Proofs (and Number Theory)

Syllabus

Final Exam Week Schedule:

  • Monday:
    • Office Hours 11am to 12pm and 1pm to 2pm,
    • MAT 141 Exam 2pm-4:30pm,
    • MAT 304 Exam 5:30pm -8pm
  • Tuesday: Not On Campus
  • Wednesday:
    • Office Hours 1pm to 3pm
  • Thursday:
    • MAT 133 Exam 8am-10:30am,
    • MAT 207 Exam 11am-1:30pm
  • Friday:
    • Office Hours 1pm to 3pm, All Work is Due By 3pm Friday, No Exceptions

 

Calendar

Your syllabus has a rough calendar of what we will be covering each class.  After each class I will post what we were actually were able to cover here:

  • …
  • 12/5 – Review and Applications (time allowing)
  • 12/2 – Euler’s \(\phi\)-function and Euler’s Theorem (pp.59-60)
  • 11/25 – Fermat’s Theorem (pp. 53-57)
  • 11/21 – Chinese Remainder Theorem (pp.50-51)
  • 11/18 – Solving equations of the form \(ax\equiv b\pmod{n}\)
  • 11/14 – Exam 2 on Proofs: Be sure you know basic definitions about divisibility and modular congruence as well as the structure of the different types of proofs. You should expect questions about
    • definitions and axioms,
    • reading proofs and identifying their type,
    • filling in justifications in an outline of a proof and writing it up in sentences,
    • proving a couple basic theorems/lemmas using an appropriate proof techniques
  • 11/11 – Solving Linear Congruences (pp.43-49)
  • 11/7 – First I want o apologize for being a little spacey recently, I will work to get us back on track. Today we reviewed problem solving strategies and looked at another example of proof of existence for primes. Be sure to not lose track of material covered before like definitions and key lemms/theorems.
  • 11/4 – We covered a few additional proof examples. We will get back to new material on Thursday 11/7. The proofs exam is moved to 11/14.
  • 10/31 – Some additional examples of indirect proofs and proofs with GCD’s, LCM’s, and primes.
  • 10/28 – We covered some more material related to GCD’s and LCM’s and then started discussing properties of prime numbers. I was a bit out of it in class and so we will review and clarify some of that content on Thursday 10/31.
  • 10/24 – Today we worked through Bezout’s Lemma: Given \(a,b\in\mathbb{Z}\) not both zero, \(d=(a,b)\) if and only if \(d=ax+by>0\) is the least positive integer linear combination of \(a\) and \(b\). Then we looked at theorems 1.41-1.43 which are proved using Bezout. For next time play with Question 1.44 on page 20 try
    • \(a=2,b=4,c=3,n=6\)
    • \(a=6,b=10,c=15,n=30\)
    • \(a=7,b=5,c=3,n=2\)
    • Choose your own \(a,b,c\) for \(n=2,5,9,12,17\); use some variety.
  • 10/21 – Today we looked at additional results having to do with greatest common divisors. In particular we looked at using the Euclidean Algorithm to find the g.c.d. of two numbers, \(d=(a,b)\), and to write it as a linear combination of those numbers, \(d=ax+by\). When we finished we were working on showing that if \(d\) was the least element of the set \(S=\{ax+by>0|x,y\in\mathbb{Z}\}\) then \(d|a\) and \(d|b\). We will finish this up next time and then show that \(d=(a,b)\) as well.
  • 10/17 – Today we went through the proof of the division algorithm for that case when \(m,n\in\mathbb{N}\). We also talked about how to extend this to all integers. Both of these required the use of the Well Ordering Principle (W.O.P.) and a little argument by contradiction.
  • 10/10 & 10/14 – We have explored direct proofs and a proof by induction by looking at properties of modular arithmetic. Also, Adam pointed out a typo in the Direct Proofs Exercise, \(|x+y|\) should be \(|x-y|\) in the second conjecture. I have fixed this and uploaded the corrected version.
  • 10/7 – We started working through the text and discussing types of proofs in Number Theory. We looked at direct proofs for theorems through Theorem 1.9. We will pick up there on Thursday. Also, I forgot to remind you that you can do corrections on the exam to earn back upto 33% of your lost points. As out of class work you need to type up the redoes; try to get those in by 10/17/2024.
  • 10/3 – Exam 1: Reading, Writing, and Analyzing
  • 9/30 – Today we reviewed some basic logic from MAT 141: Foundational Discrete Math: Discrete Math Logic in Images
  • 9/26 – Finished up Parts of Conjecture Handout and started looking at a little of what you need to do for the assignment due 10/3/2024. If you have question you should come to office hours. Monday we start the next unit.
  • 9/23 – Finish Problem Solving and Reading Mathematics (Take 2):
  • 9/19 – We practiced problem solving. We will finish the problem we were working on in the next class. I do have one note to make, for the problem on Pythagorean Triples we need to assume that \(a\) , \(b\), and \(c\) have no common factors; I apologize for forgetting to include that in our assumptions. Can you figure out why it is necessary?
  • 9/16 – Polya’s Four Step Problem Solving Process
  • 9/12 – We looked at examples from the writing math packet so that you can complete the assignment from that packet. Then we started looking at the Proofreading Mathematics examples below. We will pick up with those on Monday before discussing problem solving.
  • 9/9 – We worked through sections 2 and 3 of the Writing Up Mathematics Packet. We will look at the material in section 4 on Thursday and maybe do one of the problems in section 5 together. For the assignment due on 9/16/2014 you need to write up the scrap work and type up the final solutions for all the problems in sections 5 and 6.
  • 9/5 – Reading Mathematics (Take 1): Today we looked at the below three explanations of Newton’s Method. We discussed what made some of them more attractive or easier to follow than others. Then we spent time working our way step by step through the explanation from the Apex calculus text in order to help us understand how we should actively read a textbook in order to facilitate understanding. finally I handed out printouts of the material for the first assignment, there is a digital copy below in case you need another copy.
  • 8/29 – Today we went over the syllabus and website and then worked on the 1st Day Opening Activity.

Assignments

Unless very specifically told otherwise, all assignments must be typed and in complete sentences. Proper submission formatting counts for 10% of the assignment grade.

  1. Due 9/12/2024 – Reading Assignment: Hand in your annotated copy of chapter 1 from How to Think Like a Mathematician, and your work for problem (iv) on page 12. Recall that for this you should be completing/explaining the underlined statements and answering included questions. This assignment does not need to be typed, most of it you can complete by writing directly on a copy of chapter 1.
  2. Due 9/16/2024 – Writing Up Mathematics Assignment: for this assignment you need to write up the scrap work and type up the final solutions for all the problems in sections 5 and 6.
  3. Due 9/23/2024 – Rewriting Assignment: Rewrite these poorly written solutions:  Hey isn’t this good enough Prof?
  4. Due 9/26/2024 10/3/2024 – Conjecture Exercise : This assignment does not need to be typed, in fact much of it you can complete by writing directly on a copy of the Conjecture Exercise handout, though you might want to do scrap work on the side before writing your final solutions.
  5. Due 10/17/2024 Scaffolded Direct Proof Exercise
  6. Due 10/28/2024 Complete this Scaffolded Induction Exercise
  7. Due 11/11/2014 Complete the Scaffolded Indirect Proofs (Fixed typo so that theorem 2 is \(xy\) not \(x+y\))
  8. Due 11/25/2024 … Insert Number Theory Assignment 1 …
  9. Due 12/13/2024 by 3pm Number Theory Assignment

Exams

  1. Monday 9/30 Thursday 10/3 – Exam 1: Reading, Writing, and Analyzing
  2. Thursday 11/7 Thursday 11/14– Exam 2: Proofs
  3. Thursday 12/12 @ 11am (During Finals!) – Exam 3: Number Theory will cover:
    • Divisibility
    • Well Ordering Principle
    • Division Algorithm
    • Greatest Common Divisor and Least Common Multiple
    • Bezout’s Lemma
    • Euclidean Algorithm
    • Fundamental Theorem of Arithmetic
    • Modular Arithmetic
    • Chinese Remainder Theorem
    • Wilson’s Theorem
    • Euler’s \(\phi-function\) and Euler’s Theorem (Fermat’s Little Theorem)

Proofs Portfolio

Due 12/13/2024 by 3pm: Remember that the proofs portfolio must contain an example of:

  • direct proof,
  • proof by cases,
  • proof by contradiction,
  • proof by contrapositive, and
  • proof by induction.

For each proof you must include at least two drafts; one that you completed earlier in the semester that I commented on, and at least one revision of that draft showing how you improved it. You need to bind your proofs in a report folder.

Extra Credit

  1. Due 12/13/2024 by 3pm – Extra Credit What is Mathematics? Assignment: Read this article What is Mathematics? by Jenny Quinn (MAA Focus Vol, 42 No. 4), then write a brief reflection on your own thoughts. This should be typed, double spaced, with 1 inch margins, and about a page long. Be sure to include …
    • Comments on your past experience with math.
    • What sort of things you think about when you are “doing math.”
    • What you think math is based on your experiences and on what you personally got out of the article.

Links and Handouts

Typesetting Math