Syllabus for
Math 330 - Number Systems - Section 3
Spring 2015

Last update: April 13, 2015 - 3:45 PM

Instructor Section Meeting
Michael Fochler, WH 130
      Office hours: Mon 1:15 - 3:00,     Thu 2:30 - 4:00
      mfochler@math.binghamton.edu 		Section 03 Mon, Wed, Fri
Thu 		12:00-1:00
11:40-1:05 		OW 100B
OW 100B

NOTE that this document specifically pertains to section 3 of the course!

Home page for section 3:

The Section 03 home page is currently just a skeleton. Over time it will be the web page you visit most frequently as your home work assignments and general announcements will be posted here rather than on Blackboard.

PREREQUISITES:

Math 222 (Calculus 2): If you did not pass Math 222 this does not automatically disqualify you but you must see me asap! You need some Calculus background in the second half of the course and you may have to catch up on some topics to follow the course. You will be dropped from this course unless you can convince me that you have knowledge or are able to learn quickly about limits, continuity, power series, derivatives as limits of difference quotients and integrals as limits of Riemann sums.

Course Material:
Textbook:
    The Art of Proof: Basic Training For Deeper Mathematics, by M. Beck and R. Geoghegan (Springer, 2010).
The course will follow to a large degree the textbook but some items will be presented in a different order.
    ----------
Additional course material:
    The Beck/Geoghean text provides some exposure to sets and functions but I shall go beyond that. Here are some documents for to give you material in addition to that presented in the textbook. I have taken them from a 330 course that was taught earlier by Prof. Marcin Mazur.
   
  1. Differences in Notation See this document for some remarks, including potential difference in notation between the course, the B/G text and the articles listed above.
  2. Sets part 1: This document is a very basic introduction to sets which many of you should be able to skim through in a hurry, but you should skip nothing and be sure you understand all examples.
  3. Sets part 2: This document covers to a large degree the same material as Sets part 1: Again, many of you should be able to skim through in a hurry. and again, be sure you understand all examples.
    You can skip the following:
    • Ch. 1.11. Set Identities: Everything starting with ``Proof 2'' until the end on p.105
    • Ch. 1.13. Infinite Unions and Intersections: on p. 107: the use of a ``general index set'' J in the union of sets Ai where i is an element of J and exercises 1.14.1., 1.14.2.
    • All of Ch. 1.15. Computer Representation of a Set (i.e., the remainder of the document).
  4. Functions part 1: This document is very brief and it will probably take you more time per page to study it. You can skip chapter 2.4. Floor and Ceiling Functions. Note that I have an important comment on the use of f - 1(y) for both the inverse image (a set!) of y and the image of y under the inverse function f - 1 of f here
  5. Functions part 2: This document focuses on injective, surjective and bijective (invertible) functions. Pick up your copy of Stewart's Calculus and review the chapter on inverse functions. You will see material on injective functions (Stewart calls them one-to-one) and on inverse functions. This will help you understand the document. Skip all proofs as the important ones are given in B/G. but be sure to understand the definitions and examples and draw pictures with functions that you understand well to get a feeling for why the theorems are true.
  7. Lecture Notes: Math 330 - Additional Material ("MF") (MAJOR UPDATE on April 20) This document was previously called "Direct and Indirect Image". I am the author. For that reason it is much more likely to contain errors than the ones listed above as those have probably been vetted by many viewers before having been made accessible on the internet. Caveat emptor! This document will be added to in the future.
    There are reading instructions just after the table of contents. Be sure to look at them first as they tell you what parts of the material are optional (most of it) and which ones you must study. Note that those reading instructions themselves also will be subject to change!
  8. Modular Arithmetic by Miguel A. Lerma. This document contains background material on arithmetic modulo n. I do not plan to teach from it or use anything in there not covered in the book for quizzes and/or exams. This material is strictly for your convenience as it might help you to better understand the material from B/G ch.6.3 and 6.4.

Course content:

You will learn in this course how to think like a mathematician:

  • Understand the nature of a rigorous mathematically proof
  • Learn to write such proofs

To do so means you must acquire some knowledge in the following areas:

  • logic: direct proofs vs proofs by contradiction, logical quantifiers
  • the difference between axioms, definitions and propositions
  • sets, functions and relations
  • induction and recursion

The subject matter used to teach you the above will be primarily taken from the theory of number systems: an axiomatic approach to the properties of natural numbers, integers, fractions and real numbers will be presented. Besides number systems you will also learn about limits, continuity and how to compare the sizes of infinite sets (cardinality).

How to succeed in this course:

This is probably your first math course which is not focused on applying math theorems to solve problems such as "What is the derivative of the function ..." or "Compute the probability that ...". Rather, the focus is on reading and writing proofs for those math theorems. This requires a very high degree of mathematical abstraction and you will have to do a lot of hard work to develop the ability to do this kind of mathematical thinking.

My goal is to help you develop this ability. This requires that you attend class regularly, work through the material presented in class and do the reading and homework assignments.

It is your responsibility to keep informed of all announcements, syllabus adjustments, or policy changes, regardless of whether they were made by email, on Blackboard, or during class.

Homework:

Your homework will mostly be about writing proofs and it will account for more one third of your grade. It will be assigned and collected regularly. It is very likely that your homework will be corrected more than once, until your proofs are satisfactory. You will learn from the textbook and from my presentations how to write a proof, but here are some purely techincal requirements you should be clear about from the start:

Write your proofs very neatly. Use lined paper so that your text is written in straight lines. Leave margins of at least 1/2 inches to the left and at least 1 1/2 inches to the right. Write your homework double-spaced so you can insert some missing items in a neat and orderly fashion. I'll be spending a lot of time looking at your homework as it is and I won't have time to carefully deliberate whether your variable was a "u" or a "v".

Quizzes:

Quizzes will be given occasionally and they will not be announced in advance. The demands for clarity are the same as for your homework.
There will be a total of 5 quizzes which count for 20 points (2%) each. The worst of those quizzes will be dropped. Hence only 8% of your grade will be determined by those quizzes.

Tests:

There will be two tests. The dates are:

  • Test 1 (120 points): Wednesday, March 11
  • Test 2 (100 points): Wednesday, April 22

As you can see, the first test will be given more than two weeks before the deadline to withdraw or change your grading option to a pass/fail (Fri, March 27).

Final exam:

The final exams schedule for Spring 2015 was not published until the end of February. Here are the date and time for the section 3 final of this course:
Thursday May 14, 2015   -   Time: 10:25 AM - 12:25 PM   -   Location: AA G019
The time for all finals is set by the registrar and is not flexible. Do not make travel arrangements that will have you leave campus prior to the exam.

Makeup quizzes and exams:

If you will be absent from a test for a legitimate reason, you must notify me before the date of the test and provide documentation of your excuse. If you have a last minute emergency, you must e-mail me ASAP. Do NOT wait until the following class. A make-up will be given only if you provide valid documentation of your absence and you contact me before the following class.

There will be no make-up quizzes unless you were hospitalized or there was a family emergency of a very severe nature. You must provide valid documentation of your absence and contact me before the following class.
Note that the worst of the five quizzes will be dropped and not count towards your grade.

Special rules apply to the final exam: If you have a university conflict with other final exams (another exam scheduled for the same time, or three exams within a 24 hour period) you are entitled to have one of your exams rescheduled. If you have such a situation and wish to take the Math 330 Section 3 make-up final, you must notify me in writing (email is fine) by Friday, May 1.
The make-up final will be on Friday May 15, 2015 (time and location TBD).
If you do not notify me by the deadline it is unlikely that you will be accommodated. YOU MUST NOTIFY ME BY THE DEADLINE IF YOU WISH TO TAKE THE MAKE-UP EXAM. YOU CANNOT "JUST SHOW UP."

Grading policy:

The following is not 100% finalized but revision is not very likely:

  • Homework: 40%
  • Test 1: 12%
  • Test 2: 10%
  • Final: 30%
  • Quizzes: 8%

Academic Honesty:

All students are expected to adhere to the Student Academic Honesty Code. You are required to write your own homework solutions. You may not use another student’s work as a "model" and you must not allow other students to use your work.

Best wishes for a successful semester!

Michael Fochler