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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| Weitere Verfasser: | , |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge :
Cambridge University Press,
2012.
|
| Schriftenreihe: | Dolciani mathematical expositions.
|
| Schlagworte: | |
| Online-Zugang: | DE-862 DE-863 |
| Zusammenfassung: | 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. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs--From publisher description. |
| Beschreibung: | Title from publishers bibliographic system (viewed on 30 Jan 2012). |
| Beschreibung: | 1 online resource |
| ISBN: | 9781614442028 1614442029 |
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| 505 | 0 | |a 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 | |
| 505 | 8 | |a 2.1 Three examples2.2 Chebyshev�s inequality -- 2.3 The AM-GM inequality for three numbers -- 2.4 Guha�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 Simpson�s paradox -- 2.8 Chebyshev�s inequality revisited -- 2.9 Schur�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 | |
| 505 | 8 | |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 | |
| 505 | 8 | |a 3.4 The Steiner-Lehmus theorem3.5 Challenges -- CHAPTER 4 Using incircles and circumcircles | |
| 520 | |a 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. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs--From publisher description. | ||
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| contents | 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 Chebyshev�s inequality -- 2.3 The AM-GM inequality for three numbers -- 2.4 Guha�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 Simpson�s paradox -- 2.8 Chebyshev�s inequality revisited -- 2.9 Schur�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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| dewey-search | 515.26 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| indexdate | 2025-04-11T08:37:34Z |
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| series2 | Dolciani Mathematical Expositions ; |
| spelling | When Less is More : Visualizing Basic Inequalities / Claudi Alsina, Roger B. Nelsen. Cambridge : Cambridge University Press, 2012. 1 online resource text txt rdacontent computer c rdamedia online resource cr rdacarrier Dolciani Mathematical Expositions ; v. 36 Title from publishers bibliographic system (viewed on 30 Jan 2012). 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 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. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs--From publisher description. Inequalities (Mathematics) http://id.loc.gov/authorities/subjects/sh85065985 Visualization. http://id.loc.gov/authorities/subjects/sh85143939 Geometrical drawing. http://id.loc.gov/authorities/subjects/sh85054129 Inégalités (Mathématiques) Visualisation. Dessin géométrique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh MATHEMATICS Geometry General. bisacsh Geometrical drawing fast Inequalities (Mathematics) fast Visualization fast Alsina, Claudi. Nelsen, Roger B. Print version: Alsina, Claudi. When Less Is More : Visualizing Basic Inequalities. Washington : Mathematical Association of America, ©2014 9780883853429 Dolciani mathematical expositions. http://id.loc.gov/authorities/names/n42009859 |
| spellingShingle | When Less is More : Visualizing Basic Inequalities / Dolciani mathematical expositions. 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 Inequalities (Mathematics) http://id.loc.gov/authorities/subjects/sh85065985 Visualization. http://id.loc.gov/authorities/subjects/sh85143939 Geometrical drawing. http://id.loc.gov/authorities/subjects/sh85054129 Inégalités (Mathématiques) Visualisation. Dessin géométrique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh MATHEMATICS Geometry General. bisacsh Geometrical drawing fast Inequalities (Mathematics) fast Visualization fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85065985 http://id.loc.gov/authorities/subjects/sh85143939 http://id.loc.gov/authorities/subjects/sh85054129 |
| title | When Less is More : Visualizing Basic Inequalities / |
| title_auth | When Less is More : Visualizing Basic Inequalities / |
| title_exact_search | When Less is More : Visualizing Basic Inequalities / |
| title_full | When Less is More : Visualizing Basic Inequalities / Claudi Alsina, Roger B. Nelsen. |
| title_fullStr | When Less is More : Visualizing Basic Inequalities / Claudi Alsina, Roger B. Nelsen. |
| title_full_unstemmed | When Less is More : Visualizing Basic Inequalities / Claudi Alsina, Roger B. Nelsen. |
| title_short | When Less is More : |
| title_sort | when less is more visualizing basic inequalities |
| title_sub | Visualizing Basic Inequalities / |
| topic | Inequalities (Mathematics) http://id.loc.gov/authorities/subjects/sh85065985 Visualization. http://id.loc.gov/authorities/subjects/sh85143939 Geometrical drawing. http://id.loc.gov/authorities/subjects/sh85054129 Inégalités (Mathématiques) Visualisation. Dessin géométrique. MATHEMATICS Calculus. bisacsh MATHEMATICS Mathematical Analysis. bisacsh MATHEMATICS Geometry General. bisacsh Geometrical drawing fast Inequalities (Mathematics) fast Visualization fast |
| topic_facet | Inequalities (Mathematics) Visualization. Geometrical drawing. Inégalités (Mathématiques) Visualisation. Dessin géométrique. MATHEMATICS Calculus. MATHEMATICS Mathematical Analysis. MATHEMATICS Geometry General. Geometrical drawing Visualization |
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