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An Invitation to Pure Mathematics:
Accelerated Proofs and Problem Solving 2021
Instructor: Clark Barwick
Contents
Prev
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Contents
Prev
Up
Next
Front Matter
Preface
Acknowledgements
1
Foundations of mathematics
A first look at the theory of sets
Logical Statements
Conjunction
Intersection
Disjunction
Union
Negation
Complements
Conditionals
Logical equivalence
Proof by Contradiction
Quantifiers
Natural numbers
A first look at induction
Week 1 Exercises, due 28 September 2021
2
Maps and relations
Cartesian products
Maps
Injective and surjective
Cardinalities of finite sets
Images and preimages
Equivalence Relations
Week 2 Exercises, due 5 October 2021
3
Numbers
The integers
The rational numbers
Ordered fields
Completeness and the real numbers
Week 3 Exercise, due 12 October 2021
4
Induction and inequalities
Preliminaries on summation notation
Induction
Roots
AM/GM Inequality and Cauchy-Schwarz
Some sample exercises
Week 4 exercises, due 19 October 2021
5
Sequences and limits
Sequences and the definition of limit
Rules for Limits
Infinite limits
The Monotone Convergence Theorem
Subsequences
Cauchy sequences
Week 5 Exercises, due 26 October 2021
Week 6 Exercises, due 2 November 2021
6
Decimals and Series
Decimals
Infinite Series
The number \(e\)
Irrationality of \(e\)
Week 7 Exercises, due 9 November 2021
7
Complex Numbers
Complex Numbers
Polar form
Geometry and arithmetic of complex numbers
Exponential form
Roots of Unity
Sample exercises
Introduction
Factoring Polynomials
Real Polynomials
Root-Coefficient Theorem
Week 8 Exercises, due 16 September 2021
8
Multiplicative theory of integers
Remainders and divisibility
The GCD and the Euclidean Algorithm
Corollaries of Bezout's Identity and the Linear Combination Lemma
The Fundamental Theorem of Arithmetic (FTA)
Finding divisors via FTA
LCM and GCD via prime factorizations
Powers
Application: Pythagorean Triples*
Week 9 Exercises, due 23 November 2021
9
Modular Arithmetic
Arithmetic modulo \(m\)
Solving linear equations modulo \(m\)
\(\mathbb{Z}/m\)
Fermat's Little Theorem
Authored in PreTeXt
Acknowledgements
Acknowledgements
This text synthesizes a lot of material and approaches from previous lecturers of APPS and PPS, including: Jonas Azzam, Tony Carbery,Dougal Davis, Richard Gratwick, Milena Hering, Ana Rita Pires, and Chris Sangwin.