Math 347, Fundamental Mathematics

Fall 2014

MWF 9-9:50AM, 241 Altgeld Hall

Welcome to the course webpage for Math 347, Section B1. Here you will find some general info about the course. This is also the place to look for homework assignments, occasional course notes, and other things that might interest you. Most of this information will also be available through our course page on Illinois Compass.

ANNOUNCEMENT: The final will be on Monday, Dec. 15, 8AM-11AM in 241 Altgeld. Here are some practice problems for you to prepare. Here are the solutions. You will be allowed the front and back side of a usual piece of paper of handwritten notes, as well as a non-electronic dictionary to help with language, if needed. The exam will be cumulative: that means that anything which has been covered in class will be fair game for the final. There will not be any intended bias towards any particular part of the course (however, since this is a class about proving things, the methods of proof that we discussed in the first part of the class resonates throughout all of the other sections, and it would be good for you to make sure you know them). Extra OFFICE HOURS this week are scheduled for Friday, 10-11AM and 3-4PM.

Textbook Homework Policy Exams Homework Assignments Material by Day Various Notes Back to Main Page

Course Syllabus

Professor: Elena Fuchs
Office: 359 Altgeld Hall
Email: lenfuchs at illinois dot edu
Office hours: M 10:00AM-12PM or by appointment.

Textbook and Prerequisites:

The textbook we'll be using is "Mathematical Thinking" by D'Angelo and West (2nd edition). For the real analysis sections, we may borrow material from other books and I will provide you with copies of the relevant sections at that time.

The prerequisite for this course is Math 231.

Homeworks, Exams, and Grading:

Your grade for the course is determined as follows: 15% for homework, 25% for each of the best two of three midterms, and 35% for the final. There will be no make up exams.

Homework (along with occasional supplementary notes) will be posted here and will be due every Wednesday at 9:50AM in class, or in my office by the same day/time (if you choose the office option, you can slide it under the door). You are encouraged to work with your classmates on homework and discuss the homework problems amongst each other. However, solutions should be written up independently, and you should write down which classmates you worked with at the top of your homework. The lowest two homework scores will be dropped. Therefore the policy is that no late homework will be accepted, especially since we will sometimes discuss the solutions to the homework problems in class.

The midterms will be in class on Wednesday, September 24, Wednesday, October 15, and Wednesday, November 12. More details about these exams will be announced in class and posted here closer to the dates.

If you have a cold or flu, I ask you to please do yourself, your classmates, me, and all the people we live with a huge favor by staying home and resting, rather than coming to class! If this happens I will work hard to help you catch up on notes and things you missed in class once you are healthy!

Course Outline:

This course is meant to introduce you to writing mathematical proofs, as well as aquaint you with some math topics beyond calculus. Many of you will go on to study these topics in more depth in future courses, and the hope is that having been exposed to the topics in this course will help you.

Basic Plan:

We will try to cover Chapters 1-4, 12-13, and 6-7 in our book, which are roughly structured as follows:

Preliminaries: some basic set theory and a bit about real numbers; logical arguments and various methods of proof.
Real Analysis: a deeper look at the real numbers and discussion of convergent sequences.
Elementary number theory.

Note: The book is written in a conversational style and sometimes goes off a bit on tangents. We will likely ignore these tangents, and I will inform you when that happens. You are, however, welcome to read everything in the chapters above and beyond!
Detailed Plan:

The following is a rough outline of what we will be doing in lecture every day, along with the relevant sections in the book (note that the reading for a given lecture should be taken to mean the relevant section of the mentioned chapter). In reality, we may move faster or slower. It will be updated on a regular basis.

8/25: Natural numbers, integers, rational numbers, and real numbers. Fields and inequalities. Reading: Chapter 1.
8/27: Inequalities continued, defining sets, set notation, proving things about sets. Reading: Chapter 1.
8/29: Some set theory: notation. Reading: Chapter 1 and Professor Hildebrand's notes on sets below.
9/3: Proving things about sets; introduction to proofs and logical statements. Reading: same reading on sets as described on 8/29, beginning Chapter 2.
9/5: More on proofs and logical statements. Lots of examples. Reading: Chapter 2, M. Hutchings' notes on proofs below.
9/8: Proof by contradiction and negating statements. Possibly starting induction. Reading: Chapter 2, M. Hutchings' notes on proofs below, and possibly beginning of Chapter 3.
9/10: Induction: lots of examples and why it works. Reading: Chapter 3.
9/12: More on induction and sum/product notation. Reading: Chapter 3.
9/15: Some theorems about polynomials. Reading: Chapter 3.
9/17: Strong induction and the well ordering principle. Functions, injectivity, surjectivity. Reading: Chapters 3, part of Chapter 1 on functions, and beginning of Chapter 4.
9/19: Bijections and cardinality, countability. Reading: Chapter 4.
9/22: Review for midterm 1.
9/24: Midterm 1. Midterm 1 Solutions
9/26: Continuing bijections and cardinality, discussing countability. Reading: Chapter 4.
9/29: Bijections and cardinality continued. Reading: Chapter 4.
10/1: Countable sets and properties of bijections between infinite sets. Reading: Chapter 4.
10/3: Finishing off cardinality, beginning real analysis. Reading: Chapter 4 and 13. Note: for the real analysis section we will be filling in the material in our book with material from a different book. Relevant scans of this book will be available on Illinois Compass.
10/6: Discussing supremum and infinimum, stating Completeness axiom. Reading: Chapter 13.
10/8: More on completeness axiom, the archimedean property. Reading: Chapter 13.
10/10: Beginning talking about sequences of real numbers and what it means to be a convergent sequence. Reading: Chapter 13.
10/13: Midterm 2 review.
10/15: Midterm 2. Midterm 2 Solutions
10/17: Sequences and limits. Reading: Chapter 13 in your book, sections 7,8,9 in scan of Ross.
10/20: Sequences and limits: proofs. Reading: Chapter 13 in your book, section 8 in scan of Ross.
10/22: More examples of sequences. Reading: section 8 in scan of Ross.
10/24: Limit theorems, definition of Cauchy sequence. Reading: Chapter 14 in your book, section 9 and 10 in scan of Ross.
10/27: Class cancelled due to illness.
10/29: More on Cauchy sequences, monotone sequences, Bolzano-Weierstrass theorem. Reading: Chapter 14 in your book, sections 10, 11 in scan of Ross .
10/31: Proving Bolzano-Weierstrass Theorem and that Cauchy sequences are convergent. Reading: Chapter 14 in your book, sections 10, 11 in scan of Ross.
11/3: Series: ratio test, root test, comparison test. Reading: Chapter 14 in your book, section 14 in scan of Ross.
11/5: Wrapping up series and real analysis section of the course. Beginning number theory. Reading: beginning of Chapter 6.
11/7: Greatest common divisors, prime factorization. Reading: Chapter 6.
11/10: Review for Midterm 3.
11/12: Midterm 3.Midterm 3 Solutions
11/14: More on divisibility, Euclidean Algorithm. Reading: Chapter 6.
11/17: Fundamental theorem of arithemtic. Reading: Chapter 6.
11/19: Introducing modular arithmetic. Reading: Chapter 7.
11/21: Modular arithmetic continued: divisibility by 3 and 9. Reading: Chapter 7.
12/1: Modular arithmetic continued, divisibility by 11, powers modulo primes. Reading: Chapter 7.
12/3: Fermat's little theorem, definition of Euler's phi function. Reading: Chapter 7.
12/5: RSA cryptosystem.
12/8: Review.
12/10: Review.
12/15: FINAL at 8AM in 241 Altgeld.

Some supplementary notes:

Homework assignments:

Graded homeworks can be picked up in office hours or in class on Wednesdays. Graded midterms can be picked up in office hours.

Homework 1, due 9/3/2014
Homework 2, due 9/10/2014
Homework 3, due 9/17/2014
There is no homework due on 9/24, since our first midterm is that day. Instead, here are some practice problems for the midterm for you to try. Here are the solutions.
Homework 4, due 10/1/2014
Homework 5, due 10/8/2014
There is no homework due on 10/15, since our second midterm is that day. Instead, here are some practice problems for the midterm for you to try. Here are the solutions.
Homework 6, due 10/22/2014
Homework 7, due 10/29/2014
Homework 8, due 11/5/2014
There is no homework due on 11/12, since our third midterm is that day. Instead, here are some practice problems for the midterm for you to try. Here are the solutions.
Homework 9, due 12/3/2014
Homework 9b, due 12/3/2014
Homework 10, due 12/10/2014

Instructions in event of emergency