from complicated equations, but it should encourage them to seek the numbers surrounding
Mathematics education has undergone many changes over the last several years. Some of these changes include the key concepts all students must master and how they are taught. According to Jacob Vigdor, the concerns about students’ math achievements have always been apparent. A few reasons that are negatively impacting the productivity of students’ math achievements are historical events that influenced mathematics, how math is being taught, and differentiation of curriculum.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
Tubbs, Robert. What is a Number? Mathematical Concepts and Their Origins. Baltimore, Md: The Johns Hopkins
Barr, C., Doyle, M., Clifford, J., De Leo,T., Dubeau, C. (2003). "There is More to Math: A Framework for Learning and Math Instruction” Waterloo Catholic District School Board
‘Nature abounds with example of mathematical concepts’ (Pappas, 2011, .107). It is interesting how much we see this now we know, regarding the Fibonacci Sequence, which is number pattern where the first number added to itself creates a new number, then adding that previous number to the new number and so on. You will notice how in nature this sequence always adds up to a Fibonacci number, but alas this is no coincidence it is a way in which plants can pack in the most seeds in a small space creating the most efficient way to receive sunlight and catches the most
Fibonacci numbers are numbers in the Fibonacci sequence. In this paper, you will find out what Fibonacci numbers are related to. You will also find out how Fibonacci numbers are everywhere in the world. Though Fibonacci numbers are found in mathematical subjects, they are also found in other concepts.
Irrational numbers are real numbers that cannot be written as a simple fraction or a whole number. For example, irrational numbers can be included in the category of √2, e, Π, Φ, and many more. The √2 is equal to 1.4142. e is equal to 2.718. Π is equal to 3.1415. Φ is equal to 1.6180. None of these numbers are “pretty” numbers. Their decimal places keep going and do not end. There is no pattern to the numbers of the decimal places. They are all random numbers that make up the one irrational number. The concept of irrational numbers took many years and many people to discover and prove (I.P., 1997).
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.
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.
Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section. It falls under number theory, which is the study of whole or rational numbers. Number Theory develops theories, simple equations, and uses special tools to find specific numbers. Some topic examples from number theory are the Euclidean Algorithm, Fermat?s Little Theorem, and Prime Numbers.
...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.
When I graduated from high school, forty years ago, I had no idea that mathematics would play such a large role in my future. Like most people learning mathematics, I continue to learn until it became too hard, which made me lose interest. Failure or near failure is one way to put a stop to learning a subject, and leave a lasting impression not worth repeating. Mathematics courses, being compulsory, are designed to cover topics. One by one, the topics need not be important or of immediate use, but altogether or cumulatively, the topics provide or point to a skill, a mastery of mathematics.