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
  3.4.1 Categories
  3.4.2 Pólya’s stages
  3.4.3 Understand the problem
  3.4.4 Devise a plan
  3.4.5 Carry out the plan
  3.4.6 Looking back
4 Notes for 20 August
 4.1 Review
  4.1.1 Problem solving
  4.1.2 Categories
 4.2 Today’s goal: Problem solving principles
 4.3 Pólya’s principles
  4.3.1 Understand the problem
  4.3.2 Divise a plan
  4.3.3 Carry out the plan
  4.3.4 Examine your solution
 4.4 Two closely related tactics, guessing and making a list
  4.4.1 Guessing
  4.4.2 Example 1.3 from the text
  4.4.3 Tabling / Making a list
  4.4.4 Example 1.4 from the text: a counting problem.
  4.4.5 Different example to show bisection
 4.5 Next time: More problem solving ideas.
 4.6 Homework
5 Notes for 22 August
 5.1 Review
  5.1.1 Pólya’s problem solving principles
  5.1.2 Tactic: Guessing
  5.1.3 Tactic: Tabling / Making an orderly list
  5.1.4 Example of creating a list
  5.1.5 Tactic not from the text: Orderly creation of a partial list
 5.2 New tactic: Drawing a diagram
  5.2.1 Understanding the problem
  5.2.2 Devise a plan
  5.2.3 Carry out the plan
  5.2.4 Looking back
 5.3 Homework
6 Solutions for first week’s assignments
 6.1 Problem Set 1.1
  6.1.1 Problem 10
  6.1.2 Problem 13
 6.2 Example like 1.3 with no solution
 6.3 Problem Set 1.2
  6.3.1 Problem 5
  6.3.2 Problem 7
  6.3.3 Problem 8
 6.4 Consider solving Example 1.3 with a table
 6.5 More in Problem Set 1.2
  6.5.1 Problem 6: whoops, not assigned
  6.5.2 Problem 13
  6.5.3 Problem 18
  6.5.4 Problem 20
7 Notes for 25 August
 7.1 Review
 7.2 Draw a diagram, follow dependencies
  7.2.1 Understanding the problem
  7.2.2 Devise a plan
  7.2.3 Carry out the plan
  7.2.4 Looking back
 7.3 Look for a pattern
  7.3.1 Example: What is the last digit of {7}^{100}
 7.4 Patterns and representative special cases
  7.4.1 Sums over Pascal’s triangle
 7.5 Homework
8 Notes for 27 August
 8.1 Review
 8.2 Ruling out possibilities
  8.2.1 Logic puzzles
 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
  10.3.1 Arithmetic progressions: Problem 6
  10.3.2 Problem 9
  10.3.3 Problem 11
  10.3.4 Problem 20
 10.4 Problem set 1.4
  10.4.1 Reducing possibilities: Problem 7
  10.4.2 Logic puzzle: Problem 9
  10.4.3 Pigeonholes: Problem 12
  10.4.4 Pigeonholes: Problem 13
  10.4.5 Pigeonholes: Problem 14
 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)
  14.2.1 Problem 1
  14.2.2 Problem 2
  14.2.3 Problem 4
  14.2.4 Problem 5
  14.2.5 Problem 6
  14.2.6 Problem 27
15 Notes for 10 September
 15.1 Review
  15.1.1 Definitions
  15.1.2 Relations
  15.1.3 Operations
 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
  17.1.1 Problem 1
  17.1.2 Problem 2
  17.1.3 Problem 6
  17.1.4 Problem 13
  17.1.5 Problem 21
  17.1.6 Problem 23
  17.1.7 Why answering problem 32 would be a bad idea.
 17.2 Problem set 2.3
  17.2.1 Problem 2
  17.2.2 Problem 5
  17.2.3 Problem 11
  17.2.4 Problem 24
 17.3 Write 2 + 3 using disjoint sets.
 17.4 Illustrate 2 + 3 using Peano arithmetic.
 17.5 Problem set 2.4
  17.5.1 Problem 5
  17.5.2 Problem 10
  17.5.3 Problem 26
 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
  18.2.1 Understand the problem
  18.2.2 Devise a plan
  18.2.3 Carry out the plan
  18.2.4 Look back at your solution
 18.3 Set theory
  18.3.1 Definitions and mappings
  18.3.2 Cardinality and one-to-one correspondence
 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
  20.2.1 Converting to Decimal
  20.2.2 Converting from Decimal
 20.3 Operating on Numbers
  20.3.1 Multiplication
  20.3.2 Addition
  20.3.3 Subtraction
  20.3.4 Division and Square Root: Later
 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
  26.1.1 In general\mathop{\mathop{…}}
  26.1.2 The other example
 26.2 Into real numbers
  26.2.1 Operator precedence
 26.3 Rational numbers
 26.4 Review of rational arithmetic
  26.4.1 Multiplication and division
  26.4.2 Addition and subtraction
  26.4.3 Comparing fractions
 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
  28.2.1 Positive exponents
  28.2.2 Zero exponent
  28.2.3 Negative exponents
  28.2.4 Rational exponents and roots
  28.2.5 Irrational numbers
 28.3 Decimal expansions and percentages
  28.3.1 Representing rationals with decimals
  28.3.2 The repeating decimal expansion may not be unique!
  28.3.3 Rationals have terminating or repeating expansions
  28.3.4 Therefore, irrationals have non-repeating expansions.
  28.3.5 Percentages as rationals and decimals
 28.4 Fixed and floating-point arithmetic
  28.4.1 Rounding rules
  28.4.2 Floating-point representation
  28.4.3 Binary fractional parts
 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