Continued Fractions.

Front Cover -- Continued Fractions -- Copyright Page -- Contents -- Preface -- Chapter 1. Expansion of Rational Fractions -- 1.1 Introduction -- 1.2 Definitions and Notation -- 1.3 Expansion of Rational Fractions -- 1.4 Expansion of Rational Fractions (General Discussion) -- 1.5 Convergents and Their Properties -- 1.6 Differences of Convergents -- 1.7 Some Historical Comments -- Chapter 2. Diophantine Equations -- 2.1 Introduction -- 2.2 The Method Used Extensively by Euler -- 2.3 The Indeterminate Equation ax - by = ±1 -- 2.4 The General Solution of ax - by = c, (a, b) = 1 -- 2.5 The General Solution of ax + by = c, (a, b) = 1 -- 2.6 The General Solution of Ax ± By = ±C -- 2.7 Sailors, Coconuts, and Monkeys -- Chapter 3. Expansion of Irrational Numbers -- 3.1 Introduction -- 3.2 Preliminary Examples -- 3.3 Convergents -- 3.4 Additional Theorems on Convergents -- 3.5 Some Notions of a Limit -- 3.6 Infinite Continued Fractions -- 3.7 Approximation Theorems -- 3.8 Geometrical Interpretation of Continued Fractions -- 3.9 Solution of the Equation x2 = ax + 1 -- 3.10 Fibonacci Numbers -- 3.11 A Method for Calculating Logarithms -- Chapter 4. Periodic Continued Fractions -- 4.1 Introduction -- 4.2 Purely Periodic Continued Fractions -- 4.3 Quadratic Irrationals -- 4.4 Reduced Quadratic Irrationals -- 4.5 Converse of Theorem 4.1 -- 4.6 Lagrange's Theorem -- 4.7 The Continued Fraction for N -- 4.8 Pell's Equation, x2 - Ny2 = ±1 -- 4.9 How to Obtain Other Solutions of Pell's Equation -- Chapter 5. Epilogue -- 5.1 Introduction -- 5.2 Statement of the Problem -- 5.3 Hurwitz' Theorem -- 5.4 Conclusion -- Appendix I. Proof That x2 - 3y2 = - 1 Has No Integral Solutions -- Appendix II. Some Miscellaneous Expansions -- Solutions to Problems -- References -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.