# Math 120 Writing in the Major Paper

Clear writing is essential to mathematical communication, as you probably realize from reading better and worse mathematical texts. Good exposition is an acquired and important skill. Throughout this class, you'll received feedback on solutions to your problem sets, and you should use this to refine your ability to communicate your ideas clearly and effectively. This writing project will give you an opportunity to focus on your exposition, as opposed to absorbing new mathematical content.

Kamil Szczegot (the WIM grader) will be available for help with the writing project. You can contact him at kamil-at-math-dot-stanford-dot-edu. His office is 380-M, in the basement of the math building, and he has office hours Tuesdays 11am-12:30pm and Wed 8:30-10 am, or by appointment.

The project. Your goal is to present a readable and complete proof of the Fundamental Theorem of Finite Abelian Groups. The theorem is discussed in Section 5.2 (and in fact a more general form, the fundamental theorem for finitely generated abelian groups, is discussed there). A proof is sketched at the end of section 6.1, and there is enough there for you to work out a complete proof.

The mathematical content of this assignment is not intended to be the primary challenge. The point of this project is to concentrate on the exposition of the proof. Your paper should be 4-7 pages long; quality not length is what matters. Past papers in this class have gone on to win university writing awards, and I hope to nominate at least one paper this year too.

What you should do. Your target audience is a typical Math 120 colleague who has not yet read this section. Your target audience is not me or Kamil. If you have been frustrated by reading mathematical writing in the past (which you undoubtedly have), this is your chance to show how it should be done!

In the introduction, you should describe the theorem informally, and explain why the reader might want to know this fact, and why the theorem is important. Put it in some larger context. Give enlightening examples of the theorem in action. By the end of the introduction (but not at the start!) there should be a precise statement of the theorem. Prepare the reader for the proof by describing succinctly the main ideas and techniques used in the proof. Then give the proof itself. Be clear what results you are quoting, and try to use as little as possible from earlier in the text. The less self-contained the paper is, the less useful it is to the reader. Do not just say something like "by Theorem 4.2 of the book" --- state any invoked theorem precisely or else give it a descriptive name (such as "the First Isomorphism Theorem"). Your paper should be readable by someone who is familiar with the material of the text up until this proof, but who learned it from a different source.

You may want a brief conclusion, in which you highlight the key points of the argument, so your reader can remember them. This is an opportunity to make sure your reader has a big picture in mind. Ask yourself: what should the reader remember after reading this paper?

You don't need to define "group", "abelian group", etc.; your target is familiar with these notions, and can be assumed to have read everything up until this section of the book.

Use complete sentences. Do not use shorthand symbols and words when possible ("iff", right arrows, three dots for therefore, etc.) --- these shorthand symbols are useful for the author, and sometimes necessary during a lecture when time is in short supply, but they needlessly slow down the reader. But definitely use "usual" mathematical notation (of the sort used in the text).

Run your draft by someone else (ideally in the class).

Timeline

• November 7, noon: drafts handed in to Kamil Szczegot's mailbox. Drafts may not be handwritten.
• November 17: drafts returned with comments (you can pick them up in class November 18).
• November 21, noon: final papers handed in to Kamil Szczegot's mailbox.
• December 2: final papers should be handed back in class.

Resources

You can use whatever word processing or typesetting program you wish. The standard one used in mathematics, statistics, and other parts of science and engineering is called LaTeX, a version of Donald Knuth's famous TeX typesetting program. Implementations of LaTeX are available for free on all operating systems. A not-so-short introduction to LaTeX is availble here.

Here is a link to an article about writing mathematics well. Click on "In His Own Words".

You might want to be aware of the Hume Writing Center, which offers its services for any stage of the writing process. They have a good reputation, and if you use them, I'd be interested in hearing about your experience.

Acknowledgments. Thanks to Joan Licata for ideas from Math 171; and Kamil.