Course: Math 310W, Foundation for Higher Mathematics
Class hours: MWF 2-3 pm
Classroom: Seigle 109
Instructor: Quo-Shin Chi
Office: Room 210, Cupples I
Office Hours: MW 4-6pm
Textbook: Class Notes

We shall rigorously go through the construction of real numbers. By "rigorously" we mean we adopt the axiomatic approach to follow precise logical arguments to deduce important properties (theorems) from the chosen fundamental set-theoretic postulates (axioms), while along the way we introduce definitions to facilitate our train of thoughts. More precisely, we start with the 9 axioms of set theory to derive the Peano axioms for natural numbers to build the four operations +, -, *, / for them. We then extend from natural numbers to integers, rational numbers and finally real numbers and their corresponding operations +, -, *, / . The construction of real numbers is the most subtle of all, to which there are several approaches. To let the students be as comfortable and skillful as possible with the notion of limits that is paramount in more advanced courses, we shall adopt Georg Cantor's approach in which a real number is identified with an equivalence class of convergent Cauchy sequences of rational numbers. If we have time, we shall cover some material from elementary number theory of integers.

In addition to the above material for Math 310, we will cover the basics of typing with LaTex in the first two weeks concurrently with the above course material. All your homework sets should be typeset in LaTex; moreover, we will have three essay papers, each about six pages long, such as a report on Fermat's last theorem, etc., to write (with LaTex) during the semester.

There will be homework assignments (30%), one take-home midterm exam (30%) and the take-home final exam (40%), which constitute 80% of  your overall score, whereas the essay writings will constitute the remaining 20%. Each homework assignment will be given through email on a Friday, except possibly for a couple of exceptions, and it will be due the next Friday in class, except possibly for a couple of exceptions. The three essays will be given as the semester moves on. You should check email regularly.