Home Page for Math 1C03

Introduction to Mathematical Reasoning


This is the Course Home Page; it is not on Avenue to Learn. Do not send email from your Avenue account, as I cannot reply to it.

Table of Contents


            read this section often for updates on assignments, tests, corrections, comments, etc.

April 2 2015 If you do not have a mark posted for Midterm 2, please bring your midterm to Jamal before or after class on Tuesday. He will make a note of your mark.

April 2 2015 Here is my complete record of marks in the course. Please let me know of any discrepancy as soon as possible, and certainly before the final exam.

April 1 2015 study guide for final exam

March 30 2015 slides for Br Bolker's lecture posted on the calendar

March 26 2015 Problem sheet 9 is now posted.

March 23 2015 Quiz 8 is the LAST quiz. It is on complex numbers and solving complex polynomial equations. Solutions to Problem Sheet 8 are posted on the calendar below. Class on Tuesday March 24 will be in BSB B136.

March 16 2015 The final exam question related to the guest lectures is posted here for you to prepare.

March 16 2015 Solutions to Problem Sheet 7 posted on the calendar. Continue to work on Problem Sheet 8 this week; complex numbers will be the topic of Quiz 8, which is on Wednesday March 28. The topic for Quiz 7 this week is cardinality.

March 11 2015 Problem Sheet 8 posted on the calendar

March 10 2015 Review session today for midterm is in HH 302 3:30-5:30

March 2 2015 Slightly modifed version of Problem Sheet 7 posted on the calendar.

March 2 2015 Midterm 2 is coming up on Wednesday March 11. NOTE the slightly different time; the midterm will start at exactly 7:00!

Midterm 2 covers induction, rational and real numbers, functions. You should be able to define rational numbers (as quotients of integers),
and properties of functions including well-defined, injective, surjective, domain, range. You should be able to use the principle of induction
and strong induction to prove statements about positive integers. You should be able to calculate the periodic decimal expansion of a rational
number, and the rational form of a periodic decimal expansion. You should be able to prove assertions about functions.

March 1 2015 Problem Sheet 7 posted on the calendar. A link to a youtube video with solution to Problem Sheet 6 question 3 is posted on the calendar. Complete solutions to Problem Sheet 6 will appear tomorrow.

Feb 24 2015 Problem Sheet 6 posted on the calendar below.

Feb 21 2015: Please note that the fact that the decimal expansion of a rational number is periodic is a theorem, NOT the definition!

Feb 18 2015: Solutions to Problem Sheet 5 are posted on the calendar below. parts of Question 2 were quite challenging, so don't worry if you could not work them out. A summary of the marks so far are posted here, indexed by the last 5 digits of student id and sorted in increasing order. MSAFs are not included. Please let me know if you see an error in your marks.

Feb 9 2015: I will be in the vicinity of my office on Tuesday afternoon (except 2:30-3:30) for anyone who did not get their midterm back in tutorial and wants to come by and pick it up.

Feb 7 2015: problem sheet 5 and information about quiz 4 now posted on the calendar.

Jan 29 2015: problem sheet 4 posted on the calendar

Jan 28 2015: Review for Midterm 1: review session Tuesday Feb 3 3:30--5:30 HH 302. Come prepared with questions for Jamal.

You should know the following definitions: prime, divisible, gcd, relatively prime, congruent .
You should be able to state the following theorems:  quotient/remainder theorem (2.12), GCD chararacterisation theorem (2.24), Fermat's Little Theorem (3.42), Inductive property of the natural numbers (4.11).
You should be able to prove the numbered results from the text (also covered in lectures): 2.21, 2.27 (i), 2.28, 2.52.
You should be able to use the division algorithm to calculate quotient and remainder, the euclidean algorithm to find the greatest common divisor of two numbers, and the definition of addition and multiplication of congruence classes modulo m.
Review all the problem sheets. Think about the process of proving a statement: what are the hypotheses that you assume? what are the definitions of the terms used; how can you restate the hypotheses to give you information that you can calculate with?  what are you trying to prove (perhaps re-stated)? how can you re-write the information that you have to make it look like the information that you need?
Think about examples in all of the above. Write down an example to illustrate the theorem. Write down an example in which the hypotheses fail to illustrate why they are needed.

Jan 27 2015: Solutions to problem sheet 3 posted below.

Jan 24 2015: Information about Quiz 3 posted on the calendar below. This weekend, you should work on Problem Sheet 3. Problems 1 and 3 are practice with calculations and the definition of congruence. For problems 2 and 4, follow the outline of how we proved similar properties in class. Write down what the assumptions are telling you, think about what you are trying to prove, and then do some algebra to go from what you know to what you want. Problem 5 is a bit more complicated, but the same approach should get you through it.   We will talk about this problem in class next Wednesday.

Jan 22 2015: Midterm 1 is on Wednesday, February 4, 7:15--8:45, in T28 001. Anyone who has a scheduled conflict (evening class or exam in another class), should email me with their name, id number, and explanation for the conflict. Anyone who has a different conflict may email me and ask for special consideration to take the early write. The alternate seating for the exam will be the same evening 5:15-6:45. The location will be sent to the people whose name is on the approved list. If you want to take the early write, let me know by  Wednesday January 28.

Jan 20 2015: Solutions to Problem Sheet 2 posted on the calendar below.

Jan 18 2015: Topic for Quiz 2, definitions and theorems to know are now posted on the calendar below.

Jan 13 2015: Solutions to Problem Sheet 1 now posted on the calendar.

Jan 9 2015. You should plan to complete Problem Sheet 1 this weekend. This is not to be handed in, but you can be sure that problems on the quiz will be closely related to problems that you have practised. Feel free to ask me or Jamal if you have any questions about how to do the problems, or if you want to know if you have expressed your solutions correctly.
    In question 4, notice that there is only one reasonable definition of prime in E. We normally say that a positive integer p is prime if it is only divisible in the positive integers by itself and 1, that is, cannot be factored in the positive integers except in a trivial way. Thus p is prime in E must mean that the number p in E is only divisible in E by itself and 1, that is, cannot be factored in E.

Jan 9 2015. Jamal's office hours posted below. This weekend, work through all of Problem Sheet 1. You should be able to do all the problems. Anything that you don't understand you should ask about in tutorial on Monday Jan 13 or class on Tuesday Jan 14.

Jan 6 2015. Corrections made to course outline and Week 1 recommended problems on calendar as pointed out in class.

Jan 4 2014. Welcome to Math 1C03. I am looking forward to the class, and I hope that you are too. Tutorials start on Thursday January 8.

Instructor Information

    Dr Deirdre Haskell, HH 316 ext 27244
    Office hours: T 10:30-12:00, F 10:30-11:30
    Course meeting time TWTh 9:30-10:20 HH109
    Tutorials: one of Th 9:30-10:20 T13/106
                                M 13:30-14:20 HH305
                                M 15:30-16:20 HH305
    TA Jamal Kawach  kawachjk@math.mcmaster.ca
        office hours W 4:30-5:30, Th 4:30-6:30 in Math Help Centre


An introduction to mathematical thinking: algebra and number systems, by William Gilbert and Scott Vanstone, Pearson Prentice Hall
The textbook is required. The calendar indicates which sections we will be working on each week. Read the section ahead of time. You don't have to read and understand the details of every proof the first time through. But you should note the things that you don't understand, plan to ask about them in class or tutorial, and go back and read the material again after class and see if it makes more sense. If not, come to office hours and ask about it. As you read a proof, ask yourself "why is this true?", "why did the author choose this approach to the proof?", "how would I do it differently?".


        The weekly schedule is given below. This is tentative, and subject to change throughout the semester, as events overtake us. Check often for changes. The "week" is considered to begin with the first lecture on the Tuesday, and to end with the tutorial on the following Monday.

Topic and reading
Quiz topic (Wednesday)
Recommended problems
Week 1 Jan 5 - 9
2.5 prime numbers
read chapter 1
no quiz
Problem Sheet 1
p.20 1,3,6,17-20,41-44,45-50
solutions to problem sheet 1

Week 2 Jan 12-16
2.1, 2.2 division and euclidean algorithms
read chapter 2, sections 2.1, 2.2, 2.5
Quiz 1 is on primes and divisibility
Know the definitions of prime, relatively prime and divisibility
Problem Sheet 2
p.50 1,3,7,9,13,19
solutions to problem sheet 2

Week 3 Jan 19-23
Finish chapter 2
3.1, 3.3, 3.4 congruences, equivalence relations and modular arithmetic
read chapter 3, sections 3.1, 3.3, 3.4
Quiz 2 is on the division and euclidean algorithms
Know the definitions of relatively prime, greatest common divisor. Review the proof of the quotient/remainder theorem.
Finish working on problem sheet 2, and the problems on p.50
Problem Sheet 3
p.82 8-11,22-26,27-30,45-47
solutions to problem sheet 3

Week 4 Jan 26-30
4.1 induction
read section 4.1
Quiz 3 is on the notion of congruence. Review the definition, and the elementary properties of congruence proved in class and on Problem Sheet 3.
Problem Sheet 4
p.104 6,8-10,12,15,18,22,23,28
solutions to problem sheet 4

Week 5 Feb 2-6
4.1 induction
read section 4.1
Midterm 1Wednesday Feb 4 7:15-8:45 pm T28 001

Week 6 Feb 9-13
5.1, 5.2 rational and real numbers Quiz 4 is on induction. You should know the inductive principle of the natural numbers and be able to use it.
Problem Sheet 5
p. 106 36, 45, 57, 62, 67
p. 121 10, 12, 23, 27
solutions to problem sheet 5
Guest lecture: Dr Lozinski will speak about Actuarial and Financial Math on Tuesday Feb 10
Reading Week Feb 16-20

Week 7 Feb 23-27
6.1-6.5 functions Quiz 5 is on rational and real numbers. You should know the definitions of rational and irrational.
Problem Sheet 6
video solution to Q3
solutions to problem sheet 6

Guest lecture: Dr Protas will speak about numerical and applied mathematics on Wednesday Feb 25
Week 8 Mar 2-6
6.6 cardinality Quiz 6 is on functions. You should know the definitions of injective and surjective, and how to prove that a function has one of these properties.
Problem Sheet 7
solutions to problem sheet 7
Guest lecture: Dr Valeriote will speak about an uncomputable function
Week 9 Mar 9-13
8.1-8.5 complex numbers
Midterm 2 Wednesday March 11 7:00-8:30 T28 001
Problem Sheet 8
solutions to problem sheet 8

Week 10 Mar 16-20
8.6-8.8 complex polynomial equations
7.1 cryptography
Quiz 7 is on cardinality. You should know the definition of countably infinite and |A|=|B|.

Guest lecture: Dr Wang

Week 11 Mar 23-27
7.1, 7.2 cryptography
7.4 public key cryptography, RSA
Quiz 8 is on complex numbers.
Problem Sheet 9 Guest lecture: Dr Bronsard
Guest lecture: Dr Bolker
Week 12 Mar 30-Apr 3
7.4 more on RSA

April 3 is Easter Friday: no classes

Week 13 Apr 6-8

Guest lecture: Dr Childs