Math 412/612 Modern Algebra  Fall 2016
Math Writing and Polished Work

Overview.  Formal mathematical writing is the primary means by which mathematical knowledge is organized, shared, and preserved.  New discoveries are often initially presented orally at conferences and in small groups, but without a formal written record, this knowledge will quickly be forgotten.  One goal of this course is to give students practice with mathematical writing, which involves some special notational and formatting conventions.   For certain homework problems you will be asked to prepare a careful formal solution following the guidelines on this page.  These problems will be referred to as Polished Work.  The notation and formatting conventions should already be familiar: they are almost always followed in mathematics textbooks.

Organization.  Each polished work exercise will be completed first as a normal homework problem.  After that is reviewed by the instructor and returned, you will prepare a polished second draft.   Each polished problem should begin on its own page.  It should begin with a complete statement of the problem (which will often be a proposition to prove), as well as a problem and section number (or something similar for extra problems that do not come from the text).
You may wish to organize your work by proving lemmas that you can cite in the main proof. In this case, each lemma should have a clear statement and proof separate from the main question.

Solutions may either be handwritten or prepared with word processing software.  Word processing has some advantages: revisions and corrections are easier to produce, and the finished product is easier to read.  But if you are inexperienced using word processing software for mathematical writing, a hand written approach may be faster, at least initially.  Each student is asked to use software for at least some of the work in his or her portfolio, so that one outcome of the course will be a familiarity with this approach to mathematical writing.  More information about word processing software appears below.

The polished work is supposed to be your own work. At times, you may have to seek help from me or someone else to find a correct proof for a problem. In this case, please indicate the nature of the help in a brief statement at the top of the page. Tell who supplied the help, and what the nature of the help was. If the problem was solved in class, and if that solution made a significant contribution to your polished solution, say so.

Format.  Mathematical writing follows the usual rules of grammar, including the use of complete sentences, organization into paragraphs, correct punctuation and capitalization, etc.  In addition, there are a few conventions that are specific to mathematics:
1. (Using word processing software) Variables should be italicized, or for vectors, set in bold typeface;
2. Mathematical equations and inequalities may be included in symbolic form (although, when read aloud, they should make sense in the context of the surrounding material);
3. Equations, inequalities, and expressions may either appear in-line within the surrounding text, or may be displayed on separate lines. Displayed lines should be centered on the page, and may be numbered for reference.  Follow the format of the textbook for this.
4. All writing should appear in either normal paragraph formatting or centered displayed lines of mathematical symbols, but not a combination.  Do not introduce unusual indentation schemes.
5. Feel free to include tables or figures if appropriate.  These can be formatted as on pages 33 and 277 of the text, without any labels or captions.  On the other hand, a caption and a label can be useful if one problem solution includes more than one table or figure, and you want a way to refer to a specific figure.
6. Do not use mathematical symbols as shorthand.  For example, do not insert a ∃ in a sentence to mean there exists and do not use arrows as a substitute for words.  Logical symbols are generally only permitted as part of symbolic portrayals of formal logical propositions.  The logical symbols for element of  and subset of are permitted within running text though as a general rule, such mathematical symbols should appear only as part of a larger complete mathematical statement.  Thus, it is permitted to write "Suppose n ∊ ℤ" but not  "Suppose nthe integers."   The 'word' iff is a permitted contraction of if and only if.
Word Processing Software.    The industry standarrd for writing in mathematics and several other technical fields is LaTeX, and its variants.   Two share-ware packages for LaTeX are Lyx and Miktex.  Although I am pretty familiar with LaTeX , I know very little about these particular packages.

Another option is to use MS Word, which includes an equation editor for formatting complicated mathematical expressions.  You can find it using pull down menus as follows:   Insert   >>  Object   >>   Microsoft Equation 3.0.  (This is illustrated below for an older version of Word, but something similar should work for the newer versions).  Once you have the equation editor open, with a little experimentation you will see how to create mathematical expressions.  Feel free to ask me for assistance with this.