When Less is More : Visualizing Basic Inequalities /

The proofs in When Less is More are in the spirit of proofs without words, though most require at least a few words. The first inequalities presented in the book, such as the inequalities between the harmonic, geometric, and arithmetic mean, are familiar from analysis, but are given geometric proofs...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Otros Autores: Alsina, Claudi, Nelsen, Roger B.
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Cambridge : Cambridge University Press, 2012.
Colección:Dolciani mathematical expositions.
Temas:
Inequalities (Mathematics)
Visualization.
Geometrical drawing.
Inégalités (Mathématiques)
Visualisation.
Dessin géométrique.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
MATHEMATICS > Geometry > General.
Geometrical drawing
Visualization
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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 examples2.2 Chebyshevâ€?s inequality
  • 2.3 The AM-GM inequality for three numbers
  • 2.4 Guhaâ€?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 Simpsonâ€?s paradox
  • 2.8 Chebyshevâ€?s inequality revisited
  • 2.9 Schurâ€?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_cMore 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 theorem3.5 Challenges
  • CHAPTER 4 Using incircles and circumcircles

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