Math 330-3 Fall 2022 Number Systems Feingold SYLLABUS

Meeting Time and Contact Information

Meeting Time: MWF 11:20-12:50 WH-100B.

Office: Whitney Hall, Room 115, Office Hours: MWF 1:00 - 2:00 and by appointment.


Transition to Higher Mathematics: Structure and Proof (Second Edition), Bob A. Dumas and John E. McCarthy, 2015, (free digital textbook). Here is a link to download the book as a pdf file: Transition to Higher Mathematics - Dumas and McCarthy ISBN - 978-1-941823-03-3

Course Contents

Logic and Predicate Calculus (Quantifiers)

Relations (Equivalence Relations, Equivalence Classes, Partial Orderings) and Modular arithmetic

Proofs, Set Theory, Logic and Propositional Calculus

Natural Numbers, Induction and recursion

The Continuous, Real Numbers, Embedding the integers in the Reals

Limits and other consequences of completeness

Cardinality of Sets

The integers, divisibility, division and Euclidean algorithms.

The natural numbers, integers, rational numbers, real numbers, properties.

Rational vs irrational numbers, decimal expansions

The complex numbers and their properties.

Exams and writing assignments

There will be 3 hourly exams administered during class time, and 1 Final Exam during the scheduled Finals period. The hourlies will be worth 100 points each, and the (2-hour) Final Exam will be worth 150 points. The contents of each exam will be determined one week before the exam. The Final Exam will be comprehensive, covering the whole course. ANYONE UNABLE TO TAKE AN EXAM SHOULD CONTACT THE PROFESSOR AHEAD OF TIME TO EXPLAIN THE REASON. A MESSAGE CAN BE LEFT AT THE MATH DEPT OFFICE, 607-777-2147. NO ONE SHOULD MISS THE FINAL!

Since this course has a Harpur W designation, there is a writing requirement of at least 15 pages. There will be assignments that require students to discover and write proofs throughout the course. These can be considered homework assignments, but must be done individually with no outside help. The number and quality of this work will be taken into consideration and will affect your grade.

Exams and Quizzes

Here is a link to a Quiz 1 and its solutions.

Here is a link to a Quiz 2 and its solutions.

Here is a link to a Quiz 3 and its solutions.

Here is a link to a Quiz 4 and its solutions.

Here is a link to a Quiz 5 and its solutions.

Here is a link to a Quiz 6 and its solutions.

Here is a link to a Quiz 7 and its solutions.

Here is a link to a Quiz 8 and its solutions.

Here is a link to a Quiz 9 and its solutions.

Here is a link to a Quiz 10 and its solutions.

Here is a link to a Practice Exam 1 and its solutions.

Here is a link to Exam 1 and its solutions.

Here is a link to Exam 1 Grade Distribution_and Letter Grades.

Here is a link to a Practice Exam 2 Questions.

Here is a link to Exam 2 and its solutions.

Here is a link to Exam 2 Grade Distribution_and Letter Grades.

Here is a link to Exam 3 and its solutions.

Here is a link to Exam 3 Grade Distribution_and Letter Grades.

Exam 1: Oct. 7, 2022.

Exam 2: Nov. 7, 2022.

Exam 3: Dec. 5, 2022.

Final Exam: Wednesday, Dec. 14, 2022, 12:50 - 2:50 PM, CW-113.

Special Instructions

It is hoped that classes and exams will be held in-person, with whatever precautions are required due to COVID. Properly worn masks will probably not be required in Fall 2022. A student may test positive for COVID and have to be under quarantine, or a student may be too ill to take an exam or quiz (the flu is still a threat). To deal with those situations, I will make use of Gradescope so that students with a valid medical excuse can take an exam or quiz online remotely. That method of testing is not as secure as an in-person exam, so I have to rely on your honesty not to cheat in that situation. If I find that any cheating has occured, the penalty will be severe. Information about how to use Gradescope is in the next section.

In case the university administration cancels classes because of a snow emergency, we can still have a virtual class meeting on zoom. Panopto recordings of lectures from a previous semester are already available with the posted lecture notes, but if we have a class meeting on zoom, it will be recorded and a link made available on this webpage. Since snow cancelations are rare, I will not set up a recurring zoom meeting for this class, but I will only do one at a time if needed. An email announcing each zoom meeting would be sent in the morning including a link to the meeting.

Information About Document Submission Through Gradescope

Students who are unable to attend class in-person for exams or quizzes should find the exam or quiz on Gradescope, take it within the time limit, and submit the solutions as a single electronic pdf file. I highly recommend using CamScanner for this purpose. It is available from this website: CamScanner Website.

All submitted documents should be sent to me using Gradescope according to the directions in the following file: Gradescope Submitting Guide.

Here is the webpage for direct access to Gradescope: Gradescope login page.


For each exam a letter grade interpretation of the numerical score will be given, and the Total of all points earned will also be given a letter grade interpretation. The letter grades on the exams indicate how a student is doing relative to the rest of the class, and will be taken into consideration in interpreting the Totals. The course grade will be mainly determined by the Total points earned. Borderline cases will be subject to further adjustment based on Classwork and Homework. Any cases of cheating will be subject to investigation by the Academic Honesty Committee of Harpur College.

Classwork and Homework

We will be attempting to raise your level of mathematical sophistication. This means learning how to do more than calculations ``just like those done by the teacher or the book''. You will study my proofs in class and those in the textbook, and then you will be asked to try doing some of your own. Your participation in class will be requested at various points, and your response will affect your course grade. You will also be assigned homework, some to be written down and collected within a limited time period, some to be presented in class. Late assignments will be accepted at the discretion of the Professor. Assignments will be examined by the professor, who will record the fact that an assignment was attempted, and give some feedback on how selected problems were done. QUESTIONS ABOUT ASSIGNMENTS SHOULD BE ASKED OF THE PROFESSOR AT THE BEGINNING OF CLASS. DO NOT DEPEND ON THE PROFESSOR TO FIND AND CORRECT YOUR MISTAKES. Although homeworks and classwork will not be precisely graded, the amount and quality of your classwork and homework will be considered as a factor in determining your course grade if you are a borderline case in the Totals letter grade interpretation.

General Comments

CLASS ATTENDANCE IS ABSOLUTELY ESSENTIAL. This course aims to bridge the gap between the lower level courses and the higher level courses. The lower level courses tend to emphasize specific computational skills and material required for courses outside of the Mathematics Department, while the higher level courses require more conceptual skills and understanding of proofs and abstractions. The student will be helped to move towards that higher level through study of a series of topics which will involve proofs and exposure to fundamental ideas in logic and set theory which are the universal language of all of mathematics. Techniques of proof, such as proof by contradiction and proof by induction, will be examined and used in many different settings. The proper use of quantifiers and the properties of functions and relations on sets will be discussed, as well as applications of equivalence relations and partial orderings. We will touch on various subject areas, each of which is more deeply covered in a separate course. These subject areas include: Elementary Number Theory, Combinatorics, Graph Theory, Abstract Algebra, Analysis. But in each area the main goal is to see how mathematical concepts are logically related, and results are obtained by rigorous mathematical arguments. Students should come out of this course able to critically read and understand mathematical proofs, and to write proofs of their own. Lectures can be interrupted at any time for questions. At the start of each class be ready to ask questions about homework problems or about the previous lecture.

This page last modified on 12-6-2022.