Elements of Higher Mathematics by Frans Keune
Elements of Higher Mathematics - Table of Contents
PART-I The Rules of the Game
1. The Tower of Hanoi
2. Intuitive Set Theory
3. Structure
PART-II Foundations
4. The Natural Numbers
5. Counting
6. Iteration
7. The Integers
8. Numeral Systems
9. The Rational Numbers
PART-III Investigations and Applications
10. The Fundamental Theorem of Arithmetic
11. Combinatorics
12. Permutations
13. Modular Arithmetic
14. Quadratic Residues
15. Prime Tests and Factorization
PART-IV Completions
16. Limits
17. The Real Numbers
18. The p-Adic Numbers
PART-V Extensions
19. The Complex Numbers
20. Quadratic Extensions of Q
21. Quadratic Numbers
What You Will Learn in Elements of Higher Mathematics
"Elements of Higher Mathematics: Learning Mathematics through Numbers" by "Frans Keune" is a beginner-friendly textbook designed to guide students and self-learners through the foundations of "higher mathematics". The book starts with basic principles, including counting and logical reasoning, and gradually builds toward more advanced concepts. A central focus is the construction of the "number system", covering natural numbers, integers, rationals, reals, and complex numbers. Using "set theory" as a foundation, Keune shows how mathematics grows from simple ideas into rigorous structures, helping readers develop clear mathematical thinking and intuition.
The textbook also introduces important topics such as "combinatorics", "primality tests", quadratic reciprocity, p-adic numbers, continued fractions, and applications in number theory and "cryptography". Each concept is supported with examples, exercises, and problem-solving tasks that encourage active learning. Keune emphasizes understanding proofs and reasoning over rote memorization, making abstract ideas approachable while fostering critical thinking. This structure ensures that readers gradually gain both conceptual knowledge and practical problem-solving skills.
Designed for first-year mathematics students but also suitable for self-study, the book balances theory, applications, and computational exercises. Its clear exposition, structured progression, and interactive examples make it a strong foundation for future studies in mathematics. Freely available under a Creative Commons license, it is an excellent open educational resource for anyone starting their journey in "higher mathematics".
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.
Read or Downloadable Elements of Higher Mathematics
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