Class hours: MWF 3pm-3:50pm
Classroom: Seigle 208
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
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.
There will be homework assignments
(30%), one take-home midterm exam (30%) and the take-home final exam (40%). 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. (Exams will also be given through email.) You should check email regularly.
Last but not least, in general, leafing through the solution
manual when the homework is due will never get you to learn
mathematics; your mind's eye must take part in a decisive fashion. That
is, you must do mathematics by yourself.