Math 135 Resources (Unofficial)

Here you will find a list of resources for Math 135 that have been created over the years based on the reformulation of the course in 2012. You are welcome to repost, recast, reproduce and share these resources with anyone. So long as the new material remains free and benefits students.

Course Notes

Version 1.0 of the Math 135 course notes

Below are both typed version and the hand written versions of the notes. The typed version are intended to be a typed version of the hand written notes (but might contain typos). The handwritten notes were notes I used for my Fall 2015 classes. I've also included a typed version of all notes as well as all the handouts I used in class. A zip package of all the files is also available at the top of this page if you would like to reproduce these.

All lectures from 2 to 44

All lectures typed from 2 to 44

Handouts from all classes

All tex and source files for typed notes (Will update at end of term)

Lecture 1 [Fall version 1A] (Outline)
Lecture 1 [Winter version 1B] (Outline)
Lecture 2 (Statements and (in)equality proofs) Typed Written Notes
Lecture 3 (Truth Tables) Typed Written Notes
Lecture 4 (Divisibility) Typed Written Notes
Lecture 5 (Divisibility of Integer Combinations and Intro to Sets) Typed Written Notes
Lecture 6 (Sets) Typed Written Notes
Lecture 7 (Quantifiers) Typed Written Notes
Lecture 8 (Negating and Nesting Quantifiers) Typed Written Notes
Lecture 9 (Contrapositive) Typed Written Notes
Lecture 10 (Contradiction) Typed Written Notes

In Lecture 11, there is a mistake in the written notes. I argue that 0 < m-n < 1. However, the lower bound could be as bad as -1. This would give that m=n which is a contradiction. This is fixed in the typed notes

Lecture 11 (Division Algorithm, uniqueness, injections, surjections) Typed Written Notes
Lecture 12 (Random Proofs) Typed Written Notes
Lecture 13 (Induction) Typed Written Notes
Lecture 14 (Strong Induction) Typed Written Notes
Lecture 15 (More Induction) Typed Written Notes
Lecture 16 (Fundamental Theorem of Arithmetic) Typed Written Notes
Lecture 17 (GCD Introduction) Typed Written Notes
Lecture 18 (GCD With Remainders, Euclidean Algorithm, Back Substitution, Bezout's Theorem [GCDCT]) Typed Written Notes
Lecture 19 (Euclid's Lemma and more GCD) Typed Written Notes

Note: Lectures 20 and 21 might be swapped. I prefer doing EEA on midterm day.

Lecture 20 (Extended Euclidean Algorithm) Typed Written Notes
Lecture 21 (GCD Properties) (Thanks Amy!) Typed Written Notes
Lecture 22 (DFPF (Divisors from prime factorization) and GCDPF) Typed Written Notes
Lecture 23 (Linear Diophantine Equations Part 1) Typed Written Notes
Lecture 24 (Linear Diophantine Equations Part 2 and Congruences Introduction) Typed Written Notes
Lecture 25 (Congruences Part 1 [Properties]) Typed Written Notes
Lecture 26 (Congruences Part 2 [Divisibility, Division and CISR]) Typed Written Notes
Lecture 27 (Linear Congruences) Typed Written Notes
Lecture 28 (Integers Modulo m) Typed Written Notes
Lecture 29 (Inverses and FLT) Typed Written Notes
Solution to Practice Problem in Lecture 29
Lecture 30 (More FLT and CRT) Typed Written Notes
Lecture 31 (More CRT and Splitting the Modulus) Typed Written Notes
Lecture 32 (Splitting the Modulus and Introduction to Cryptography) Typed Written Notes
Lecture 33 (RSA) Typed Written Notes
Lecture 34 (Introduction to Complex Numbers) Typed Written Notes
Lecture 35 (Complex Number Arithmetic and Conjugates) Typed Written Notes
Lecture 36 (Modulus and Polar Coordinates) Typed Written Notes
Lecture 37 (Polar Multiplication and De Moivre's Theorem) Typed Written Notes
Lecture 38 (More De Moivre's Theorem and Complex Exponentials) Typed Written Notes
Lecture 39 (Complex nth Roots) Typed Written Notes
Lecture 40 (Introduction to Polynomials) Typed Written Notes
Lecture 41 (Division Algorithm for Polynomials, Remainder Theorem, Factor Theorem(s)) Typed Written Notes
Lecture 42 (Fundamental Theorem of Algebra) Typed Written Notes
Lecture 43 (Rational Roots Theorem) Typed Written Notes
Lecture 44 (Conjugate Roots Theorem, Real Quadratic Factors and Real Factorization) Typed Written Notes
Lecture 45 (Cardinality)
Lecture 46 (2009 Exam Review 1)
Lecture 47 (2009 Exam Review 2)

Symbol Cheat Sheets

Theorems Cheat Sheets

Here, we attempt to break down the most important theorems by week. It is recommended that you always keep the week 4 reference sheet handy as these theorems will be used throughout the course. The weekly theorem sheets contain overlaps which are intentional and are an attempt to help you organize the course into important subsections as well as guide you for weekly assignments. As a caveat, not all instructors may follow the theorem order listed here and should be treated with caution.

Videos

These videos were created beginning in 2015 when I was first hired at the University of Waterloo. I view these more as extra practice problems as opposed than I do as a full content review. Feedback and errors can be sent to my email address.
Notes: Some of these videos (eg. the Sept. 24th videos) are pencasts (click on the picture to play the video). I found these work best on Google Chrome.

Week 1 Show p and p+1 are prime implies p=2. pdf
Week 1 Negating an implication and logical equivalence. pdf
Week 2 If and only if question and proof.
Week 2 A set equality.
Week 2 Another set equality. pdf
Week 2 A calculus example of nesting and negating quantifiers. pdf
Week 3 An example of uniqueness. pdf
Week 3 An example of an injective and surjective function. pdf
Week 4 An example of Strong Induction. pdf
Week 4 An example of finding a closed form expression and proof by Mathematical Induction. pdf
Week 5 Examples of summations. pdf
Week 5 Euclidean Algorithm example. pdf
Week 5 A sample of a proof of a gcd fact. pdf
Week 6 Examples of GCD Part 1. pdf
Week 6 Examples of GCD Part 2. pdf
Week 6 Divisors From Prime Factorization example. pdf
Week 7 Linear Diophantine Equation and EEA example. pdf
Week 7 Congruent Iff Same Remainder example. pdf
Week 8 A Fermat's Little Theorem Example. pdf
Week 8 Congruence Equations. pdf
Week 9 RSA. pdf
Week 9 Revisit CRT. pdf
ERRATA: In this next video, I incorrectly simplify \sqrt{24 - 12\sqrt{3}} to 2\sqrt{8 - 3\sqrt{3}} when it should be 2\sqrt{6 - 3\sqrt{3}}. I've corrected this in the pdf but left the video unaltered (it doesn't change much of the content).

Week 10 Polar Coordinates. pdf
Week 10 Complex nth Roots. pdf
Week 11 Fundamental Theorem of Algebra. pdf
Week 11 Long Division Over R and Zp. pdf png png2
Week 11 Long Division Over R and Zp Part 2. png3

Carmen's Core Concepts

These videos are my weekly review videos for Math 135. Note that different classes might have covered topics in a different order but the majority of the content is fairly consistent. These videos highlight what I think were the important concepts for the week. Again if you didn't cover all the same topics as me this week that is not a sign to be worried. The idea of these videos is to provide you with a brief summary of what happened during the week so that you can get a nice reminder and hopefully help you on your assignments and while studying for finals. These videos are not meant to replace lectures in any way and are meant as a supplement to lectures.
PS. Sorry about the lighting in the first video.