Instructor: Jeffrey Diller (click for contact info, general policies, etc.)
Time and place: MWF 1:55-2:45 PM, Hayes-Healy 229.
Help Session: Monday evenings 7-8(ish) PM in Hayes-Healy 125.
Office Hours: Tuesdays 5:30-7:00 PM.
Textbook: An Introduction to Mathematical Thinking. by William Gilbert and Scott Vanston. We'll also rely heavily on supplementary notes that I provide for download on the schedule page. Reimburse me as your conscience dictates. That's a joke by the way.
I put a couple of other books on reserve in the math library. I just found the book Numbers: A Very Short Introduction
by Peter Higgins a couple of months ago, and I have not yet read
it. But it looks very much like it discusses all the stuff we'll
be covering in a less rigorous but more entertaining fashion. If
you look at it yourself, I'll be very curious to know what you
think. The book Mathematical Thinking
by D'Angelo and West is another textbook that has been used a lot for
(other incarnations of) this course. It's more ambitious than
ours, even if you add in my supplementary notes. and the style is
also quite different.
Why this course: Up til now, most of your mathematics courses have likely emphasized examples, computation, and intuitive understanding of mathematics. This course will emphasize careful mathematical arguments. By addressing questions about familiar things like numbers (Are there finite or infinitely many prime numbers? Do all rational numbers have rational square roots?) and sets (What does it mean for a set to have ``infinitely many'' elements? Do all sets with infinitely many elements have the same size?), we will see how it is that one justifies statements in mathematics. In a nutshell, the subject of this course is numbers, and its goal is to help you understand, invent, and present proofs.
What we'll cover: Course content falls roughly into four categories. We'll definitely cover the first three, though the first will be somewhat dispersed among the other two. The fourth category is a sort of grab bag that we'll reach into as much as we can.
Basic material concerning sets and proofs: methods of proof, relations, functions, cardinality.
The integers: ring axioms, order and induction, divisibility and factorization, Euclidean algorithm, congruences, rational numbers.
Analysis: least upper bound property, sequences, convergence, decimal expansions.
Other topics, time permitting: e.g. RSA encryption scheme, complex numbers, fundamental theorem of algebra.
In terms of the textbook, we will cover the following in more or less the order listed: chapter 2, sections 4.1 and 4.3, chapter 3, chapter 5 (we'll definitely need notes here, since this one is far to brief for our purposes), and sections 6.1-6.6. It'd be nice to spend time on chapters 7 and 8, too.
How you will be evaluated:
Homework: assigned and collected every Wednesday, worth 30% of your final grade. I encourage you to collaborate with each other on homework assignments. In fact, on each assignment, you may collaborate with up to one other person and turn in a single, jointly prepared set of solutions. Since I assign only a small fraction of the number of problems that you face in classes such as Calculus, I expect you to take special care in writing up your solutions well. If the grader takes off points for sloppy presentation, she's only doing her job. On a similar note, if you want feedback from the grader, you should allow space for this to happen. As a general rule of thumb, you should alot at least half a page for short solutions and at least a page for longer ones. Note that I assign homework a week before it's due and expect you to take advantage of all that time. It would definitely not be a good idea to wait til the last minute to start your homework.
Midterm Exams: given in class on Wed 2/29 and Wed 4/18, each worth 20% of your final grade.
Final Exam: Wed 5/9 from 4:15-6:15 PM, comprehensive and worth 30% of your final grade.