Jason Riedy
Fall semester, 2008
These pages are available as PDF, either as one growing PDF document for the
entirety or as individual documents for each session’s notes.
If you have difficulties viewing these or have particular accessibility needs, please mail
me at jason@acm.org.
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
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