#### Department of Mathematics and StatisticsBinghamton University

How to write a proof:
"Where shall I begin?" he asked. "Begin at the beginning," the King said, "and stop when you get to the end."
– Lewis Carroll , Alice in Wonderland

# Math 330: Number Systems

## (a.k.a. Introduction to Higher Math, or simply Proofs)

### Section 4 · Zaslavsky · Spring 2024

Schedule and homework | Announcements | Advice | Term Project | Syllabus

### Instructor

Tom Zaslavsky
Official web page, personal web page.
Office: WH-216
Office Hours: M W F 3:00 - 5:00, or make an appointment.
E-mail: zaslav@math.binghamton.edu. E-mail is good; use it often.

### Class Schedule

M W F 2:50 - 4:10 in WH-100B.

#### Attendance policy

You are expected to attend all class meetings. The maximum number of absences permitted in this course is 5 (five). Excessive tardiness will count as absence. If you are compelled to be absent, such as by illness, I will expect you to promptly inform me so I will know what's going on. [University Bulletin]

I will not expel you from the class for absences beyond five, but your grade will be lowered.

### Prerequisite

Calculus II (C- or better in Math 227 or Math 230). If you do not satisfy this prerequisite, you may be dropped from the class.

### Description

Careful discussion of the integers, the rational numbers and the real numbers, including a thorough study of induction and recursion. Countable and uncountable sets. The methodology of mathematics: basic logic, the use of quantifiers, equivalence relations, sets and functions. Methods of proof in mathematics. Training in how to discover and write proofs.

Click on the link for a more detailed syllabus. The syllabus is not all there yet; it will be developed as we go through the course.

### Textbook

The Art of Proof: Basic Training for Deeper Mathematics, by Matthias Beck and Ross Geoghegan, Springer, New York, 2010.

The book is essential; get it. Advice and corrections to the textbook (this is important) are on the advice page.

### All official announcements and assignments are given on this Web site.

I will usually mention them in class and by e-mail, but the Web site is the main place to look. Stay up to date.

### What You Should Learn From This Course

The subject matter and especially the methods, including how to read, understand, and write proofs.

You accomplish this by attempting all the assignments and not falling behind. (Falling behind in this class is worse than usual. Everything depends on what came before. Believe it.)

This is a writing emphasis course, and therefore there will be a lot of writing. I will be grading your work on (W) the quality of the writing (clarity and grammar) as well as on (M) the content (the logic and math). You need all that for writing proofs; if your writing is not clear, your proof cannot be correct.

 60% Homework (the main part of the course), and some announced and unannounced quizzes. 15% Midterm Exam, to be scheduled in March, in class, before the course withdrawal deadline of March 25. 25% Final Exam, scheduled for Wed. May 08, 2024, 3:15-05:15 p.m. in CW 106.

When calculating your course grade there is one more rule: if your homework score is an F then your course grade is an F, regardless of other scores.

SUNY Central requires me to announce the grade percentage breakpoints in advance; therefore, here is my compliance with SUNY regulations:
A xx.x-100.0, A- yy.y-(xx.x-.1), etc.
SUNY does not want to admit that some courses can't have numbers fixed in advance. What I can tell you is that you do not need to get an A on every problem or solve every problem at all (no one ever does) to get an A in the course. You need to do well overall. Good luck!

Of course, you are expected to obey the Student Academic Honesty Code. This means no copying and no plagiarism. Plagiarism means using any source other than your own work and our textbook without stating your source and what you got from it.

### Homework Assignments

See the schedule page for the assignments. Assignments will be announced as the term progresses.

Written work must be handed in on paper. I do not accept electronic submissions. Remove all stubs! Don't hand in messy papers; they must be neatly written or printed. I will encourage people to learn how to type mathematics using LaTeX, a mathematical document system that is in wide use in math and science. A good guide to LaTex is the online wikibook LaTeX.

Most of your written work will be proofs. Proofs are expected to be complete and correct--but not necessarily the first time! I allow a total of 4 (four) submissions of each proof. That gives you three rewrites; you have 3 weeks (from the due date) to complete them. I will return your work to you with the first major error indicated. You will have to figure out what's wrong and fix it. It's smart to come to talk with me if you get stuck.

I also expect the proofs to be well written. Writing matters: not only because this is a (W)riting course, but because math must be written clearly and precisely to be seen to be correct.

• Proofs: I usually use the "red line" method. I'll read down until I find a significant error. I'll underline the error (it may be something missing) and stop there. You should figure out what's wrong and fix it. If you can't figure it out and your friends can't help, please do come and ask. That's what office hours are for (and if you can't get to them, make an appointment to see me at some other time).
• Writing: I will mark errors by circling them, including minor ones like grammatical and punctuation errors. Here are some common ones. Don't worry; you'll get better as the term progresses.
• RUN-ON: A run-on sentence should be two or more separate sentences. It probably combines thoughts that should be separate.
• FRAG: A sentence fragment. Every sentence needs a subject and a verb. No exceptions. (Oops.)
• PUNC: Punctuation and capitalization.
• Use a CAPITAL letter at the beginning and a PERIOD (.) at the end of every sentence. That's how I identify different sentences.
• Grades: I will give a grade (A, B, ...), especially when the problem is basically solved or when the work is way off base. If you get a lower grade than you like, such as B, then you may submit a rewrite (up to 3 rewrites). If you get no grade, I want to see a rewrite. Your rewrite grade can be higher or lower than the original grade.
Some problems will get two grades:
• (M) marks the mathematics grade.
• (W) denotes the writing grade.
Meaning of letter grades: You can convert them to numbers to see what the differences are:
A = 8, B = 6, C = 4, D = 2, F = 1, not submitted = 0.
The symbols +/− are small adjustments to the letter.
• You are not expected to answer every problem in order to get an A for the course. If you answer most of the problems very well (and do well on the other work), you will have a good grade, possibly an A.

Schedule and homework | Announcements | Advice | Term Project | Syllabus