Contents
I  Introduction
1 Syllabus
 1.1 Concepts of Modern Mathematics I
 1.2 Goals
 1.3 Instructor: Jason Riedy
 1.4 Text
 1.5 Grading
 1.6 On homework
 1.7 Submitting homework
2 Syllabus schedule
II  Notes for chapters 1 and 2
3 Notes for 18 August
 3.1 Syllabus and class mechanics
 3.2 Introductions
 3.3 First ”homework”
 3.4 Problem solving
4 Notes for 20 August
 4.1 Review
 4.2 Today’s goal: Problem solving principles
 4.3 Pólya’s principles
 4.4 Two closely related tactics, guessing and making a list
 4.5 Next time: More problem solving ideas.
 4.6 Homework
5 Notes for 22 August
 5.1 Review
 5.2 New tactic: Drawing a diagram
 5.3 Homework
6 Solutions for first week’s assignments
 6.1 Problem Set 1.1
 6.2 Example like 1.3 with no solution
 6.3 Problem Set 1.2
 6.4 Consider solving Example 1.3 with a table
 6.5 More in Problem Set 1.2
7 Notes for 25 August
 7.1 Review
 7.2 Draw a diagram, follow dependencies
 7.3 Look for a pattern
 7.4 Patterns and representative special cases
 7.5 Homework
8 Notes for 27 August
 8.1 Review
 8.2 Ruling out possibilities
 8.3 The pigeonhole principle
 8.4 Mathematical reasoning
 8.5 Next time: structures and kinds of proofs
 8.6 Homework
9 29 August: Review of previous notes
10 Solutions for second week’s assignments
 10.1 Patterns: The 87th digit past the decimal in 1/7?
 10.2 Patterns: Units digit of \mathbf{{3}^{100}}
 10.3 Problem set 1.3
 10.4 Problem set 1.4
 10.5 Inductive or deductive?
11 Notes for 1 September
 11.1 Review
 11.2 Proof
 11.3 Direct proof
 11.4 Proof by contrapositives
 11.5 Homework
12 Notes for 3 September
 12.1 Proof review
 12.2 Inductive proof
 12.3 Starting with set theory
 12.4 Language of set theory
 12.5 Basic definitions
 12.6 Translating sets into (and from) English
 12.7 Next time: Relations between and operations on sets
 12.8 Homework
13 Notes for 8 September
 13.1 Review
 13.2 Relations and Venn diagrams
 13.3 Translating relations into (and from) English
 13.4 Consequences of the set relation definitions
 13.5 Operations
 13.6 Homework
14 Solutions for third week’s assignments
 14.1 Induction: Sum of first n integers
 14.2 Problem set 2.1 (p83)
15 Notes for 10 September
 15.1 Review
 15.2 From sets to whole numbers
 15.3 Homework
16 Notes for 12 September
 16.1 Review
 16.2 Addition of whole numbers
 16.3 Subtraction of whole numbers
 16.4 Multiplication of whole numbers
 16.5 Monday: Division and exponentials
 16.6 Homework
17 Solutions for fourth week’s assignments
 17.1 Problem set 2.2
 17.2 Problem set 2.3
 17.3 Write 2 + 3 using disjoint sets.
 17.4 Illustrate 2 + 3 using Peano arithmetic.
 17.5 Problem set 2.4
 17.6 Illustrate 2 ⋅ 3 using Peano arithmetic. You do not need to expand addition.
 17.7 Illustrate (1 ⋅ 2) ⋅ 3 = 1 ⋅ (2 ⋅ 3) using a volume of size six.
18 Notes for the fifth week: review
 18.1 Review
 18.2 Problem solving
 18.3 Set theory
 18.4 Operations and whole numbers
19 First exam and solutions
III  Notes for chapters 3, 4, and 5
20 Notes for the sixth week: digits, bases, and operations
 20.1 Positional Numbers
 20.2 Converting Between Bases
 20.3 Operating on Numbers
 20.4 Homework
21 Solutions for sixth week’s assignments
 21.1 Problem set 3.1
 21.2 Problem set 3.2
 21.3 Problem set 3.3
 21.4 Problem set 3.4
22 Notes for the seventh week: primes, factorization, and modular arithmetic
 22.1 Divisibility
 22.2 Primes
 22.3 Factorization
 22.4 Modular Arithmetic
 22.5 Divisibility Rules
 22.6 Homework
23 Solutions for seventh week’s assignments
 23.1 Problem set 4.1
 23.2 Two diagrams
 23.3 Problem set 4.2
 23.4 A familiar incomplete integer
24 Notes for the eighth week: GCD, LCM, and ax + by = c
 24.1 Modular arithmetic
 24.2 Divisibility rules
 24.3 Greatest common divisor
 24.4 Least common multiple
 24.5 Euclidean GCD algorithm
 24.6 Linear Diophantine equations
 24.7 Homework
25 Solutions for eighth week’s assignments
 25.1 Problem set 4.3
 25.2 Computing GCDs
 25.3 Computing LCMs
 25.4 Linear Diophantine equations
26 Notes for the ninth week: ax + by = c, fractions
 26.1 Linear Diophantine equations
 26.2 Into real numbers
 26.3 Rational numbers
 26.4 Review of rational arithmetic
 26.5 Complex fractions
 26.6 Homework
27 Solutions for ninth week’s assignments
 27.1 Diophantine equations
 27.2 Problem set 6.1
 27.3 Problem set 6.2
 27.4 Problem set 6.3
28 Notes for the tenth week: Irrationals and decimals
 28.1 Real numbers
 28.2 Exponents and roots
 28.3 Decimal expansions and percentages
 28.4 Fixed and floating-point arithmetic
 28.5 Homework
29 Second exam and solutions
30 Third exam, due 1 December
31 Third exam solutions
32 Final exam
IV  Resources
33 Math Lab
34 On-line
 34.1 Educational Standards
 34.2 General mathematics education resources
 34.3 Useful software and applications