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. These
notation
and formatting conventions should already be familiar: they are almost
always followed in mathematics textbooks.

Most students are accustomed to writing problem solutions with the idea that the reader (namely, the teacher) will know how the problem is supposed to be done. For that situation, you simply want to demonstrate that you followed the correct steps. Formal math writing has a different objective: To communicate to someone an analysis you have performed to reach certain conclusions. You cannot assume your reader already knows how to solve whatever problem or prove whatever theorem is being discussed, and your goal is to make an argument that not only communicates your approach, but convinces the reader that your approach is valid. This is a valuable life skill, and is applicable is all sorts of writing, whether or not there is any mathematics involved. But because mathematical arguments can be made unnaturally exact and precise, it is generally clear whether they are valid or not. So, if you want to develop your abilities to write coherently, logically, and convincingly, mathematics is a good subject to practice in.

In formal math writing, it is never acceptable to say (or to imply) "you know what I mean." Your job is to say exactly what you mean. You must explain clearly what your reasoning is in a way that makes it evident that your reasoning is valid. At times, you may cite results from class or the reading to establish the correctness of your steps. At other times your steps will be standard algebraic or arithmetic methods, and you may assume the reader knows those.

Organization. Each problem in your formal problem set submission should begin on its own page. The left margin should be at least 3 inches wide for me to write comments. Also leave white space between paragraphs for the same purpose. Each polished problem should begin with a complete statement of the problem (which will often be a proposition to prove), as well as the problem number. 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 formal problem sets, 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.

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:

Samples. Here is a sample formal problem solution written by hand, with annotations in red to highlight the format and style. Here is one produced by word processing software, with annotations in yellow boxes.

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, and there may be others. Although I am pretty familiar with LaTeX , I know very little about these particular packages. Learning to use some version of LaTeX is probably worthwhile for math majors, and you may wish to experiment with this type of system. But it is not required and should not distract you from the primary objectives of the course.

Another option is to use MS Word, which includes an equation editor for formatting complicated mathematical expressions. In my 2010 version of Word, I start the equation editor by selecting an option from the insert menu, as shown below.

Something similar should work for newer versions of word. 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. If you use word, you can manually italicize variables and use font properties to create subscripts and exponents that appear in the running text. For anything more complicated than that, you should use the equation editor, either in-line or centered on a separate line.

Most students are accustomed to writing problem solutions with the idea that the reader (namely, the teacher) will know how the problem is supposed to be done. For that situation, you simply want to demonstrate that you followed the correct steps. Formal math writing has a different objective: To communicate to someone an analysis you have performed to reach certain conclusions. You cannot assume your reader already knows how to solve whatever problem or prove whatever theorem is being discussed, and your goal is to make an argument that not only communicates your approach, but convinces the reader that your approach is valid. This is a valuable life skill, and is applicable is all sorts of writing, whether or not there is any mathematics involved. But because mathematical arguments can be made unnaturally exact and precise, it is generally clear whether they are valid or not. So, if you want to develop your abilities to write coherently, logically, and convincingly, mathematics is a good subject to practice in.

In formal math writing, it is never acceptable to say (or to imply) "you know what I mean." Your job is to say exactly what you mean. You must explain clearly what your reasoning is in a way that makes it evident that your reasoning is valid. At times, you may cite results from class or the reading to establish the correctness of your steps. At other times your steps will be standard algebraic or arithmetic methods, and you may assume the reader knows those.

Organization. Each problem in your formal problem set submission should begin on its own page. The left margin should be at least 3 inches wide for me to write comments. Also leave white space between paragraphs for the same purpose. Each polished problem should begin with a complete statement of the problem (which will often be a proposition to prove), as well as the problem number. 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 formal problem sets, 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.

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:

- (Using word processing software) Variables should be italicized, or for vectors, set in bold typeface;
- 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);
- 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 assigned readings for this.
- 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.
- Feel free to include tables or figures
if
appropriate. These should be centered and may include a
label and/or caption if you wish. This is useful when one
problem solution includes more
than one table or figure, and you want a way to refer to a specific
figure. But sometimes no label or caption is needed, and you can
refer to the the figure below
or above without any
confusion.

- 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 n ∊ the integers." The 'word' iff is a permitted contraction of if and only if.

Samples. Here is a sample formal problem solution written by hand, with annotations in red to highlight the format and style. Here is one produced by word processing software, with annotations in yellow boxes.

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, and there may be others. Although I am pretty familiar with LaTeX , I know very little about these particular packages. Learning to use some version of LaTeX is probably worthwhile for math majors, and you may wish to experiment with this type of system. But it is not required and should not distract you from the primary objectives of the course.

Another option is to use MS Word, which includes an equation editor for formatting complicated mathematical expressions. In my 2010 version of Word, I start the equation editor by selecting an option from the insert menu, as shown below.

Something similar should work for newer versions of word. 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. If you use word, you can manually italicize variables and use font properties to create subscripts and exponents that appear in the running text. For anything more complicated than that, you should use the equation editor, either in-line or centered on a separate line.