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

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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