Elements of mathematics : a problem-centered approach to history and foundations / / Gabor Toth

• Record Nr.: UNISA996466389803316
• Autore: Tóth Gábor <1964->
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-75051-5
• Descrizione fisica: 1 online resource (534 pages)
• Collana: Undergraduate Texts in Mathematics
• Disciplina: 510
• Soggetti: Mathematics - History; Història de la matemàtica; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: Intro -- Preface -- Why This Book? -- Audience -- The Historical Context -- In Closing: Gelfand's Teaching Legacy -- Acknowledgment -- Contents -- 0 Preliminaries: Sets, Relations, Maps -- 0.1 Sets -- Exercises -- 0.2 Relations -- Exercise -- 0.3 Maps and Real Functions -- Exercises -- 0.4 Cardinality -- Exercises -- 0.5 The Zermelo-Fraenkel Axiomatic Set Theory* -- Exercise -- 1 Natural, Integral, and Rational Numbers -- 1.1 Natural Numbers -- Exercises -- 1.2 Integers -- Exercises -- 1.3 The Division Algorithm for Integers -- Exercises -- 1.4 Rational Numbers -- Exercises -- 2 Real Numbers -- 2.1 Real Numbers via Dedekind Cuts -- Exercises -- 2.2 Infinite Decimals as Real Numbers -- Exercises -- 2.3 Real Numbers via Cauchy Sequences -- Exercises -- 2.4 Dirichlet Approximation and Equidistribution* -- Exercises -- 3 Rational and Real Exponentiation -- 3.1 Arithmetic Properties of the Limit -- Exercises -- 3.2 Roots, Rational and Real Exponents -- Exercises -- 3.3 Logarithms -- Exercises -- 3.4 The Stolz-Cesàro Theorems -- Exercises -- 4 Limits of Real Functions -- 4.1 Limit Inferior and Limit Superior -- Exercise -- 4.2 Continuity -- Exercise -- 4.3 Differentiability -- Exercises -- 5 Real Analytic Plane Geometry -- 5.1 The Birkhoff Metric Geometry -- Exercise -- 5.2 The Cartesian Model of the Birkhoff Plane -- Exercises -- 5.3 The Cartesian Distance -- Exercise -- 5.4 The Triangle Inequality -- Exercise -- 5.5 ; Lines and Circles -- Exercises -- 5.6 Arc Length on the Unit Circle -- Exercise -- 5.7 The Birkhoff Angle Measure -- Exercises -- 5.8 The Principle of Shortest Distance* -- Exercises -- 5.9 π According to Archimedes* -- Exercise -- 6 Polynomial Expressions -- 6.1 Polynomials -- Exercises -- 6.2 Arithmetic Operations on Polynomials -- Exercises -- 6.3 The Binomial Formula -- Exercises -- 6.4 Factoring Polynomials -- Exercises.; 6.5 The Division Algorithm for Polynomials -- Exercises -- 6.6 Symmetric Polynomials -- Exercises -- 6.7 The Cauchy-Schwarz Inequality -- Exercises -- 7 Polynomial Functions -- 7.1 Polynomials as Functions -- Exercises -- 7.2 Roots of Cubic Polynomials -- Exercises -- 7.3 Roots of Quartic and Quintic Polynomials -- Exercise -- 7.4 Polynomials with Rational Coefficients -- Exercises -- 7.5 Factoring Multivariate Polynomials -- Exercises -- 7.6 The Greatest Common Factor -- Exercise -- 8 Conics -- 8.1 The General Conic -- Exercise -- 8.2 Parabolas -- Exercises -- 8.3 Ellipses -- Exercises -- 8.4 Hyperbolas -- Exercises -- 9 Rational and Algebraic Expressions and Functions -- 9.1 Rational Expressions and Rational Functions -- Exercises -- 9.2 The Partial Fraction Decomposition -- Exercises -- 9.3 Asymptotes of Rational Functions -- Exercises -- 9.4 Algebraic Expressions and Functions, Rationalization -- Exercises -- 9.5 Harmonic, Geometric, Arithmetic, Quadratic Means -- Exercises -- 9.6 The Greatest Integer Function -- Exercises -- 10 Exponential and Logarithmic Functions -- 10.1 The Natural Exponential Function According to Newton -- Exercises -- 10.2 The Bernoulli Numbers* -- Exercise -- 10.3 The Natural Logarithm -- Exercises -- 10.4 The General Exponential and Logarithmic Functions -- Exercise -- 10.5 The Natural Exponential Function According to Euler -- Exercises -- 11 Trigonometry -- 11.1 The Unit Circle S vs. the Real Line R -- Exercise -- 11.2 The Sine and Cosine Functions -- Exercises -- 11.3 Principal Identities for Sine and Cosine -- Exercises -- 11.4 Trigonometric Rational Functions -- Exercises -- 11.5 Trigonometric Limits -- Exercises -- 11.6 Cosine and Sine Series According to Newton -- Exercise -- 11.7 The Basel Problem of Euler* -- Exercise -- 11.8 Ptolemy's Theorem -- Exercise -- Further Reading -- Index.