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Prime Numbers, 2004.
This paper discusses the history and theories relating to prime numbers.
1,625 words (approx. 6.5 pages), 6 sources, MLA, $ 52.95
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Abstract
This paper explains that prime numbers have upset humanity for the last 2,300 years, ever since Euclid proved that there was infinity. The paper presents Euclid's proof. The author points out that, in the 19th century, the Russian, Tchebychef ,made significant contributions to the study of prime numbers by concluding that if the B(x) from Legendre's equation had a limit, then that limit had to be 1. The paper relates that Bernhardt Riemann, in 1859, continued Euler's work on series of numbers and made fundamental new discoveries within the prime numbers domain.
From the Paper
"Let x be a positive real number and let us define ? (x) as the number of primes less or than equal to x. ? (x) is a function and will be studied as such. For small values of x, we can intuit the values for ? (x). Thus, for example, if x is 10, then the primes less then 10 are 2, 3, 5 and 7, so that ? (10) is 4. Similarly, ? (24) = 9. Our concern is for very large values of x."
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"The Foundations of Arithmetic", 2004.
A review of Gottlob Frege's much-discussed book, "The Foundations of Arithmetic".
1,522 words (approx. 6.1 pages), 3 sources, MLA, $ 50.95
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Abstract
This paper explains how Frege's book is an influential and valuable insight into the philosophy of mathematics. It shows how Frege delves deeply into, not only an understanding of numbers, but also looks into much larger questions surrounding meaning and truth. Frege argues against the idea that arithmetic is based on psychology and, instead, notes that logic is the main underpinning of arithmetic.
From the Paper
"Friedrich Ludwig Gottlob Frege was born in Germany in 1848. He got his doctoral degree in G?ttingen, and quickly wrote his post-doctoral thesis, and became a university professor. During his long and fruitful academic career, the tireless Frege worked extensively to build up the philosophical foundations of mathematics and science. Among his important contributions is the invention of an artificial language called Begriffsschrift, which was based on logical notation (Frege Biography). Frege wrote extensively, publishing a number of other influential papers during his lengthy career, including Grundgesetze der Arithmetik, Volume I (1893), Logic (1897), Sources of Knowledge of Mathematics and the Mathematical Natural Sciences (1924/5), A Brief Survey of my Logical Doctrines (1906), and Thought (1918) (Frege Biography)."
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Problems in Statistics Education, 2004.
Critical analysis of a current problem in the field of statistics education.
2,681 words (approx. 10.7 pages), 20 sources, MLA, $ 80.95
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Abstract
This paper examines some of the ways to teach statistics that will best overcome some of the main problems that students encounter while learning statistics and offers solutions to these problems.
From the Paper
"Students do not normally encounter statistics until they are in college--at least not on any kind of practicable level--unless they are in extremely advanced mathematics classes at their high school. Even so, not every high school offers statistics as a course, while almost every college does. Teaching and learning statistics is problematic for most college students and teachers because to learn and understand statistics, it is necessary to first have a grasp of some of the properties and features of higher mathematics. Many college students do not have these skills upon entering college, and many professors assume that they do have these skills when beginning to teach a statistics course."
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The Pascal's Triangle, 2004.
This paper discusses the life of Blaise Pascal and Pascal's Triangle.
1,210 words (approx. 4.8 pages), 4 sources, APA, $ 41.95
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Abstract
This paper explains that the mathematical formula known as "Pascal's Triangle" was simultaneously discovered centuries before Pascal by the Chinese and the Persians; it was even mentioned by Omar Khayyam centuries before Pascal. Pascal, however, one of the world's most famous mathematicians, was the first "modern" mathematician to realize the true potential of the formula and apply it. The author points out that Pascal's Triangle contributed to the understanding of probabilities, which led to the development of "average gain" or "probable gain" formulas that are still used extensively in business and industry. The paper relates that there is one problem with Pascal's formula: as the numbers increase, the triangle takes much longer to solve, and the formula becomes ungainly, but mathematicians have learned to cope with the formula and have created alternates that let them work with the numbers more effectively. Formula included.
From the Paper
"The mathematical formula known as "Pascal's Triangle" has long been attributed to the great mathematician and philosopher, Blaise Pascal, who lived in France during the 17th century. Pascal only lived to be thirty-nine years old, but during his lifetime, he made significant achievements in mathematics and philosophy, and may be most well known for the mathematical formula of Pascal's Triangle, which he did not invent, but has long received credit for inventing. Pascal was a bright child, who created the first known type of automatic calculator at the age of nineteen, and invented the modern-day barometer before he turned thirty-one."
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Omar Khayyam, 2003.
A description of the life and works of the famous Persian Omar Khayyam.
2,505 words (approx. 10.0 pages), 8 sources, MLA, $ 76.95
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Abstract
This paper presents an overview of the life of Omar Khayyam, born on 18 May 1048 at Nishapur, the provincial capital of Khurasan. The writer explores all aspects of his amazing life, as painter, mathematician, musician, writer and philosopher. The paper begins with his early life in Persia through to his death in Nishapur on 4th December 1131. The writer believes that Omar Khayyam was an outstanding astronomer and astrologer and his contributions to this field are invaluable still today. The paper includes a number of drawings of the man and examples of his writing.
From the Paper
"Omar Khayyam was well known as a poet, philosopher, mathematician, astronomer and physician. His full name was Ghiyath al-Din Abu?l-Fath Omar ibn Ibrahim Al-Nishapuri al-Khayyami. A literal translation of the name al-Khayyami means ?tent maker? which maybe derived from his father?s trade or he may have practiced this skill at one time. Khayyam played on the meaning of his own name when he wrote; ?Khayyam, who stitched the tents of science, Has fallen in grief?s furnace and been suddenly burned, The shears of Fate have cut the tent ropes of his life, And the broker of Hope has sold him for nothing!?."
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Integrating the Internet, 2004.
This paper highlights the facts and information of mathematics curricula and takes a broader look at the use of Internet technology in mathematical learning.
1,859 words (approx. 7.4 pages), 5 sources, MLA, $ 59.95
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Abstract
This paper explains the position that the Internet is on the brink of playing a role in sustaining inquiry-based mathematical classrooms, as well as to caution educators away from guidelines that have been established as unprofitable. The paper begins by unfolding wide themes that have surfaced from the work in relation to mathematical educational employment of the Internet, and then inspects how these themes occupy themselves in a particular classroom. The paper ends with the consideration of how scientific developments in education should introduce high-quality, skilled teachers, who are prepared to make use of these novel technologies to encourage student learning.
From the Paper
"Educational philosophy has developed all through the last decade of research. This research has made analysts believe that the Internet will not give an easy way to enhanced education; the analysts have come to consider that people-to-people relations and particularly face-to-face contact play a vital responsibility in education. Alternatively, a number of analysts have turned out to be even more overwhelmed by the influence of the technology and remain persuaded that this authority will eventually be controlled for the development of education. Whilst technology develops rapidly, on the other hand, the human aptitude to recognize, formulate, as well as integrate these changes develops gradually. Access to remote resources-- peers, images, experts, texts, teachers, as well as data--is quickly turning out to be a commonplace, nevertheless, the consideration of how to make superior utilization of these capitals is barely gradually emerging. The time necessary for the growth, modification, and acceptance of suitable novel pedagogies might be a decade or more (Bruce & Rubin, 1993)."
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William Gosset, 2004.
A biography of the statistician William Gosset and his lifetime achievements, including the t-test.
1,102 words (approx. 4.4 pages), 5 sources, MLA, $ 38.95
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Abstract
This paper looks at the life of William Gosset, who worked as a chemist in the Guinness brewery in Dublin in 1899 ,and who also carried out crucial experiments on statistics. It explores how the conditions of brewing gave Gosset an insight to work as a statistician and how he took his data from the different examples of brewing to experiment, which was the best combination of factors. In particular, it examines how these experiments led to the invention of the t-test to calculate and manage small samples for quality control in brewing and how, under the name "Student", Gosset developed the form of the t distribution by a combination of mathematical and empirical work with random numbers on the basis of the early application of the Monte Carlo method.
From the Paper
"In 1903, Gosset, came up with methods that could calculate standard errors. In 1904 he wrote on the brewing of beer. After reading this new report written by William Gosset, Karl Pearson consulted Gosset and also they met Pearson in July of 1905. They discussed the developments and reports for a long time. Pearson, helped Gosset understand the theory of standard errors in less than two hours. Gosset after understanding the procedure went back to the brewery and practiced those methods to develop something new for the next year. The meeting was successful because Pearson motivated Gosset to take up the study of the law of error."
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?The Proof?, 2004.
A review of the video, "The Proof".
898 words (approx. 3.6 pages), 1 source, MLA, $ 31.95
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Abstract
This paper discusses the video, "The Proof," a NOVA episode aired on PBS, which presents a look at one man's obsession with proving or disproving a theory, Fermat's Last Theorem, written over two hundred years ago and never proved. Specifically, it summarizes and reviews the video, with a focus on what the video tells us about how people learn to do mathematics. It looks at how "The Proof" is more than just a video about solving a complex mathematical problem and how it is a story of determination, setting goals, and finding out that solutions come from many different places and ideas.
From the Paper
"The program then delves into how Wiles began obsessing about the "proof" when he was ten years old, and began a lifelong process of proving Fermat's Theorem. While the story is clearly mathematical, it becomes more than that during the course of the story. It becomes a tale about a man who cannot let go of his obsession, and how to creatively find the solutions to complex problems, whether they are mathematical or not. One mathematician in the show talks about making "good mistakes," and how difficult it is. This is the key to learning about mathematics, and solving mathematical problems. You will make mistakes. Learning how to make "good" mistakes is quite difficult. However, if you can learn from your mistakes, or your mistakes lead you in another direction, they are valuable, and can keep you always learning about mathematics, and other complex problems."
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Stephan Korner's "The Philosophy of Mathematics", 2004.
Summary and review of Stephan Korner's "The Philosophy of Mathematics: An Introductory Essay".
1,091 words (approx. 4.4 pages), 2 sources, MLA, $ 38.95
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Abstract
The first part of this paper expounds on Stephan Korner's discussion, in "The Philosophy of Mathematics, of the nature of mathematics, and the three main schools of thought relating mathematics to philosophy. The paper continues with a discussion on logicism and why it provides the clearest way to look at mathematical concepts and the best way to explain mathematical philosophy.
From the Paper
"Mathematics is an indispensable science that justifies and confirms many aspects of other scientific subject matter. Mathematics relies on conclusions not assumptions and evidence is required to confirm theoretical entities as true. Of course the debates exist as to which school of thought holds the most validity. Mathematical realism will always be different to each of these philosophical schools and arguments can be found to both support and reject each school of thought."
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Rationality, 2004.
Discussion on the human ability to practice rational thinking.
1,701 words (approx. 6.8 pages), 4 sources, APA, $ 55.95
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Abstract
This paper examines two of the systematic mistakes that humans tend to make when they make decisions that they are likely to consider to be rational. These include mistakes or inclinations toward both pessimistic and optimistic biases.
From the Paper
"Except, of course, that we?re not. But it is true that humans are relatively bad at purely rational thinking. This should not perhaps be surprising to us: We are not, after all, computers, which are far better than are humans at making rational decisions and providing rational calculations about situations. This is not entirely a bad thing: Humans have apparently (though the process of evolution) sacrificed the ability to make perfectly rational calculations for the ability to excel at what those who are trying to teach computers to think like humans call fuzzy thinking. We are good, for example, at being able to read another person?s internal emotional state by the tilt of their eyebrows but we are relatively bad at calculating the odds of whether to take another card in blackjack ? to the unending enrichment of the Las Vegas casinos."
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