Much of the information here can also be found in the course syllabus.

The purpose of this course is to develop techniques and skills for writing in the mathematical sciences. This will include three main objectives:

1. writing proofs,
2. writing expository essays on mathematical topics,
3. combining exposition and proofs to produce math research papers.

Although you will not be expected to produce your own research in this class, you will have assignments for which the writing style should mimic such work.

Grades

20% of your final grade will come from class participation. The remaining 80% will be based on your coursework.

Class Participation

Attendance will be taken during each class. Much of our class time will consist of discussing and evaluating fellow students' work. Your participation in this discussion will be noted and used to calculate your final grade.

Turning in Work

1. All assignments must be done in LaTeX, which is a mathematical typesetting program. It is each student's responsibility to learn how to install and use LaTeX on a computer of their choosing. See below for more info.
2. Unless otherwise stated, assignments are due on Wednesday at 8 AM and turned in via e-mail. Late assignments will be given half credit.
3. You must attach both the .tex file and the compiled .pdf file.
4. Send your assignments to the following e-mail: math300wed@gmail.com. This e-mail should be used only for turning in assignments. For anything else, use my UIC e-mail above.

Assignments

All assignments are due at 8AM on the date specified.

• Wednesday, January 22: proof 1
  Prove that the square root of two is irrational.
• Wednesday, January 29: essay 1
  Write an introduction to the concept of a derivative. Assume the reader is a student enrolled in a calculus course, and has already learned about limits.
• Wednesday, February 5: proof 2
  Let f be a function from the set of real numbers to itself. Prove that if f is differentiable at a real number c, then f is continuous at c.
• Wednesday, February 12: proof 3
  Let S be a partially ordered set, with the additional property that every chain s₀ ≤ s₁ ≤ s₂ ≤ ... has an upper bound in S (i.e. there is some t in S such that s_n ≤ t for all n). Suppose C is a countably infinite subset of S such that for every u,v in C there is some w in C such that u ≤ w and v ≤ w. Prove that C has an upper bound in S.
• Wednesday, February 19: rough draft of research paper 1
  This is a non-technical research paper about a famous mathematician. I would like you to include

  • a short biography of their life, including mathematical training and career,
  • an analysis of their work, focusing on its importance relative to the historical context,
  • a discussion of how their work impacts current mathematical research,
  • a bibliography of your sources.

  I don't expect you to read and understand the specifics of the person's individual research. Just try to have a general grasp of the area of their work, famous theorems they proved, etc...
  
  You will choose one of the following mathematicians:

  • Georg Cantor
  • Paul Erdos
  • Leonhard Euler
  • Evariste Galois
  • Carl Friederich Gauss
  • Kurt Godel
  • Benoit Mandelbrot
  • Emmy Noether
  • Bernhard Riemann
  • Alan Turing
  • Karl Weierstrauss

  E-mail me your top three choices.
• Wednesday, February 26: essay 2
  Discuss a mathematical paradox. Include a short description, why it is a paradox, and possible ways of resolving it. Examples of paradoxes include Zeno's paradox, the liar's paradox, and Russell's paradox.
• Wednesday, March 5: research paper 1
  This is a non-technical research paper about a famous mathematician. I would like you to include

  • a short biography of their life, including mathematical training and career,
  • an analysis of their work, focusing on its importance relative to the historical context,
  • a discussion of how their work impacts current mathematical research,
  • a bibliography of your sources.

  I don't expect you to read and understand the specifics of the person's individual research. Just try to have a general grasp of the area of their work, famous theorems they proved, etc...
  
  You will choose one of the following mathematicians:

  • Georg Cantor
  • Paul Erdos
  • Leonhard Euler
  • Evariste Galois
  • Carl Friederich Gauss
  • Kurt Godel
  • Benoit Mandelbrot
  • Emmy Noether
  • Bernhard Riemann
  • Alan Turing
  • Karl Weierstrauss

  E-mail me your top three choices.
• Wednesday, March 12: essay 3
  The following is a list of open problems in mathematics. Choose one and write a newspaper article announcing that the problem has been resolved. Be as creative as you want with the circumstances, but make sure to include a description of the problem that can be understood by a general audience.

  • Riemann Hypothesis
  • P versus NP
  • Erdos' conjecture on arithmetic progressions
  • Is the Euler-Mascheroni constant irrational?
• Wednesday, March 19: proof 4
  Prove that if \(n>0\), then \(\displaystyle\frac{(2n)!}{n!^2}\) is an even integer.
• Wednesday, April 2: essay 4
  There is an island, on which every person has either brown eyes or blue eyes. The people on the island live according to the following rules.

  • Everyone on the island is capable of perfectly logical reasoning.
  • No one knows the color of their own eyes.
  • Every person on the island can see the eyes of any other person.
  • There is no possible way for someone to see the color of their own eyes.
  • No person can communicate, in any way, to any other person about anything regarding eye color.

  Every day at 11:00 AM the people of the island get together in the square and talk to each other. Everyone has the chance to see everyone else. At noon, anyone who knows the color of their own eyes must leave the island forever. Everyone would witness this happening (if it ever did happen).
  
  One day, a single visitor, with blue eyes, arrives on the island. She spends a day with the islanders and makes eye contact with each one of them, but makes no mention of eye color. Then, at 10 AM the next day, she gets back in her boat to leave. Just before she sails away she says, "It was very nice to see other people with the same color eyes as me." All of the islanders hear this comment.
  
  What, if anything, happens on the island because of this?
• Wednesday, April 9: proof 5
  Given a positive integer n, suppose S is a subset of {1, 2,..., 2n} with |S| = n + 1. Prove that there are distinct a,b in S such that a divides b.
• Wednesday, April 16: proof 6
  Prove that there are infinitely many different lines that go through the origin and are also tangent to the curve y = sinx.
• Wednesday, April 23: rough draft of research paper 2
  Choose one of the following topics to write a research-style paper about. I will supply you with an outline of what to include. You will organize it however you wish, as long as you include all relevant definitions and theorems (with proofs). I encourage you to take a look at published math papers to get an idea of how your paper should look. In particular, the American Mathematical Monthly publishes articles on topics that are usually more accessible for undergraduates. You can browse this journal (and others) through MathSciNet with your UIC NetID.
  Topics

  • primes congruent to 1 modulo 4
  • sum of reciprocals of perfect squares
  • Euler-Mascheroni constant and the harmonic series
  • perfect numbers
  • cardinality and countable sets
  • the Golden Ratio
  • properties of e
  • properties of π
  • tilings of a checkerboard
  • Pythagorean triples

  Once again, the topics are first-come first-served. E-mail me your top three choices.
• Wednesday, April 30: no assignment due, we will continue discussing the final research paper.
• Wednesday, May 7: research paper 2
  Choose one of the following topics to write a research-style paper about. I will supply you with an outline of what to include. You will organize it however you wish, as long as you include all relevant definitions and theorems (with proofs). I encourage you to take a look at published math papers to get an idea of how your paper should look. In particular, the American Mathematical Monthly publishes articles on topics that are usually more accessible for undergraduates. You can browse this journal (and others) through MathSciNet with your UIC NetID.
  Topics

  • primes congruent to 1 modulo 4
  • sum of reciprocals of perfect squares
  • Euler-Mascheroni constant and the harmonic series
  • perfect numbers
  • cardinality and countable sets
  • the Golden Ratio
  • properties of e
  • properties of π
  • tilings of a checkerboard
  • Pythagorean triples

  Once again, the topics are first-come first-served. E-mail me your top three choices.

Class Handouts

These handouts consist of notes and comments we will discuss in class. If you miss the class, the notes in the handouts may be unclear. You are encouraged to email me if you have questions.

• 1/22 Handout
• 1/29 Handout
• 2/19 Handout
• 2/26 Handout
• 3/12 Handout
• 4/16 Handout

Writing Center

The English department runs a writing center, making an appointment is suggested. The website is here.

Math Learning Center

Located in 430 SEO. Tutoring is usually available for lower-level courses such as calculus, but you should be able to find help with basic proofs as well.

LaTeX Info

Installation: PC or Mac. Once you have LaTeX installed you are free to use your own file format. Or you can use this template (right-click and select 'Save link as...').

Other Stuff

• LaTeX template (right-click and select 'Save link as...').
• Some sample proofs (pdf), and the .tex file.
• Detexify can be helpful in figuring out how to type LaTeX symbols.
• A list of LaTeX symbols.
• Example of a list of references, similar to what you might find in a published journal article. pdf file, and the .tex file.
• Instructions on how to get to MathSciNet
  • Go to the Math Department homepage
  • Click the small "AMS" button located in the bottom right-hand corner of the screen. (show/hide picture)
    
  • If you are on a UIC computer (e.g. in the library) it should take you directly to MathSciNet. Or you may need to sign in with your UIC NetID and password.
  • Once you are at the MathSciNet page, click the tab for "Journals". Here you can search for journals, e.g. American Mathematical Monthly. (show/hide picture)
    
  • If you search for American Mathematical Monthly then you should click the first link that appears in the search list. That will take you to some info on the journal, including a link to JSTOR. (show/hide picture)
    
  • The JSTOR link will take you to an archive of volumes and articles that you can browse.