A general guide to math research

Here are some pointers for writing in Math:

- Use complete sentences, correct grammar, and correct punctuation.
- Don't submit your first draft. Read it, get someone else to read it, and find out where it's unclear.
- Shorten your writing. The fewer words you make your reader read, the better, provided all the ideas are still presented clearly. After writing a sentence, examine it for phrases that have already occurred earlier in the text and try to eliminate them.
- Don't expect mathematical formulas and figures to speak for you. Refer explicitly to each one and tell the reader what you want the reader to get from it.
- Many mathematical adjectives and nouns have precise mathematical meanings, and an English synonym will not serve as a replacement. For example, "element" and "part" are not interchangeable when referring to an element of a set.
- Look at examples of writing proofs well in the book, and try to emulate the style.
- Global organization of a proof is important. Tell the reader what you are about to do, and then do it. Use paragraphs to delineate the parts of a proof.
- Define all symbols before using them.
- Start each sentence with a word, not a mathematical symbol.
- Two mathematical expressions or formulas in a sentence should be separated by more than just a space or by punctuation; use at least one word.
- When proving a statement of the form P=>Q don't mistakenly prove Q=>P by starting with Q and deducing something from it.
- Words have meanings: be aware of them. For example, an equation has an equal sign in it.
- Never say "it is easily verified that ..." or "it is easy to see that ...". The temptation to write such a phrase indicates you know some further justification is required, so provide something with content.
- Don't use abbreviations.

Grayson, D. R. (n.d.). *Math 248, section D1*. Retrieved from http://www.math.uiuc.edu/~dan/Courses/2003/Fall/248/