When Less is More: Visualizing Basic Inequalities
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.
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Representing positive numbers as lengths of segments
Representing positive numbers as areas or volumes
Inequalities and the existence of triangles
Employing nonisometric transformations
Այլ խմբագրություններ - View all
acute algebraic altitudes AM-GM inequality angle Application arithmetic mean assume base bounded Cauchy-Schwarz inequality Challenge Chapter circle concave consequence consider construct convex Create cyclic denote diagonals distance draw equality equilateral equivalent establish example existence extend formula function geometric give given graph gray greater hence holds illustrated in Figure implies inscribed inside isosceles joining Lemma lengths less lies line segment locate Mathematical maximum measure mediant method n-gon Note obtain parallel path perimeter polygon positive numbers problem proof prove quadrilateral radius real numbers rectangle region relates respectively result right triangle rotate shown side lengths sides similar similarly slope square tangent Theorem trapezoid triangle ABC vectors vertex vertices visual volume Vs(s yields