Number Theory and the Queen of Mathematics

(ProQuest: ... denotes formula omitted.)IntroductionGeometry is an integral component of secondary mathematics curriculum. From my experience in a mathematics teacher preparation program, I have seen a real push to connect geometry to other areas of mathematics. Secondary geometry can often be presented without clear or any connections to other areas of mathematics. One main purpose of this paper is to explore geometry and its rightful connection to other areas of mathematics, specifically number theory. Such strong emphasis is placed on drawing connections to number theory because of its intrinsic value in enhancing understanding of mathematical concepts. Learning number theory has positive ramifications for students, making the transition from arithmetic to an introduction to algebra. It helps "students develop better understandings of the abstract conceptual structure of whole numbers and integers," and it has important algebraic characteristics, which relate to variables and mathematical reasoning (Campbell & Zazkis, 2006, p.28). Another purpose of this paper is to explore not only number theory as it relates to geometry, but also the basic history of number theory. Number theory has a beauty, accessibility, history, formal and cognitive nature, and intrinsic merits all to its own (Campbell & Zazkis, 2006, p. 13). For the sake of all the intrigue of number theory, I have a desire to learn more about it to fuel my own pursuit of a better understanding of effectively teaching mathematics, but also to use it to encourage and engage students in their personal pursuit of mathematical understanding.Because of the purposes of this paper, this research was compiled and organized from a mathematics education perspective. Although, in order to meet the purposes of this paper, the history of number theory and its history of interaction of geometry (This is more of the history of the acknowledgment of number theory and its interaction of geometry.) will play an important role. Despite this, however, this research was not compiled strictly from a historical perspective, but rather from a mathematics education perspective, with a historical perspective that is inherently a part of it.Ancient Geometry-An Early look at the Union of Geometry and Number TheoryGeometry consistently played an important role in the mathematics of early civilizations. The discovery and study of approximately 500 clay tablets from the area once known as Mesopotamia indicate that the Babylonians were no exception. Babylonians' interest in geometry is evident. One tablet, Plimpton 3221-a tablet of what appears to be pythagorean triples-indicates that the Babylonians knew of the relationship between the sides of a right triangle. Plimpton 322, dated to the mid-18th century B.C., has been the subject of research and study for several decades (Robson, 2001, p. 170). Plimpton 322 appears to be incomplete; there is an evident break along its left edge. Also, this tablet does not include any scratch work that would shed light on the methods of the Babylonians' pythagorean triple generation. The question on the minds of researchers is how did the Babylonians generate this list of pythagorean triples?The significance of answering this question relates to the study of the origins of Babylonians generation of pythagorean triples. To study these origins, it is imperative to unveil our eyes of our modern and cultural views embedded into our mathematical understanding and try to see Babylonian mathematics through their cultural veil. In answering this question, it will be important to acknowledge Babylonian culture. There are significant distinctions between the Babylonians' culturally view of mathematics and our modern view. Also, in answering this question, an important union of two mathematical branches, geometry and number theory, will reveal itself. Modern mathematical views and Babylonian mathematical views are distinct without question. …