When Less is More: Visualizing Basic InequalitiesMAA, 09 ապր, 2009 թ. - 181 էջ 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. |
Բովանդակություն
Representing positive numbers as lengths of segments | 1 |
Chapter | 2 |
Using incircles and circumcircles | 4 |
Chapter | 8 |
Representing positive numbers as areas or volumes | 19 |
Chapter 9 | 28 |
Inequalities and the existence of triangles | 43 |
Using reflections | 73 |
Using rotations | 81 |
7 | 93 |
1 | 106 |
Notation and symbols | 169 |
Այլ խմբագրություններ - View all
Common terms and phrases
a₁ acute triangle Alsina altitudes AM-GM inequality angle bisectors Application Applying the AM-GM arithmetic mean b₁ Cauchy-Schwarz inequality Challenge Chapter Chebyshev's inequality circle circumcenter circumcircle circumradius concave convex cos² cyclic denote the area denote the lengths diagonals equality Erdős-Mordell establish example function graph gray triangle harmonic mean hence Heron's formula illustrated in Figure inequality is equivalent inradius inscribed isoperimetric theorem Lemma line segment m-gon Mathematical max(a mean inequality mean square mediant n-gon Nelsen Padoa's inequality parallelogram polygon positive numbers problem prove quadrilateral radius Ravi substitution real numbers rectangle right triangle Schur's inequality secant line semiperimeter side lengths similarly superellipse tangent line Theorem trapezoid triangle ABC triangle given triangle inequality triangle is equilateral triangle with sides vectors vertex vertices visual proof yields