Excursions in Number Theory, Algebra, and Analysis

This textbook is a modern rendering of a course taught by Ken Ireland in 1972, focusing on the theoretical underpinnings of undergraduate mathematics. It includes excursions into number theory, algebra, and analysis, with 200 'Dialing In problems' for active enquiry. The chapters develop theory to illuminate observations and roadblocks, and examples, exercises, historical anecdotes, and asides add motivation and context. It is suitable for a senior capstone, professional development for practicing teachers, or independent reading, and assumes a background in real analysis and abstract algebra.

Format: Hardback
Length: 177 pages
Publication date: 29 March 2023
Publisher: Springer International Publishing AG

This textbook, originating from a course taught by the late Ken Ireland in 1972, is a comprehensive exploration of the theoretical foundations of undergraduate mathematics. With a focus on interrelationships and hands-on experience, the course aimed to provide students with a modern rendition of Ireland's path of discovery.

The textbook is replete with surprising connections, deep insights, and brilliantly curated invitations to engage in problem-solving at just the right moment. This journey weaves a rich body of knowledge that is ideal for those pursuing studies in mathematics or teaching the subject.

To facilitate active learning, the book begins with a pool of 200 'Dialing In problems,' which serve as a catalyst for inquiry throughout the course. The subsequent chapters develop theory to provide illumination for the observations and roadblocks encountered in these problems, situating them within the broader mathematical landscape. Topics covered include polygons and modular arithmetic, the fundamental theorems of arithmetic and algebra, irrational, algebraic, and transcendental numbers, and Fourier series and Gauss sums.

A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Regular return trips to the Dialing In problems are encouraged, offering opportunities for students to put theory into practice and make lasting connections along the way.

This textbook is suitable for a wide range of audiences, including senior capstone courses, professional development for practicing teachers, and independent reading. It assumes a background in real analysis and abstract algebra, although the most important prerequisite is a willingness to engage with the material and embark on a mathematical journey of discovery.

In conclusion, this textbook offers a comprehensive and engaging exploration of the theoretical underpinnings of undergraduate mathematics. With its surprising connections, deep insights, and problem-solving invitations, it provides students with the tools they need to excel in their studies and pursue careers in mathematics or teaching.

Weight: 594g
Dimension: 181 x 262 x 17 (mm)
ISBN-13: 9783031130168
Edition number: 1st ed. 2023