Number and Proof

Beschreibung

Number and Proof invites readers on a rigorous and engaging journey from the most familiar objects in mathematics, the natural numbers, to some of the central ideas of modern number theory. Written for undergraduate students in mathematics, science, and philosophy, the book develops proof-writing skills through active exploration while asking deeper questions: What are numbers? What does it mean to prove something? And how certain can mathematical knowledge really be? Drawing on history, logic, and philosophy as well as mathematics itself, the text illuminates the foundations underlying arithmetic and mathematical reasoning.

Beginning with the Peano axioms and the construction of the integers, the book guides readers carefully through the development of rigorous arguments before introducing the classic results of elementary number theory. Along the way, students encounter prime factorization, modular arithmetic, Diophantine equations, quadratic reciprocity, and finally the RSA public-key cryptosystem. Extensive exercises, many designed to encourage discovery and conjecture before formal proof, make the text equally suitable for classroom use and independent study.

Distinctive in both scope and style, the text treats mathematics not merely as a collection of results but as a human intellectual enterprise shaped by millennia of debate about logic, certainty, and the nature of knowledge. By connecting foundational questions with beautiful mathematical ideas and practical applications, the author offers readers an accessible introduction to proof, a rewarding encounter with number theory, and an invitation to explore the wider landscape of mathematics.

Number and Proof invites readers on a rigorous and engaging journey from the most familiar objects in mathematics, the natural numbers, to some of the central ideas of modern number theory. Written for undergraduate students in mathematics, science, and philosophy, the book develops proof-writing skills through active exploration while asking deeper questions: What are numbers? What does it mean to prove something? And how certain can mathematical knowledge really be? Drawing on history, logic, and philosophy as well as mathematics itself, the text illuminates the foundations underlying arithmetic and mathematical reasoning.

Beginning with the Peano axioms and the construction of the integers, the book guides readers carefully through the development of rigorous arguments before introducing the classic results of elementary number theory. Along the way, students encounter prime factorization, modular arithmetic, Diophantine equations, quadratic reciprocity, and finally the RSA public-key cryptosystem. Extensive exercises, many designed to encourage discovery and conjecture before formal proof, make the text equally suitable for classroom use and independent study.

Distinctive in both scope and style, the text treats mathematics not merely as a collection of results but as a human intellectual enterprise shaped by millennia of debate about logic, certainty, and the nature of knowledge. By connecting foundational questions with beautiful mathematical ideas and practical applications, the author offers readers an accessible introduction to proof, a rewarding encounter with number theory, and an invitation to explore the wider landscape of mathematics.

Autor*in

Lawrence Susanka

Themen in »Number and Proof«

undergraduate number theory Law of the Excluded Middle Peano axioms linear Diophantine equations prime factorization Fermat's Little Theorem Chinese Remainder Theorem primitive roots Lagrange's Theorem Wilson's Theorem Euler's Theorem Gauss's Theorem polynomial congruences quadratic reciprocity public key encryption

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