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)



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



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)

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

1st week           

2nd week           

3rd week

4th week

5th week

6th week           

7th week

8th week

9th week

10th week

11th week







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