Elements of Higher Mathematics by Frans Keune

Book 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

About this book :-
"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 Detail :-
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 Author :-
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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