Roots to Research: A Vertical Development of Mathematical Problems
Certain contemporary mathematical problems are of particular interest to teachers and students because their origin lies in mathematics covered in the elementary school curriculum and their development can be traced through high school, college, and university level mathematics. This book is intended to provide a source for the mathematics (from beginning to advanced) needed to understand the emergence and evolution of five of these problems: The Four Numbers Problem, Rational Right Triangles, Lattice Point Geometry, Rational Approximation, and Dissection. Each chapter begins with the elementary geometry and number theory at the source of the problem, and proceeds (with the exception of the first problem) to a discussion of important results in current research. The introduction to each chapter summarizes the contents of its various sections, as well as the background required. The book is intended for students and teachers of mathematics from high school through graduate school. It should also be of interest to working mathematicians who are curious about mathematical results in fields other than their own. It can be used by teachers at all of the above mentioned levels for the enhancement of standard curriculum materials or extra-curricular projects. -- Book cover.
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algebraic number assume Borsuk's Borsuk's conjecture canonical forms compute congruent number Consequently convex polygonal convex set Corollary corresponding countable cube defined Dehn invariant denominator diameter dihedral angles divides edge elements elliptic curve equal equation equidecomposable equidissectable example Exercises Exercise fact Farey sequence finite number Four Numbers Game hyperplane hypotenuse implies inequality infinitely many rational integer points integer right triangle interior intersection irrational number lattice points lattice polygon lattice square lattice triangles Lemma line segment linear mathematics Minkowski's theorem modp nonnegative integers nonzero number of lattice number theory pairwise disjoint parallelogram partition Pick's theorem plane polyhedra polyhedron polynomial positive real number primitive Pythagorean triple Proposition prove Pythagorean triple quadruple rational numbers rational points rational right triangle rectangle region relatively prime right triangle rotation satisfying Section sides square free strictly decreasing subsets sum of squares Suppose tetrahedra vectors vertex vertices zero