Instructor: Jeffrey Diller (click for contact info, etc)
Time and place: MWF 11:40-12:30 PM, Hayes-Healy 129.
Math Bunker help: Sun-Thurs 7-9 in Hayes-Healy basement (recently former math library) or virtually via gather.town (url and pwd available in the contact file in Google Drive). The math bunker provides peer-help for proof-based math courses and is staffed by upper class math majors with a lot of experience with these. This has been a popular resource the past several years, and I highly encourage you to use it. If nothing else, you can go work on your homework in good & sympathetic company.
Office Hours: (Note change to time and place as of 2/10) Mondays 5-6:30 PM. In person in Bond Hall 114 and virtually by Zoom—see contact file for url.
Textbook: An Introduction to Mathematical Thinking by William Gilbert and Scott Vanston. We'll also rely heavily on supplementary notes (available in our Drive folder) that I provide. Reimburse me as your conscience dictates. That's a joke by the way.
If you’re interested in another take on this material, the book Mathematical Thinking: Problem-Solving and Proofs 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 somewhat 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, representation in different bases, Euclidean algorithm, congruences, rational numbers.
Analysis: least upper bound property, sequences, convergence, continuity.
Other topics, time permitting: e.g. RSA encryption scheme, complex numbers, fundamental theorem of algebra, platonic solids, continued fractions.
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 (note that because of the covid-compressed semester, I’m counting the hwk for a bit more and the final for a bit less than is my habit for this class):
Homework: assigned and collected every Wednesday (by 5 PM please), worth 35% 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. It is NOT ok, however, to copy solutions from other (pairs of) students. 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. 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: in class on Wed March 17 and Wed March 28, each worth 20% of your final grade.
Final Exam: (Note change—corrected on 2/20/21) Monday May 17 from 1:45-3:45, comprehensive and worth 25% 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.