Writing Mathematics Papers

Transcription

1 Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not fined tuned for you, but it still sould be useful. Matematics writing is different from ordinary writing and arder in addition to all te requirement of ordinary good writing, tere are additional constraints and conventions in matematics. An additional constraint is tat matematics follows muc more demanding rules of logic tan ordinary discourse, and your writing must follow and display tis logic. (However, tis does not mean tat tere is only one rigt way to present a matematical argument.) Some of te additional conventions are tose for defining new concepts and tose for organizing te material troug teorems and examples. Titles Every paper sould ave one. It sould be informative witout being too long. Coosing a good title is not so easy. Often a mat paper inges on a concept tat is only defined witin te paper itself, so to use te name of tat concept in te title will convey no meaning at all. Incidentally, wereas in some literary fields an obscure word-play is considered a good title, it is generally not well regarded in matematics. Teorems A mat paper is not complete witout some teorems and proofs. Eac teorem in your paper sould be igligted I suggest indenting it, putting extra vertical space around it, and using a different type style if tat s available to you. (Oter formats are allowed; browse and see.) Wy igligt teorems (and oter key items suc as examples)? Because matematical arguments can be so complex. Formal statements of teorems (and oter key items) serve as toucstones, like aving an outline witin te text body. Tey also makes it easy for te reader to refer back to key specifics. For te same reasons, for eac proof in your paper you sould indicate clearly were it begins and were it ends. Definitions Definitions are a major way tat mat writing differs from general writing. Most disciplines don t deal wit definitions nearly as muc as mat tey don t need to be so precise nor do tey deal so regularly wit situations outside common experience. Along wit te need to define words (derivative) comes te need to introduce notation ( d ): if you use a new dx concept frequently, you need a sortand way to refer to it. How and were definitions sould be presented. For te most part, you will be tempted to define a word immediately after you first used it. Tis sows lack of planning and lack of tinking about wat will read best. Wen tis policy leads to definitions being parceled out in twos and trees in te early part of te paper, it works fine. But if it means you interrupt a difficult, key proof late in te paper to give two more definitions, it is disruptive and confusing.

2 Writing Matematics page 2 Before you start writing a mat paper, you sould make a list of all te definitions you tink you need, and ten try to see to it tat tey all get introduced relatively early. (Of course, you will not yet realize some of te definition you will need, so after te first draft you will ave to go back and add more definitions.) Te extreme version of tis approac is to put all te definitions in a definitions section near te beginning. Tis isn t always best eiter: a long section of definitions can be deadly boring. Wen you give a definition, you can do it in-line (witin a paragrap), but te word or prase being defined sould be igligted, by underlining, by italic print, or by boldface. Boldface is now te most common format, since underlining is not common in typeset material and italic already as a use in bot mat and ordinary writing to indicate empasis. Here s an example of a definition: A prime number is a positive integer wit no positive integer divisors oter tan 1 and itself. Anoter format is to display definitions just like teorems are displayed. Te display format sould be reserved for te most important definitions. For instance, in Mat 4 te definition of derivative migt be displayed, but te definition of polynomial (if you need it at all) can be done in-line. Once you define a word, don t ever substitute anoter word wic is a synonym in ordinary Englis. For instance, if you ave defined slope, don t afterwards sometimes say steepness. Te wole point of matematics definitions is temporarily to give ordinary words some extra precision needed for a matematical discussion. But tere may be several different precise meanings tat are matematically useful and tat one can attac to te same looser ordinary concept. A trained matematical reader, upon seeing steepness in your paper after you ave defined slope, will assume you are using bot words because you need two different refinements of te ordinary slope concept. Suc a reader will ten attempt to find were you defined steepness and will get frustrated. Among te metods of indicating tat a word is being defined, I did not include quotes. Quotes may be used to indicate tat a word is being used in a somewat special way, but no declaration is tereby made tat tis special way applies for te rest of te paper. Here is an incorrect use of quotes: Te initial case P(n 0 ) of an induction is called te basis. Tis is incorrect because basis is being defined. Here is a correct use: In order to verify te inductive step, we first simplify n k=1 k2 to ( n 1 k=1 k2) + n 2. Te point is, it is a bit of a surprise to call tis a simplification (altoug it does simplify in te sense of leading to a solution) and we don t claim we will always use simplify in tis sense. Local Definitions. So far I ave been talking about global definitions, tose tat apply trougout your paper. Most global definitions introduce words or notation. In contrast, local definitions apply only briefly, say, to te current example or current paragrap. Usually local definitions are definitions of symbols, for instance, te f in let f(x) = x 2. Local definitions don t require special igligting, altoug tey sould be displayed if tey involve complicated formulas (see te section on displays later). In any event, te item being defined sould always come on te left-and side of te equation; e.g., let x 2 = f(x) is wrong. Wy?

3 Writing Matematics page 3 Because wen an equal sign is used wit let to make a definition, te equal sign as a special meaning: te quantity on te rigt is being assigned to te quantity on te left. Tis is quite different from regular equality, were it makes no difference if you write a = b or b = a. Examples Examples are like definitions, in tat tey can appear in-line or be igligted by indentation and extra spacing. For a very brief example, te in-line metod is fine. However, a lengtier example sould be displayed, especially if it is a key item of your work. Not only does tis format draw attention to your example, but is also makes it easy to refer back to later (e.g., see Example 3). If an example is a sample problem, make clear were te solution begins and ends. Figures Figures can be extremely elpful. Eac figure sould be numbered, referred to by number, and (preferably) be positioned in te paper sortly after te first reference to it. (Anoter convention is to put all figures at te end. If you use tis alternative, please say so te first time you reference a figure.) I d prefer eac figure to ave a caption as well; e.g., Te steeper te line, te greater te slope. If you know ow to use computer software to produce a good figure and place it into your paper, great, but I m perfectly appy if you insert your figures by and. You ll probably need more space tan you first tink. Footnotes Generally, footnotes are not used in matematics papers. First, te superscripts used to number tem migt be confused wit exponents. Second, it used to be tat matematical symbols weren t always available in te smaller type used for footnotes. (Some mat texts, including yours, do use footnotes occasionally, but publised paper still almost never do.) So wat do you do? If you need to refer to an outside source in te text, write [x] (were x is te number of te reference in te reference list at te end). For instance, if Goldstein et al. (te text for Mat 4) is reference 1 in te list at te end of your paper, you migt write: Te following application is treated in [1], or Te following is treated in Goldstein et al [1]. If you want to refer to a specific place in your reference, a teorem or example number is at least as valuable as a page number: Te following application is treated in [1, Tm 3.5]. Wen to Give Credit Te conventions for giving credit to oters are somewat different in matematics tan in general academic writing. Briefly put, direct quotes must be credited, but paraprase generally need not be. Furtermore, mat papers rarely include direct quotes. Tus, wile a umanities paper is full of footnotes giving credit, in a matematics paper te credits are less frequent (and tey appear in text, not in footnotes, as explained earlier). Tis is all elaborated below. Pay close attention, since failure to give credit wen it sould be given amounts to plagiarism and is a major academic sin.

4 Writing Matematics page 4 Rigtly or wrongly, matematics is regarded as aving an existence independent of te words used to describe it. Tus Goldstein may describe te Cain Rule teorem in sligtly different words tan anyone else, but tat doesn t give Goldstein any special credit. If you use te Cain Rule, and you learned about it from te Goldstein text, you sould not reference Goldstein at te point were you introduce te Cain Rule. Tis would be an embarrassment to Goldstein and is coautors, because tey would never claim credit for te Cain Rule temselves. Tis same principle applies to definitions; don t reference Goldstein for te definition of derivative. Note. Teorems ave two sorts of names: descriptive names, suc as Cain Rule ; and sequential names, suc as Teorem 6 and Limit Rule II. A text gives a descriptive name only if it as become widely used, so you can use te name Cain Rule too. But sequential names are specific to an individual text. Tus, if you must refer to Limit Rule II, you ave to reference Goldstein and give a page number oterwise readers won t ave a clue wat rule you are referring to. But wy make your readers look tis up wen you could simply state te rule in your paper? Te independent existence perspective also explains wy matematicians rarely quote eac oter directly your original words are generally not perceived to ave an advantage over my paraprase. Tis attitude is quite different from tat in umanities, were te nuances in somebody s verbalization of an idea may make all te difference. (If you feel you sould noneteless use a direct quote, ten write: As Smit says [3, p. 43] and ten display te quote as an indented paragrap.) Te original discoverer of a matematical result or metod is given credit for it, and if te result is fairly recent, te paper in wic te result first appeared must be referenced. However, if te result is classical, e.g., Euler s grap teorem, it suffices to call it Euler s Teorem; no reference to a source is necessary. (In matematics, a result is classical if it is a well known and already appears in many written sources; occasionally results proved as recently as 20 years ago ave already become classical.) So muc for giving credit for definitions and teorems. Sould examples be credited? Again, te answer is usually no. If te example is very unusual and involved, give credit. Also, if you use te exact numbers and words of an example from some book, ten give credit; but in general you souldn t be quoting problems verbatim anyway. In order to write about an example wit understanding you sould internalize it and make it your own before using it. And if you ave internalized it, you can recreate it (peraps wit different numbers) witout looking it up. Bibliograpy format. Several disciplines seem to ave special conventions for references. Te style used for most matematics is different from te style I was taugt in ig scool and from te style in te Cicago Style Manual (followed by some oter disciplines). Here are standard mat examples for journals and books. 1. J. E. Skeat, Holomorpic functions in 2 variables, J. Complex Analysis 31 (1966) D. Rosen and E. Mullins, Calculus wit Probability, Boden & Quigley, Piladelpia, 1968.

5 Writing Matematics page 5 (In te first, Holomorpic Functions in 2 Variables was an article appearing on pages of te Journal of Complex Analysis, Volume 31, wic volume appeared in 1966.) Fonts for Mat A font is a style of type, e.g., roman or italic. In books, letters representing matematical quantities, suc as a, x and f(z), are not set in te same roman font as ordinary text. Te default for mat letters is italic, as just done. However, certain quantities are traditionally printed in oter fonts, for instance, te vector x is usually boldface, te angle θ is Greek, large sets S are often script capitals, and certain autors like to use old-german letters, like R. More generally, if you ave several quantities, and eac as an Englis name beginning wit M, you migt want to represent eac of tem matematically by te single letter m. Ten you need to use a different font for eac meaning. Te previous paragrap describes wat books do. Wat sould you do? If you ave access to a good word-processing system, you will ave italic fonts available and you sould use tem. If you don t ave italics (for instance, you are using a traditional typewriter), ten put extra blank space around matematical letters and expressions. For instance, Consider te variables x and y in te equation y = 2x+3. Suc extra space is especially important around te letter a, since it could be eiter an Englis word or a mat symbol. Wile letters in matematical expressions are in italic, punctuation (like parenteses, brackets and exclamation points) are not. Tis makes writing a little tedious wen (as wit a basic Macintos system) all you ave are ordinary roman and ordinary italic fonts. In ordinary italic, punctuation is slanted too, so you ave to go back and fort from roman to italic just to write a single equation. Wat you really need are mat italic fonts, but none of te standard Macintos fonts are mat italic. A basic Mac system does ave subscripts and superscripts; learn ow to employ tem. Warning: Tere is a special problem wen Macintos MacWrite documents containing italic are printed wit a laser printer. Wen you sit at a Mac and type italic f followed by roman (, te slanted f will overlap te ( on te screen. You will be inclined to fix tis by putting a space between tem. If you do tis te laser output won t look rigt tere will be too muc space between te f and te (. Te problem is tat te laser output and te screen image are not exact duplicates. Fortunately, tere is a fairly simple fix. First, always write you mat papers using a laser font. Te fonts named after cities (Geneva, Cicago, etc.) are not laser fonts; most of te oters available wit MacWrite (Times Roman, Courier, Helvetica) are laser fonts. Since Geneva is te default font, you ave to pull down te font window to cange te font before you start writing. I especially recommend switcing to Times Roman. Second, don t correct te overlaps you see on te screen. Wen you print tings out, it will look fine. I suspect te same problem, and te same solution, obtain if you use te word-processing software Word on a Macintos, but I aven t tried it.

6 Writing Matematics page 6 Handwriting Matematics. If your metod of producing your paper gives you no special capabilities for printing matematics, you will need to insert te more complicated matematical displays of your paper by and. Even if you ave a computer wit mat italic fonts, complicated matematical displays are still ard to set. If you aven t set any complicated matematics by computer before, it s not wort it for you to spend ours learning ow to do it now. (On te oter and, any practicing matematician sould probably learn at least one suc system eventually, preferably te most powerful, wic currently is TEX.) If a display is going to take a lot of time to set, ten leave a blank space in your paper and write it in by and. At first, you will probably grossly underestimate te amount of space you need for and insertion. But if te rest of your paper is written on a computer, it will be easy to output anoter copy wit more space. Displays Any long expression wit matematical symbols is displayed centered on a line by itself, wit extra vertical space around it. Tere are tree reasons for displays. First, if an expression is particularly important, you draw attention to it by displaying it. Second, longer matematical expressions tend to be tall. To fit someting like 5 dx 2 x 2 in-line requires adding + 1 disconcerting interline space (see it?), or else making te symbols disconcertingly small, like tis: 5 2 dx x Tird, it is distracting to break an expression across a line. You can ypenate words, but mat expressions are usually longer tan words and don t bear ypenation well. So if an expression would ave to be broken if it appeared in te middle of a paragrap, it sould instead be put in a display. For instance, if you wis to say y = d dx (x2 + 3x) witin a line, it would be bad form to break tis after te + and terrible form to break it after. According to some writers, it is even bad form to break it after =. d dx A display sould be numbered if you find tat you refer to it anywere oter tan witin a few lines of it. Te numbering can appear eiter on te left, as in (1) y = d dx (x2 + 3x), or on te rigt, as in y = d dx (x2 + 3x), (2) but be consistent. Just below a display you can refer to it as te previous display, but two pages later it s too long-winded to refer to te tird display on page 7. Besides, if you modify your paper, you will ave to rewrite suc a reference completely. Tat s wy we use numbering.

7 Writing Matematics page 7 Multiline Displays. In most cases, tese sould be lined up so tat te main connectives (usually an =, but maybe or = ) fall rigt on top of eac oter: Similarly, one writes 2x + 1 = 2 + 3, 2x = 4, x = 2. x 2 + 2x < x 2 + 2x + 1 = (x+1) 2. (3) (4) Notice tat display (3) as expressions on bot sides of te equal sign on eac line, but display (4) as only a rigt-and side after te first line. Wen a display could ave been written as one long line (e.g., a = b < c = d = e), but is written in several lines for empasis or because it won t all fit on one line, tis rigt side only format is appropriate. Te fact tat calculations can be written as one long display witout words doesn t mean tat tey sould be written tis way. Matematics does not consist of calculations alone! Very elementary algebra, as in (3), need not be explained, but more complicated calculations sould be, especially if te calculations are justified by matematics tat te reader is just learning. Tere are two ways to provide te explanation. One (your text s way) is to put it before, between and after te lines of calculation. For instance, suppose you ave defined f(x) = x 2, and want to sow tat [f(x+) f(x)]/ = 2x+. You could write: Substituting te definition f(x) = x 2, f(x+) f(x) = (x+)2 x 2 Ten expanding te first square, combining like terms, and finally canceling te common factor, we obtain. f(x+) f(x) = (x2 + 2x + 2 ) x 2 2x + 2 = = 2x +. An alternative approac is to put all te reasons in comments on te rigt: f(x+) f(x) = (x+)2 x 2 [def. of f(x)] = (x2 + 2x + 2 ) x 2 [expand] 2x + 2 = [combine terms] = 2x +. [divide]

8 Writing Matematics page 8 Actually, for you seniors all of tese steps are standard enoug tat you could skip te commentary. But if you were a calculus student you souldn t skip it. In your own papers, any calculations tat weren t immediately obvious wen you first saw tem in your reading sould be explained. Punctuation Punctuation in displays. Notice te commas and te period in display (3). Matematics is written in sentences, and tis display is a sentence consisting of a sequences of clauses, te equations. Tus te clauses are separated by commas (some writers would use semicolons) until te sentence ends wit a period. However, tere is an alternative convention (followed by about alf te publisers in te US) wic leaves off punctuation at te end of displays. Personally, I m not so concerned about te commas (te separation into different lines tells me to pause) but I do feel strongly about te period. Sometimes a sentence does not end at te end of te display; tat is, I am supposed to keep reading in order to understand wy te display makes sense. On te oter and, if te display ends is a period, tat is a signal tat I am supposed to figure out for myself wy te display is legitimate. In oter words, te presence or lack of a period at te end of a display tells me weter tis is a point at wic I need to stop and digest wat as just been said. Since it is very important for te autor to give te reader cues like tat, te punctuation at te end of a display is crucial. In display (4) tere is no comma after te first line and tere never sould be. Wy? Punctuation in text. Consider te sentence If Mary goes, Jon, Sam and Alice will too. Te word ten is missing, but te meaning is clear and so suc an elipsis is quite acceptable Englis. However, if we leave ten out from If x = 3, ten 4, 5 and 6 are te next cases. we get If x = 3, 4, 5 and 6 are te next cases. Now we ave a major ambiguity: wic numbers are possible values for x and wic are te next cases? Moral: never separate matematical expressions tat are distinct (not merely part of te same list) by commas alone. Similarly, don t separate distinct matematical expressions by space alone. For instance, altoug te sentence We will call te value of P P(n) is perfectly reasonable in meaning, it is ard to read. Are P and P(n) being multiplied? So rewrite te sentence to get some words between P and P(n): Tis value of P we will call P(n). Paragrap and Sentence Divisions In reading student papers, I often find cases were a paragrap sould ave been broken

9 Writing Matematics page 9 up, or te last sentence of one paragrap belonged at te start of te next. Remember, writing isn t formatted into paragraps to look nicer on te page or to indicate were you took a deep breat. Paragraping is used to separate te development of one idea from te next. Similarly, a sentence sould make one point. If tere are several different points, tey sould go in several sentences. In a proof, a good rule of tumb is tat a sentence sould give just one deduction. A paragrap in a proof sould represent a major segment of a proof. For instance, if a proof of A B is sort, ten one paragrap sould be A= B and te oter B= A. If, owever, A= B is long, ten it too sould be broken in parts. For instance, if you are presenting a long proof tat ten devote one paragrap to Euler cycle = [connected & even degree], Euler cycle = connected, and anoter to Euler cycle = even degree. Two Common Mistakes Te first mistake is not to rely on matematical symbols enoug. If, for instance, you refer to te same function more tan once, give it a name (let f(x) = x 2 ). Even if you are just discussing functions in general, you need to give te generic function a name if you are going to say anyting about it. (e.g., for any function f(t), f (t) measures te rate at wic...). More generally, if you are working towards te precise formula in some definition or teorem, it may be wise to state te formula early on, before you ave even explained all te symbols tat occur in it. Tis way at least you ave someting concrete to refer to as you work troug your explanation. Oterwise you bog down in vague verbiage. Te second mistake is to rely on matematical symbols too muc. Tis appens wen you present several lines of computation witout any commentary and was discussed in te section on displays. Revising Papers Tis is touger tan you migt tink. If you ave to add or move some definitions, or add or move a teorem, you will ave to look over te wole paper to see wat else is affected. Given te tigt structure of good matematics writing, a cange in one place can and sould require adjustments in many oter places. Indeed, tere is even a riddle/joke about tis. Riddle: wat is te order in wic you write te capters of a mat book? Answer: 1,2,1,2,3,1,2,3,4,...

10 Writing Matematics page 10 Miscellaneous Please number all your pages. Tis is especially elpful if you sould staple tem out of order (wic appens) or if I get tem out of order wile reading (tis also appens). If you ave any questions or comments on te above, don t esitate to ceck wit me. If your question/comment is a good one (and most of tem are), I will send it and my reply to all your classmates via computer (unless you ask me to keep your inquiry and my response private). Finally, my writing isn t perfect eiter. If you find fault wit te writing above, please point tis out to me. We can all benefit from eac oter s criticism. end