Guidelines for Effective Mathematical Writing

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2. In The Name Of God MathWriting

3. Saeede Alinezhad Fateme zhendijani MathWriting

4. Different kinds of writing • Expository • Technical • Persuasive • Narrative • etc. MathWriting

5. General Guidelines on Mathematical Writing • Know your audience. • Be clear and concise. • Idiosyncrasies of mathematical writing MathWriting

6. Be clear and concise • Never use synonyms for technical terms. • Short and direct sentences often work best. • Don't worry too much about the rhythm of your technical writing • Eliminate unnecessary words. MathWriting

7. Idiosyncrasies of mathematical writing • The personal pronoun "I" is rarely used in technical writing. • If you don't like using passive voice, try using "we.“ • Use mathematical symbols to enhance the clarity and precision of your writing, not to make your writing look cool. MathWriting

8. The main steps to guide your writing of a math paper • First, tell the reader what the problem is. Make sure the name of the problem or proof is stated in the title of the paper and then simply state what you are going to solve. Example ☞ Title: Addition of numbers to equal four. Problem: In this paper we will solve for the summation of 2 + 2 MathWriting

9. The main steps to guide your writing of a math paper(cont.) 2. Then, before any explanation, state in WORDS the answer to your problem. Example ☞ The summation of two plus two is equal to four, meaning that when two is added to two, the answer is four with no remainder. MathWriting

10. The main steps to guide your writing of a math paper(cont.) • Discuss any assumptions you made and any formulas you will use. Example ☞ Using the basic notion of addition……(then I would state the definition of addition and perhaps give an abstract example of this such as x + y = z.) MathWriting

11. The main steps to guide your writing of a math paper(cont.) 4.Next, tell how the problem will be approached. Explain how you will do each step of the problem and tell why. Example ☞ By simply adding two whole numbers two and two and using the basic rule of addition I will determine the answer to be four. This is a crucial step for higher level math. My example does not need much explanation, but complex problems and proofs need to be explained clearly and concisely. I’ll stop my example here since it is pretty clear. MathWriting

12. The main steps to guide your writing of a math paper(cont.) 5. Label all diagrams, tables, pictures, etc. The reader needs to know exactly what he or she is looking at. Make a title that is clear, concise, and understandable! 6. Define terms and variables, explain formulas and their derivation (where did they come from?), make sure your math is correct! MathWriting

13. Some notes on grammar 1.Sentence 2.Avoid errors of grammar and meaning 3.Avoid sources of confusion and ambiguity 4.Points of principle 5. Make your mathematics flow with your text MathWriting

14. Sentences • Simple Statements • Commands • Compound sentences • Present participles :a warning MathWriting

15. Simple Statements • Each simple statement is about a single subject • The sentence has a verb which belongs to the subject. • The verb forms part of the predicate, which describes what the subject is doing/has done /is going to do or what state the subject is in. MathWriting

16. Commands • A command need not have a subject. (some people say the subject is implied) Examples : ☞ • “Let G be a group”. Compare with, “Let us take G to be a group.” • “Consider a set X”. Compare with, “We shall consider a set X.” MathWriting

17. Compound sentences • Each compound sentence is a string of clauses, each of which has its own subject/predicate pair .These are joined by conjunctions (e.g., and ,or) or appropriate punctuation. • A compound sentence should be able to be broken down into simple statements or commands without losing its sense. MathWriting

18. Present participles • The present participle is the -ing form of a verb, e.g., having ,being ,doing ,calculating ,etc . It is easy to misuse these at the start of a sentence. Examples of misuse: ☞ • Substituting(3)into(4), the integral becomes π²/4. MathWriting

19. Avoid errors of grammar and meaning • Use “less” and “fewer” correctly: “less” is for continuous variables, “fewer” is for whole numbers. ☞“The polynomial f(x) has fewer roots than g(x).” ☞“The value of this root is less than the other roots.” MathWriting

20. Avoid errors of grammar and meaning • Use “which” and “that” correctly. That restricts while whichamplifies and informs ☞“Consider the values of f(x), which are positive,” informs us that all the values of f(x) are positive. ☞“Consider the values of f(x) that are positive,” asks us to restrict our attention to the positive values only MathWriting

21. Avoid sources of confusion and ambiguity • Avoid using“if”outside of the implication “if…then…”. It can be replaced by “whether” to achieve greater clarity. Make sure you use the “if…then…” construction properly. ☞ Misuse:“We will see if the zero of f(x) is positive, then we will take the square root ,”says that we will take the square root regardless. Correct:“We will see whether the zero of f(x) is positive. If it is [then]we will take its square root.” MathWriting

22. Avoid sources of confusion and ambiguity • Make explicit all quantifiers in equations ☞ misuse:“Consider x²+y²=1 ,x< y”, Exactly what is meant here? ☞correct :“Consider x²+y²=1 for x < y”. MathWriting

23. Points of principle • Avoid contractions (it’s ,don’t, can’t, etc) in formal written work. • Avoid abbreviations like “i.e.” (use “that is”), ”e.g.” (use “for example”) and “etc.” (use “and so on”). Use them correctly if you use them at all and punctuate them properly. • Small whole numbers (less than ten )are usually written in text ☞example: “The first four theorems are of no use to us” MathWriting

24. Points of principle (cont.) • Prefer the active voice to the passive voice It helps keep the reader awake Passive: “The equation was solved using the standard method.” Active: “ We solved the equation using the standard method.” Passive: “ An example is now given to demonstrate the point.” Active: “ Let us now give an example to demonstrate the point.” • Capitalize proper nouns : Hermitian matrix ,Lagranges theorem ,the Heidelberg method MathWriting

25. Points of principle (cont.) • Avoid using logical symbols in text. Bad :”We consider x²-1 ∀ x ∈ Z”☒ Good :”We consider x²-1 for all integers x,” or “…for all x∈ Z.”☑ • Do not mix up symbols with text. Bad:” For all real numbers >0 …“☒ Good:” For all real numbers greater than zero …,”☑ or “For all x ∈ R with x>0 ..” MathWriting

26. Points of principle (cont.) • Do not start a sentence with a symbol. Bad:”G is a group with prime order.”☒ Good:” The group G has prime order”☑ • Use sparingly the following words : “actually” ,”in fact”, “very thing” ,”interesting” ,”Most”, “nice” ,”quite”. MathWriting

27. Make your mathematics flow with your text • Any sentence, whether or not it contains mathematical symbols, should obey the rules of grammar. The symbols all represent words or phrases, and it should be possible to translate the entire sentence into grammatical, well- punctuated English. Be particularly careful about this when dealing with sentences containing displayed equations. The display should not affect the grammar or punctuation. MathWriting

28. A mathematical writing checklist • Is your paper neatly typed? • Has the paper been proofread? In college, sloppy work is not appreciated. Do check over everything. • Is there an introduction? • Did you state all of your assumptions? • Are the grammar, spelling, and punctuation correct? • Is the writing clear and easy to understand? MathWriting

29. A mathematical writing checklist (cont.) • Are the mathematical symbols used correctly? • Are the words used correctly and precisely? • Are the diagrams, tables, graphs, and any other pictures you include clearly labeled? • Is the mathematics correct? • Did you solve the problem? MathWriting

30. Good Phrases to Use in Math Papers • Therefore (also: so, hence, accordingly, thus, it follows that, we see that, from this we get, then ) • I am assuming that (also: assuming, where, M stands for; in more formal mathematics: let, given, M represents ) • show (also: demonstrate, prove, explain why, find ) MathWriting

31. Good Phrases to Use in Math Papers (cont.) • (see the formula above ). (also: (see *), this tells us that . . . ) • if (also: whenever, provided that, when ) • notice that (also: note that, notice, recall ) • since (also: because ) MathWriting

32. Miscellaneous comments • Use mathematical notation correctly. As you learn to write more complicated formulas, it is all too easy to leave out symbols from formulas. Learn how to use symbols properly! • Try to write as simply and directly as possible. No one likes to read ponderous pretentious prose. • Don't turn in pages of unreadable scribbles to your professor. MathWriting

33. Miscellaneous comments (cont.) • Use language precisely and correctly. Make sure that the words you use really mean what you think they mean. • While it is a good idea to type your paper, you may have to leave out the formulas and insert them by hand later. MathWriting

34. Mathematical Ideas into Writing • Organizing your paper. • Writing for your audience. • Defining variables and formulas. • Using pictures in mathematics. MathWriting

35. THE END … • “Bad thinking never produces good writing” (Lamport). • Good writing promotes good thinking … MathWriting

36. books on mathematical and technical writing • Nicolas Higham, Handbook of Writing in the Mathematical Sciences, 2nd ed, SIAM • Lyn Dupre, BUGS in Writing: A Guide to Debugging Your Prose, 2nd ed, Addison-Wesley • Steven Krantz, A Primer of Mathematical Writing, AMS • Donald Knuth, Tracy Larrabee, Paul Roberts, Mathematical Writing, MAA Notes Number 14 (great, but hard to find) MathWriting

37. References: • A Guide to Writing Mathematicsby Dr. Kevin P. Lee. • Writing Math Research Papers: Enrichment for Math Enthusiasts, Dr. Robert Gerver • http://www.calumet.purduue.edu MathWriting

38. Any Question? MathWriting