MATHEMATICS V2000, Fall 2016

Introduction to Higher Mathematics

Room: 407

Time: 2:40-3:55, Tuesday and Thursday

Instructor: Michael Harris

Office Hours: Tuesday 10-11, Thursday, 11-12 (exceptionally 10-11 September 15)

and by appointment, room number 521

Teaching Assistant: J. Li (see Courseworks site for contact information)

Provisional syllabus: Each of the topics listed below will occupy roughly 1-2 weeks of course time.

1. Sets and functions

2. Relations and modular arithmetic

3. Propositional logic

4. Mathematical induction

5. Divisibility

6. Sequences, limits, and continuity

7. Cardinality (finite and infinite sets)

8. Real numbers

9. Complex numbers (time permitting)

What the course is about: Math V2000 is two courses in one: an introduction to some of the basic notions of theoretical mathematics and an introduction to the methods of formal mathematical reasoning. By the end of the course students should be able to read and write simple proofs and to use the language of sets, relations, functions, and functions appropriately. They should also be familiar with the basic properties of integers, limits, and real numbers, and with the notion of cardinality of finite and infinite sets.

Prerequisites: There are no formal prerequisites. Most students will have had Calculus III and/or Linear Algebra, but these are not requirements.

Much of the course is devoted to material most students will have seen before. Thus the emphasis is mainly on thinking carefully about familiar notions rather than on acquiring new notions.

Textbook: Dumas and McCarthy, Transition to Higher Mathematics (available online from the authors as a pdf file)

Supplementary text: Reading, Writing, and Proving, by Ulrich Daepp and Pamela Gorkin (available through the Columbia network). You may also find it useful to consult the handout Introduction to mathematical arguments by Michael Hutchings (on Professor Thaddeus's home page).

It is strongly recommended that you read the chapters of the textbook before the class; chapters from the Daepp-Gorkin book are suggested for additional background.

Class and reading schedule (tentative)

Date	Topics (chapters in Dumas-McCarthy)	Supplementary Text (Daepp-Gorkin)
Sept 6, 8	Logic and sets (1.1-1.2)	Chapters 1, 2, 6-8
Sept 13, 15	Functions, relations (1.3-1.7, 2.1-2.3)	Chapters 10, 14, 4
Sept 20, 22	Modular arithmetic, propositional logic (2.5, 3.1-3.2)	Chapters 27, 3
Sept 27, 29	Propositional logic, continued (3.2-3.4)	Chapter 4
Oct 4, 6	Logic and quantifiers (3.4) and review	Review the chapters listed above
Oct 11, 13	First midterm, review of midterm, proof strategies (3.5)
Oct 18, 20	Proofs by induction, polynomials (4.1-4.3)	Chapter 18
Oct 25, 27	Divisibility, Fermat's little theorem (7.1-7.4)	Chapters 28, 27 (again),
Nov 1, 3	Sequences, limits, continuity (5.1-5.2)	Chapters 19, 20
Nov 10	Review
Nov 15, 17	Second midterm; cardinality (6.1-6.2)	Chapter 22
Nov 22	Cardinality, real numbers (6.3-6.4, chapter 8)	Chapters 23, 12
Nov 29, Dec 1	Real numbers (chapter 8)	Chapter 13
Dec 6, 8	Complex numbers (chapter 9) and review

Midterms: October 11, November 15 (in class)

Final: date to be announced

Practice exams and notes

Homework will be graded and will count for 30% of the final grade. Late homework will be penalized, unless the student has a valid excuse. On the other hand, students may resubmit revised homework for regrading — before solution sheets are posted, of course — and either the original grade or the average of the two grades (whichever is higher) will be recorded.

The lowest grade (of 11 assignments) will be dropped.

Grades will be computed as follows:

Homework: 30%

Final exam: 35%

Bonus homework or exposition: 5%

Midterms: 15% each

Homework assignments

1st week (due September 15)

2nd week (due September 22)

3rd week (due September 29)

4th week (due Tuesday October 11) (NOTE THE DATE!)

5th week (due October 20)