Mathematics Starts Here by Aguadé , Jaume;

Long description:

This textbook compiles lessons from the Foundations of Mathematics course at the Autonomous University of Barcelona. It is designed to introduce students beginning their mathematics studies to the major overarching ideas in the field—such as symmetry, structure, equivalence, abstraction, generalization, analogy, and computability. It also aims to familiarize students with the structure of mathematical discourse—how mathematics is written—and to support them in the challenging task of developing the rigor and creativity required of a professional mathematician.

In this work, readers will encounter the big ideas and fundamental principles that shape mathematical discourse, including logic, group theory, Peano's axioms, Russell's paradox, the prime number theorem, RSA cryptography, Euler's formula, and more. After each part, a set of carefully selected exercises helps students to gain greater exposure to—and confidence in—the rigors of mathematics.

Guided by aesthetic and playful principles, this book offers a gentle, passionate, and joyful approach to some of the most essential concepts in mathematics. It provides enjoyment for teachers, students, and anyone with an inclination toward the subject, and is particularly appealing to high-achieving high school students seeking to delve into college-level mathematics.

Table of Contents:

Part I-Logic, Numbers, and Structure of Mathematical Discourse.- Propositional Logic.- First-Order Logic.- The Natural Numbers: Peano's Axioms and Induction.- The Natural Numbers: Recursion.- Mathematical Discourse.- Gödel's Theorems.- Exercises on Logic and Natural Numbers.- Part II-Set Theory.- The ZFC Axioms.- Products, Relations, Functions.- The Quotient Set.- Finite, Infinite, Infinities.- Exercises on Sets and Functions.- Part III-A Glimpse into Group Theory.- Permutations.- The Concept of a Group.- The Alternating Group.- Conjugation, Normal Subgroups and Kernels.- The Cube Group.- Exercises on Groups.- Part IV-Arithmetic.- The Integers and the Rationals.- Divisibility.- Z is a PID.- The Prime Numbers.- Modular Arithmetic.- The Rings Z/(m) and the Finite Fields.- Euler's phi function.- Public-Key Cryptography.- Exercises on Arithmetic.- Part V: Polynomials.- Polynomials: Basic Concepts.- k|x| Closely Resembles Z.- Multiplicity, Irreducible Polynomials and Roots of Unity.- Solving Polynomials Equations.- Every Nonconstant Polynomial Has a Root.- Exercises on Polynomials.- Part VI-Complex Numbers.- Three Definitions of the Complex Numbers.- The Fundamental Theorem of Algebra.- The Most Beautiful Formula: e^in+1=0.- Exercises on Complex Numbers.- Index.