Preface; Introduction; 1. Representing positive numbers as lengths of segments; 2. Representing positive numbers as areas or volumes; 3. Inequalities and the existence of triangles; 4. Using incircles and circumcircles; 5. Using reflections; 6. Using rotations; 7. Employing non-isometric transformations; 8. Employing graphs of functions; 9. Additions topics; Solutions to the challenges; Selected open challenges for visualizing inequalities; Symbols and notation; References; Index.
Introduces the richness and variety of inequalities in mathematics using illustration and visualisation.
Claudi Alsina received his B.A. and Ph.D. in mathematics from the University of Barcelona. His post-doctoral studies were at the University of Massachusetts, Amherst. He is Professor of Mathematics at the Technical University of Catalonia and has developed a wide range of international activities, research papers, publications and hundreds of lectures on mathematics and mathematics education. His latest books include Associative Functions: Triangular Norms and Copulas with M. J. Frank and B. Schweizer, WSP, 2006; Math Made Visual. Creating Images for Understanding Mathematics with Roger B. Nelsen, MAA, 2006; Vitaminas Matemáticas and El Club de la Hipotenusa, Ariel, 2008. Roger B. Nelsen received his B.A. in mathematics from DePauw University in 1964 and his Ph.D. in mathematics from Duke University in 1969. Roger was elected to Phi Beta Kappa and Sigma Xi. His previous books include Proofs Without Words: Exercises in Visual Thinking, MAA 1993; An Introduction to Copulas, Springer, 1999 (2nd edition 2006); Proofs Without Words II: More Exercises in Visual Thinking, MAA, 2000; and Math Made Visual: Creating Images for Understanding Mathematics (with Claudi Alsina), MAA, 2006.
The book will be valuable for secondary and collegiate teachers as
a source of inspiration for preparing lessons. It is also
recommended for self study to students having adquate knowledge of
geometry and a sufficiently rigorous mind."" - Zentralblatt fur
Mathematik
""The Dolciani Mathematical Expositions series has once again
provided an excellent text in the publication When Less is More:
Visualizing Basic Inequalities. Alsina and Nelsen offer readers
visual arguments for many historical inequalities, allowing a
deeper understanding of those inequalities. Perhaps even more
significant than the deeper meaning of these particular
inequalities is the fact that the authors present a methodology for
producing such visual arguments. Each of the nine chapters features
a different approach and concludes with ""challenge"" problems
(with solutions) to support the understanding of the chapter's
method. The text is highly recommended for instructors as a course
supplement and could prove an excellent undergraduate text for a
course in inequalities."" - Matthew J. Haines (Minneapolis, MN)
Mathematical Reviews
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