Irrationality and transcendence in number theory. (2022)
- Record Type:
- Book
- Title:
- Irrationality and transcendence in number theory. (2022)
- Main Title:
- Irrationality and transcendence in number theory
- Further Information:
- Note: David Angell.
- Authors:
- (Mathematics), Angell, David
- Contents:
- 1. Introduction. 1.1. Irrational Surds. 1.2. Irrational Decimals. 1.3. Irrationality of the Exponential Constant. 1.4. Other Results, and Some Open Questions. Exercises. Appendix: Some Elementary Number Theory. 2. Hermite’s Method. 2.1. Irrationality of er . 2.2. Irrationality of π. 2.3. Irrational values of trigonometric functions. Exercises. Appendix: Some Results of Elementary Calculus. 3. Algebraic & Transcendental Numbers. 3.1. Definitions and Basic Properties. 3.2. Existence of Transcendental Numbers. 3.3. Approximation of Real Numbers by Rationals. 3.4. Irrationality of (3) : a sketch. Exercises. Appendix 1: Countable and Uncountable Sets. Appendix 2: The Mean Value Theorem. Appendix 3: The Prime Number Theorem. 4. Continued Fractions. Definition and Basic Properties. 4.2. Continued Fractions of Irrational Numbers. 4.3. Approximation Properties of Convergents. 4.4. Two important Approximation Problems. 4.5. A "Computational" Test for Rationality. 4.6. Further Approximation Properties of Convergents. 4.7. Computing the Continued Fraction of an Algebraic Irrational. 4.8. The Continued Fraction of e. Exercises. Appendix 1: A Property of Positive Fractions. Appendix 2: Simultaneous Equations with Integral Coefficients. Appendix 3: Cardinality of Sets of Sequences. Appendix 4: Basic Musical Terminology. 5. Hermite’s Method for Transcendence. 5.1. Transcendence of e. 5.2. Transcendence of π. 5.3. Some more Irrationality Proofs. 5.4. Transcendence of ea .5.5. Other Results.1. Introduction. 1.1. Irrational Surds. 1.2. Irrational Decimals. 1.3. Irrationality of the Exponential Constant. 1.4. Other Results, and Some Open Questions. Exercises. Appendix: Some Elementary Number Theory. 2. Hermite’s Method. 2.1. Irrationality of er . 2.2. Irrationality of π. 2.3. Irrational values of trigonometric functions. Exercises. Appendix: Some Results of Elementary Calculus. 3. Algebraic & Transcendental Numbers. 3.1. Definitions and Basic Properties. 3.2. Existence of Transcendental Numbers. 3.3. Approximation of Real Numbers by Rationals. 3.4. Irrationality of (3) : a sketch. Exercises. Appendix 1: Countable and Uncountable Sets. Appendix 2: The Mean Value Theorem. Appendix 3: The Prime Number Theorem. 4. Continued Fractions. Definition and Basic Properties. 4.2. Continued Fractions of Irrational Numbers. 4.3. Approximation Properties of Convergents. 4.4. Two important Approximation Problems. 4.5. A "Computational" Test for Rationality. 4.6. Further Approximation Properties of Convergents. 4.7. Computing the Continued Fraction of an Algebraic Irrational. 4.8. The Continued Fraction of e. Exercises. Appendix 1: A Property of Positive Fractions. Appendix 2: Simultaneous Equations with Integral Coefficients. Appendix 3: Cardinality of Sets of Sequences. Appendix 4: Basic Musical Terminology. 5. Hermite’s Method for Transcendence. 5.1. Transcendence of e. 5.2. Transcendence of π. 5.3. Some more Irrationality Proofs. 5.4. Transcendence of ea .5.5. Other Results. Exercises. Appendix 1: Roots and Coefficients of Polynomials. Appendix 2: Some Real and Complex Analysis. Appendix 3: Ordering Complex Numbers. 6. Automata and Transcendence. 6.1. Deterministic Finite Automata. 6.2 Mahler’s Transcendence Proof. 6.3 A More General Transcendence Result. 6.4. A Transcendence Proof for the Thue Sequence. 6.5. Automata and Functional Equations. 6.6. Conclusion. Exercises. Appendix 1: Alphabets, Languages and DFAs. Appendix 2: Some Results of Complex Analysis. Appendix 3: A Result on Linear Equations. 7. Lambert’s Irrationality Proofs. 7.1. Generalised Continued Fractions. 7.2. Further Continued Fractions. Exercises. Appendix: Some Results from Elementary Algebra and Calculus. Hints for Exercises. Bibliography. Index. … (more)
- Edition:
- 1st
- Publisher Details:
- Boca Raton : Chapman & Hall/CRC
- Publication Date:
- 2022
- Extent:
- 1 online resource, illustrations (black and white)
- Subjects:
- 512.73
Irrational numbers
Transcendental numbers
Number theory - Languages:
- English
- ISBNs:
- 9781000523782
9781000523737
9781003111207 - Related ISBNs:
- 9780367628376
- Notes:
- Note: Includes bibliographical references and index.
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- Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
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- Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.
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- Available online (eLD content is only available in our Reading Rooms) ↗
- Physical Locations:
- British Library HMNTS - ELD.DS.658326
- Ingest File:
- 07_030.xml