When Less is More: Visualizing Basic Inequalities (Dolciani Mathematical Expositions)
Claudi Alsina
Language: English
Pages: 164
ISBN: 0883853426
Format: PDF / Kindle (mobi) / ePub
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.
Linear Algebra and Its Applications (4th Edition)
Inside Calculus (Undergraduate Texts in Mathematics)
Graph Theory (Graduate Texts in Mathematics, Volume 244)
Visualizing Elementary and Middle School Mathematics Methods
Corollary 4.4. If Q is bicentric, then .a C b C c C d /2 8pq. Proof. 8pq D 2.4ac C 4bd / Ä 2Œ.a C c/2 C .b C d /2 D .a C b C c C d /2 . ✐ ✐ ✐ ✐ ✐ ✐ “MABK007-04” — 2009/2/17 — 11:49 — page 67 — #13 ✐ ✐ 67 4.4. Some properties of n-gons 4.4 Some properties of n-gons We begin with some results about triangles. Theorem 4.6. Among all triangles having a specified base and opposite vertex angle, the isosceles triangle has the greatest inradius. α (α + π)/2 β/2 γ/2 Figure 4.11. Proof.
convex. Thus for concave functions, the inequality in (8.1) is reversed, and a chord connecting any two points on the graph of y D f .x/ lies below the graph. When the function is twice differentiable, convexity or concavity is easily established by examining the sign of the second derivative. Application 8.7. A property of concave functions. Let f be a concave function defined on Œ0; b/, where b can be finite or 1, with f .0/ D 0. Then for any in (0, 1) and x 0; f .x/ Ä f . x/. See Figure 8.11.
than Jensen’s inequality) to establish these inequalities. Application 8.10. The perimeter and circumradius of a triangle In triangle ABC, let a; b; c denote the sides, and R the circumradius. In Figure 6.11, the angle marked ] at the circumcenter has the same measure as †A, so sin A D a=2R, or a D 2R sin A. Similarly, b D 2R sin B and c D 2R sin C . Using (8.3) yields a C b C c D 2R .sin A C sin B C sin C / Ä 6R sin 3 p D 3 3R: As a bonus, we have shown that for triangles inscribed in a
and B < b1 , then as seen in Figure 9.2, we have q q p p a12 A2 C b12 B 2 D c 2 .A C B/2 Ä .a1 C b1 /2 .A C B/2 : ✐ ✐ ✐ ✐ ✐ ✐ “MABK007-09” — 2009/1/13 — 21:50 — page 139 — #3 ✐ ✐ 139 9.2. Majorization Figure 9.2. Now set A2 D a22 C a32 C C an2 and B 2 D b22 C b32 C C bn2 . This establishes the first inequality below and the second follows from Minkowski’s inequality (8.7) with p D 2: q q a12 a22 an2 C b12 b22 bn2 s Âq Ã2 q Ä .a1 C b1 /2 a22 C C an2 C b22 C C bn2 r .a1 C b1 /2 p D .a1
7.4). 9.2 Majorization Any book on inequalities would be incomplete without a discussion of the theory of majorization. Although the theory does not make extensive use of visual techniques it does have important applications in geometry. As ✐ ✐ ✐ ✐ ✐ ✐ “MABK007-09” — 2009/1/13 — 21:50 — page 140 — #4 ✐ 140 ✐ CHAPTER 9. Additional topics A. Marshall and I. Olkin note in the Preface to their seminal work Inequalities: Theory of Majorization and its Applications [Marshall and Olkin,
