Instructor: Jeffrey Diller (click for contact info, etc)
Time and place: MWF 8:20-9:10 in Hayes-Healy 129.
Math Bunker help: Sunday-Thursday, 7pm-9pm (also 9pm-10pm on Wednesdays) in Hayes-Healy basement (room in the SE corner). The math bunker provides peer-help for proof-based math courses and is staffed by upper cle to work on your homework in good company.
Office Hours: Mondays 5-6:30 PM in Hayes-Healy 125. Note: this room is technically reserved til 5 PM for math dept colloquia. Most weeks I expect it’ll be empty by 5, but if you arrive and find the room occupied, just hang on outside til it empties out and/or I arrive.
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. The preass 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 thervious sentence is a joke, by the way.
Some other (purely optional) sources:
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.
Infinite Descent: An Introduction to Pure Mathematics, by Clive Newstead. This is a fairly recent effort but looks to me like a promising (not to mention, at present free) addition to the already fairly large collection of textbooks for courses like this. A bonus feature is that the book teaches Latex along the way. This is the software nearly all mathematicians use to write up their work (e.g. my class notes). Studies have shown that typing up your homework solutions in Latex makes them look 50% more correct.
Numbers: A Very Short Introduction, by Peter Higgins. This is more of a just-for-fun book. More history, less proof.
If you look at any of these, I’d like to hear what you think. The second one in particular looks to me like a contender for textbook in future instances of this course.
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 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 Thursday, 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.
Also, 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 6 and Wed Nov 17, each worth 20% of your final grade.
Final Exam: 8-10 AM, Monday, Dec 13; 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.