Introduction
to Higher Mathematics
Room: 407
Time: 2:403:55, Tuesday and Thursday
Instructor: Michael Harris
Office
Hours: Tuesday 1011, Thursday,
1112 (exceptionally 1011 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 12 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
DaeppGorkin book are suggested for additional background.
Class and reading schedule (tentative)
Date 
Topics (chapters
in DumasMcCarthy) 
Supplementary Text (DaeppGorkin) 
Sept 6, 8 
Logic and sets (1.11.2) 
Chapters 1, 2, 68 
Sept 13, 15 
Functions, relations (1.31.7, 2.12.3) 
Chapters 10, 14, 4 
Sept 20, 22 
Modular arithmetic, propositional logic (2.5, 3.13.2) 
Chapters 27, 3 
Sept 27, 29 
Propositional logic, continued (3.23.4) 
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 (4.14.3) 
Chapter 18 
Oct 25, 27 
Divisibility, Fermat's little theorem (7.17.4) 
Chapters 28, 27 (again), 
Nov 1, 3 
Sequences, limits, continuity (5.15.2) 
Chapters 19, 20 
Nov 10 
Review 

Nov 15, 17 
Second midterm; cardinality (6.16.2) 
Chapter 22 
Nov 22 
Cardinality, real numbers (6.36.4, 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
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)
6th week (due Tuesday October 25) (NOTE THE DATE!)
7th week (due October 27)
8th week (due November 3)
9th week (due November 17)
10th week (due December 1)
11th week (due December 8)
Solution
sheets
Return to Michael Harris's home
page