Time and place: MWF 8:20-9:10 in Hayes-Healy 229.
Weekly schedule: dates, topics covered, and further comments
Instructor: Jeffrey Diller (click for contact info, etc)
Office Hours: Mondays 5-6:30 PM in Hayes-Healy 129.
Math Bunker help: 7-9 PM Sun-Thurs in Hayes-Healy basement (room in the SE corner). The math bunker provides peer-help for proof-based math courses and is staffed by advanced undergraduate math majors. Go for answers to specific questions or even better, just to work on your homework in good company.
Textbook: A Bridge to Advanced Mathematics by Cioba and Linde, together with some additional notes I wrote myself for this class
Why this course: Up til now, most of your math classes have probably emphasized examples, computation, and intuitive understanding. 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 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 (appendices A2-A5).
The integers (algebra): ring axioms, order and induction, divisibility and factorization, representation in different bases, Euclidean algorithm, congruences, rational numbers (chapters 1 and 2, 3.1-3.2).
The real numbers (analysis): least upper bound property, sequences, convergence, continuity (4.4-4.5, 5.1-5.5).
Other topics (time permitting): e.g. RSA encryption scheme, complex numbers, fundamental theorem of algebra, decimal expansions of real and rational numbers, continued fractions. Etc. I’ll entertain requests.
How you will be evaluated:
Homework: assigned and collected every Wednesday, worth 25% of your final grade. I’m planning to grade some problems (1 or 2 a week) myself with the aim of having you revise and resubmit your solutions, gradually developing a portfolio of really well-written mathematics. Some other guidelines:
I encourage you to collaborate with each other on homework assignments. It is NOT ok, however, to copy solutions from other students or anywhere else.
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, he's only doing his job.
Please turn in your homework solutions on paper, even if you, say, write/type them all out on a laptop or ipad. To make the grader’s life easier and allow for feedback, please put each homework problem on a separate page. If there are several parts to a problem, you can keep those together.
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: in class on Wed Oct 4 and Wed Nov 15, each worth 20% of your final grade.
Final Exam: Thursday Dec 14 from 8-10 AM, Hayes-Healy 229 (same as lecture); worth 35% of your final grade.
Further Policies, Disclaimers and Fine Print
Unsolicited advice: please be very brave about asking questions. The big majority of people (including many mathematicians) worry that they’ll seem stupid when they ask about something in a math lecture. Please ignore this worry--even if the reason you’re asking is that your attention drifted for a bit and you missed a point. Most often, questions reassure the lecturer that the class is paying attention, and half your classmates are confused about the same thing you are.
Honor Code: abide by it. If you’re wondering whether or not something you’re thinking of doing is acceptable you should ask me about it.
Late homework, missed exams: I do not accept homework late, though I might consider discounting late assignments if the situation merits it. If, for some suitably dire reason, you need to miss an exam, you should clear it with me in advance if possible and be prepared to document the reason for missing.
Using the internet as a resource: this is generally fine with the exception that you are not allowed to seek or discuss solutions to particular homework problems on the web.
Attendance: I don’t formally take attendance, but I do notice engagement and absence over time and will feel free to take that into account when assigning final grades.