Elements of Higher Mathematics by Frans Keune
Elements of Higher Mathematics - Table of Contents
PART I & II: RULES & FOUNDATIONS
- 1. The Tower of Hanoi
- 2. Intuitive Set Theory
- 3. Mathematical Structure
- 4. The Natural Numbers and Peano Axioms
- 5. Counting and Iteration
- 6. Iteration Mechanics
- 7. The Integers Construction
- 8. Numeral Systems
- 9. The Rational Numbers
PART III: INVESTIGATIONS & APPLICATIONS
- 10. Fundamental Theorem of Arithmetic
- 11. Combinatorics Principles
- 12. Permutations and Symmetric Groups
- 13. Modular Arithmetic and Congruences
- 14. Quadratic Residues and Reciprocity
- 15. Prime Tests and Factorization Algorithms
PART IV & V: COMPLETIONS & EXTENSIONS
- 16. Limits and Sequences
- 17. The Real Numbers Completion
- 18. The p-Adic Numbers and Valuations
- 19. The Complex Numbers Field
- 20. Quadratic Extensions of Q
- 21. Quadratic Numbers and Ring Structures
What You Will Learn in Elements of Higher Mathematics
Elements of Higher Mathematics by Frans Keune is an authoritative, mathematically rigorous textbook published by Springer that introduces undergraduate students to the structural foundations of modern pure mathematics. This comprehensive volume guides readers through set theory axioms, Peano arithmetic, numeral systems, the Fundamental Theorem of Arithmetic, combinatorics, modular congruences, quadratic residues, primality testing, limit completions, p-adic numbers, and quadratic field extensions with exceptional structural logic and clarity.
Designed for undergraduate mathematics majors, secondary math educators, and self-study learners, this foundational text bridges intuitive problem-solving and rigorous axiomatic analysis. Keune details step-by-step mathematical proofs for induction principles, permutation cycles, real number completions, p-adic valuations, and field adjunctions over the rational numbers. Whether you are studying algebraic number theory, transitioning from calculus to rigorous analysis, or exploring discrete algebraic structures, this book offers an indispensable roadmap.
Recognized globally for its clear pedagogical layout and emphasis on number construction, it stands as one of the best introductory books on higher mathematics and number foundations. By connecting basic puzzle logic directly to advanced field extensions and topological completions, it serves as an essential tool for mastering foundational mathematics and proof techniques with confidence.
Book Details & Specifications
Title:
Elements of Higher Mathematics by Frans Keune
Publisher:
Radboud University Press
Year:
2024
Pages:
502
Type:
PDF
Language:
English
ISBN-10 #:
9493296032
ISBN-13 #:
978-9493296039
License:
CC BY-NC-ND 4.0
Amazon:
Amazon
About the Author: Frans Keune
The author Frans Keune
is a Dutch mathematician and professor at "Radboud University Nijmegen", known for his expertise in "higher mathematics" and decades of teaching. He specializes in algebra, number theory, and the foundations of mathematics, making abstract concepts clear and accessible for students. His work, "Elements of Higher Mathematics", introduces fundamental ideas such as "proof techniques", "sets and functions", "mathematical logic", and structures in modern mathematics. The text balances theory with clarity, preparing students for further exploration in abstract mathematics.
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