When Less Is More.

cover -- copyright page -- title page -- Contents -- Preface -- Introduction -- Inequalities as a field of study -- Inequalities in the classroom -- CHAPTER 1 Representing positive numbers as lengths of segments -- 1.1 Inequalities associated with triangles -- 1.2 Polygonal paths -- 1.3 n-gons inside m-gons -- 1.4 The arithmetic mean-geometric mean inequality -- 1.5 More inequalities for means -- 1.6 The Ravi substitution -- 1.7 Comparing graphs of functions -- 1.8 Challenges -- CHAPTER 2 Representing positive numbers as areas or volumes -- 2.1 Three examples -- 2.2 Chebyshev's inequality -- 2.3 The AM-GM inequality for three numbers -- 2.4 Guha's inequality -- 2.5 The AM-GM inequality for n numbers -- 2.6 The HM-AM-GM-RMS inequality for nnumbers -- 2.7 The mediant property and Simpson's paradox -- 2.8 Chebyshev's inequality revisited -- 2.9 Schur's inequality -- 2.10 Challenges -- CHAPTER 3 Inequalities and the existence of triangles -- 3.1 Inequalities and the altitudes of a triangle -- A triangle and its altitudes -- Existence of a triangle given a, b, and h_a -- Existence of a triangle given a, h_b, and h_c -- More inequalities for the three altitudes -- Altitudes, sides and angles -- 3.2 Inequalities and the medians of a triangle -- Existence of a triangle given m_a, m_b, and m_c -- Existence of a triangle given a, b, and m_a -- Existence of a triangle given a, b, and m_c -- Existence of a triangle given a, m_a, and m_b -- 3.3 Inequalities and the angle-bisectors of a triangle -- Existence of a triangle given a, h_a, and w_a -- Existence of a triangle given a, h_b, and w_c -- Ordering of sides and angle-bisectors -- 3.4 The Steiner-Lehmus theorem -- 3.5 Challenges -- CHAPTER 4 Using incircles and circumcircles.

Inequalities permeate mathematics, from the Elements of Euclid to operations research and financial mathematics. Yet too often the emphasis is on things equal to one another rather than unequal. While equalities and identities are without doubt important, they don't possess the richness and variety that one finds with inequalities. The objective of this book is to illustrate how use of visualization can be a powerful tool for better understanding some basic mathematical inequalities. Drawing pictures is a well-known method for problem solving, and we would like to convince you that the same is true when working with inequalities. We show how to produce figures in a systematic way for the illustration of inequalities; and open new avenues to creative ways of thinking and teaching. In addition, a geometric argument can not only show two things unequal, but also help the observer see just how unequal they are. The concentration on geometric inequalities is partially motivated by the hope that secondary and collegiate teachers might use these pictures with their students. Teachers may wish to use one of the drawings when an inequality arises in the course. Alternatively, When Less Is More might serve as a guide for devoting some time to inequalities and problem solving techniques, or even as part of a course on inequalities.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.