**Course Description** -- MTHT 313 (Section Number 23217) -- Fall 2014

**Instructor:** Louis H. Kauffman

**Office: **533 SEO

**Phone: **(312) 996-3066

**E-mail: ** kauffman@uic.edu

**Web page: **http://www.math.uic.edu/~kauffman

**Office Hours**: 3PM to 4PM on Monday, Wednesday, Friday or by appointment.

**Course Hours**: 12:00PM to 12:50PM on Monday, Wednesday, Friday in 315 LH.

This is a first course in Mathematical Analysis, the foundations of real numbers and the foundations of calculus.

The textbook is "Understanding Analysis" by Stephen Abbott.

** First Exam ** will be on Wednesday, October 8, 2014.
This exam will cover Chapter 1 and Chapter 2 including Section 2.6, but not beyond this Section.

** First Exam ** See Copy of First Exam.

** Second Exam ** See Copy of Second Exam.

** Third Exam ** See Copy of Third Exam.

** First Quiz ** See Copy of Quiz1.

** Second Quiz ** See Copy of Quiz2.

** Third Quiz ** See Copy of Quiz3.

** Fourth Quiz ** See Copy of Quiz4.

** Fifth Quiz ** See Copy of Quiz5.

** Sixth Quiz ** See Copy of Quiz6.

** Sample Final Exam ** See Sample Final.

** FINAL EXAM ** will be on Tueday, December 9, 2014. It will be held from 8AM to 10AM in 315 TH.
A review session for the final exam will be available in 533SEO on Monday, December 8, 2014 from 3:30PM to 5:00PM

** Second Exam ** will be on Monday, November 3, 2014.
This exam will cover Chapter 1 and Chapter 2 including Section 2.7, but not beyond this Section.
Use the first exam and the quizzes to review for the second exam.

**First Assignment:** Obtain a copy of the textbook. Read Chapter 1, particularly sections 1.1 and 1.2. Do the exercises
(all of them) on pages 11 and 12. Hand in these exercises on Friday of the second week.
Discussion Problem: Let x_{n+1} = 1/x_{n} + x_{n}/2 for n = 1,2,3... and x_{1}= 1. Show that
the sequence {x_{n}: n =1,2,3,...} converges to the square root of two.

**Second Assignment:** All exercises on page 17 and 18. All exercises on pages 27,28,29.
Homework is due on Friday, September 19, 2014,

**Third Assignment:** Read all of Chapter 2. All exercises on pages 43 and 44.
All exercises on pages 49,50. Homework is due on Friday, October 3, 2014.

**Fourth Assignment:** All exercises on pages 53,54 and 57,58 and 61,62.
This homework is due on Friday, October 17,2014.

**Fifth Assignment:** All exercises on pages 67,68,69.
This homework is due on Friday, October 31,2014.

**Sixth Assignment:** Pages 82-84. Problems 3.2.1,3.2.2,3.2.3,3.2.5,3.2.7,3.2.8,3.2.9,
3.2.12, 3.2.13.
This homework is due on Wednesday, November 12, 2014.

**Seventh Assignment:** Pages 87-89. Problems 3.3.2,3.3.3,3.3.4,3.3.5,3.3.6,3.3.7,3.3.8,3.3.9.
Read the remainder of Chapter 3.
This homework is due on Friday, November 21, 2014.

** Third Exam ** will be on Monday, November 24, 2014.
This exam will cover Chapter 1 Chapter 2 (except for 2.8) and Chapter 3 through Section 3.3.
You are also responsible for all problems on all quizzes including Quiz Number 6.

**Eighth Assignment:** Pages 92-96. Problems 3.4.1 - 3.4.10, 3.5.1-3.5.3, 3.5.8-3.5.9
Read the remainder of Chapter 3. We will discuss this homework in the week of December 1-5.
Parts of it will be on the Final Exam. You do not have to hand in this homework assignment.
Read forward into the rest of the book, comparing what you see with what you know from calculus.
We will discuss that as well in the last week.

See Cantor-Schroeder-Bernstein This is a proof of the Cantor-Schroeder-Bernstein Theorem.

See Least Upper Bound This is the Wiki Article about the Least Upper Bound Axiom and its conseqeunces.

See Ordinals This is the Wiki Article about the Transfinite ordinal numbers. In class we discussed how the set of all countable ordinals is by logical necessity uncountable! This first uncountable ordinal is called Aleph_1. Aleph_0 is the first infinite ordinal and it "is" the natural numbers. Cantor's continuum problem was to know if Aleph_1 has the same cardinality as the real numbers R. The results of Goedel and Cohen show that Cantor's Continuum Problem (as it is called) is independent of the axioms for Standard Set Theory.

See ZFC This is the Wiki Article about the axioms for Zermelo-Frankel Set Theory.

See Calculus. This is a set of notes on calculus using infinitesimals.

See Peano Arithmetic for an approach to axiomatic arithmetic via the Peano axioms for the natural numbers. In these notes we use nested marks (right angle brackets) to represent natural numbers. These notes will evolve. The present version just presents the axioms and some commentary.

See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. We will discuss the problems in class when appropriate.

See Landau's Foundations for a working out of the Peano axioms from a classic book by Edmund Landau. Landau wrote his book in 1929 and you should read the introduction for the student to experience his very upright attitude toward learning mathematics.

See ContFrac. This is a calculator on the web that converts numbers to their continued fracations. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See VanPoorten. This is an excellent article on continued fractions.

See Recreations. This is an access page to programs and mathematical recreations.