Word-smithing is a much greater percentageof what I am supposed to be doing in life
than I would ever have thought.DONALD KNUTH [6, p. 54]
Writing a Math Phase Two Paper
Steven L. Kleimanwith the collaboration of Glenn P. Tesler
c©February 2, 2005
Abstract. We discuss the kind of writing that’s appropriate in a paper submitted to the Mathdepartment to complete Phase Two of MIT’s writing requirement. First, we review the generalpurpose of the requirement and the specific way of completing it for the Math department.Then we consider the writing itself: the organization into sections, the use of language, andthe presentation of mathematics. Finally, we give a short example of mathematical writing.
1. Introduction. MIT established the writing requirement to ensure that its gradu-ates can write both a good general essay and a good technical report. Correspondingly,the requirement has two phases, which engage students at the beginning and towardthe end of their undergraduate careers. The requirement is governed by an institutecommittee, the Committee on the Writing Requirement (CWR). The requirement is ad-ministered by the Office of the Dean of Students and Undergraduate Education, whichworks in cooperation with the individual departments on Phase Two. The general in-formation given here about the requirement is taken from the MIT Bulletin and theCWR’s brochure [3], which are the official sources.
To complete Phase One, students must achieve a suitable score on the College BoardAchievement Test or Advanced Placement Examination, pass the Freshman Essay Eval-uation, pass an appropriate writing subject in Course 21 and be certified by the instruc-tor, or write a satisfactory five page paper for any MIT subject, Wellesley exchangesubject, or UROP activity. In level, format, and style, a Phase One paper should belike a magazine article for an informed, but general, readership. Papers are judged ontheir logical structure, language and tone, technical accuracy, and mechanics (grammar,spelling, and punctuation) by the instructor of the subject and by evaluators for theOffice of the Dean of Students and Undergraduate Education. A paper judged not ac-ceptable may be revised and resubmitted twice. Students must complete Phase One bythe middle of their third semester at the Institute.
To complete Phase Two, students must receive a grade of B or better for the qualityof writing in a cooperative subject approved by the student’s major department, receivea grade of B or better in one of several advanced classes in technical writing, or writea satisfactory ten-page paper for any MIT subject or UROP activity approved by themajor department. A student with two majors needs only complete the requirementin one department. In level, format, and style, a Phase Two paper should be like
1
2 MIT Undergraduate Journal of Mathematics
a formal professional report. Thus a term paper or laboratory report may have tobe reworked substantially before it is acceptable as a Phase Two paper. The paperis judged by its supervisor primarily for the technical content and by departmentalevaluators primarily for the quality of the writing. Students must complete Phase Twoby the end of registration day of their last semester; otherwise, they must petition theirdepartments and the CWR. Petitions for permission to enroll in a writing subject areroutinely approved; petitions to submit a late paper are approved only when there areexceptional circumstances.
In the Department of Mathematics, there is no cooperative subject, and most stu-dents write a paper to satisfy Phase Two. These students may also receive three unitsof credit by signing up for 18.098, Independent Activities. Each year in the spring,the department collects the papers, and publishes them here in the MIT UndergraduateJournal of Mathematics.
A Phase II paper normally begins as a term paper for a mathematics class, but everypaper must have an MIT supervisor and include some technical mathematics. Whenthe student and the supervisor feel the paper is ready, the student picks up a coversheet, which is available in the Undergraduate Mathematics Office, Room 2–108. Thestudent fills out the top, and gives the sheet to the supervisor, who must vouch for thepaper’s technical accuracy. The student then submits the paper and the cover sheet tothe departmental coordinator. The paper must be submitted by the start of IAP if thestudent intends to graduate the following June.
After a paper is submitted, the math department’s coordinator reads it for thequality of the writing, and determines whether or not the paper is acceptable as it stands.If the paper needs improvement (most do), then the coordinator and the department’sWriting TA discuss the paper. The TA contacts the student and sets up an appointmentto discuss the areas requiring further work. The student submits further revisions tothe TA, and when the paper is ready, it is resubmitted to the coordinator. Often, thecoordinator works directly with the student. Thus, not only is the paper improved, but,more importantly, the student learns how to write better. The process is tutorial.
This paper is a primer on mathematical writing, especially the writing of shortpapers. Indeed, this paper itself is intended to be a model of format, language, and style.Mathematical writing is primarily a craft, which any student of mathematics can learn.Its aim is to inform efficiently. Its basic principles are discussed and illustrated here.Some of these principles are simple matters of common sense; others are conventionsthat have evolved from experience. None need be followed slavishly, but none shouldbe broken thoughtlessly. When one is broken, the break may stand out like a sorethumb—just as unconventional spelling does. However, the writing itself should fadeinto the background, leaving the information to be conveyed out front. Abiding bythese principles will not cramp anyone’s style; there’s plenty of room for individualvariation. The various principles themselves are discussed more fully in a number ofworks, including the following works on which this primer is based: Alley’s down-to-earth book [1], Flanders’ article [4] and Gillman’s manual [5] for authors of articles forMAA journals, the notes [6] to Knuth’s Stanford course on mathematical writing, andMunkres’ brief manual of style [7].
In Section 2, we discuss the normal way a short mathematical paper is broken intosections. We consider the purpose and content of the individual sections: the abstract,the introduction, the several sections of the main discussion, the conclusion (which israre in a mathematical work), the appendix, and the list of references. In Section 3,
Writing a Math Phase Two Paper 3
we deal with “language,” that is, the choice of words and symbols, and the structuringof sentences and paragraphs. We consider seven goals of language: precision, clarity,familiarity, forthrightness, conciseness, fluidity, and imagery. We discuss the meaningof these goals and how best to meet them. Sections 2 and 3 are based mainly onAlley’s book [1]. In Section 4, we deal with a number of special problems that arise inwriting mathematics, such as the treatment of formulas, the presentation of theoremsand proofs, and the use of symbols. The material is drawn from all five sources citedabove. In Section 5, we give an illustrative sample of mathematical writing. We treatthe two fundamental theorems of calculus, for the most part paraphrasing the treatmentin Apostol’s book [2, pp. 202–204]; we state and prove the theorems, and explain theirsignificance. Finally, in the appendix, we deal with the use of such terms as lemma,proposition, and definition, which are common in treatments of advanced mathematics.
2. Organization. Most short technical papers are divided up into about a half-dozensections, which are numbered and titled. (The pages too should be numbered for easyreference.) Most papers have an abstract, an introduction, a number of sections ofdiscussion, and a list of references, but no formal table of contents or index. On occasion,papers have appendices, which give special detailed information or provide necessarygeneral background to secondary audiences. Normally, the abstract is three-to-six lineslong; the list of references has three-to-nine entries; and each remaining section fillsone-to-three pages.
In some fields, papers routinely have a conclusion. This section is not present simplyto balance the introduction and to close the paper. Rather, the conclusion discusses theresults from an overall perspective, brings together the loose ends, and makes recom-mendations for further research. In mathematics, these issues are almost always treatedin the introduction, where they reach more readers; so a conclusion is rare.
Sectioning involves more than merely dividing up the material; you have to decidewhat to put where, what to leave out, and what to emphasize. If you make the wrongdecisions, you will lose your readers. There is no simple formula for deciding, becausethe decisions depend heavily on the subject and the audience. However, you muststructure your paper in a way that is easy for your readers to follow, and you mustemphasize the key results.
The title is very important. If it is unclear or misleading, then it will not attractall the intended readers. A strong title identifies the general area of the subject and itsmost distinctive features. A strong title contains no secondary details and no symbols.A strong title is concise—rather short and to the point.
The abstract is the most important section. First it identifies the subject; it repeatswords and phrases from the title to corroborate a reader’s first impression, and it givesdetails that didn’t fit into the title. Then it lays out the central issues, and summarizesthe discussion to come. The abstract includes no general background material andpreferably no symbols. It just summarizes the contents. The abstract allows readersto decide quickly about reading on. Although many will decide to stop there, thepotentially interested will continue. The goal is not to entice all, but to inform theinterested efficiently. Remember, readers are busy. They have to decide quickly whetheryour paper is worth their time. They have to decide whether the subject matter is ofinterest to them, and whether the presentation will bog them down. A well-writtenabstract will increase the readership.
The introduction is where readers settle into the “story,” and often make the final
4 MIT Undergraduate Journal of Mathematics
decision about reading the whole paper. Start strong; don’t waste words or time. Yourreaders have just read your title and abstract, and they’ve gained a general idea of yoursubject and treatment. However, they are probably still wondering what exactly yoursubject is and how you’ll present it. A strong introduction answers these questions withclarity and precision, but in nontechnical terms. It identifies the subject precisely, andinstills interest in it by giving details that did not fit into the title or abstract, suchas how the subject arose and where it is headed, how it relates to other subjects andwhy it is important. A strong introduction touches on all the significant points, andno more. A strong introduction gives enough background material for understandingthe paper as a whole, and no more. Put background material pertinent to a particularsection in that section, weaving it unobtrusively into the text. A strong introductiondiscusses the relevant literature, citing a good survey or two.
Finally, a strong introduction describes the organization of the paper, making explicitreferences to the section numbers. It summarizes the contents in more detail than theabstract, and it says what can be found in each section. It gives a road map, whichindicates the route to be followed and the prominent features along the way. This roadmap is essentially a table of contents in a paragraph of prose. It is always placed at theend of the introduction to ease the transition into the next section.
The body discusses the various aspects of the subject individually. In writing thebody, your hardest job is developing a strategy for parceling out the information. Everypaper requires its own strategy, which must be worked out by trial and error. Thereare, however, a few guidelines. First, present the material in small digestible portions.Second, don’t jump haphazardly from one detail to another, and don’t illogically makesome details specific and others generic. Third, try to follow a sequential path throughthe subject. If such a path doesn’t exist, simply break the subject down into logicalunits, and present them in the order most conducive to understanding. If the units areindependent, then order them according to their importance to the primary audience.
There are three main reasons for dividing the body into sections: (1) the divisionindicates the strategy of your presentation; (2) it allows readers to quickly and easilyfind the information that interests them; and (3) it gives readers restful white space,allowing them to stop and reflect on what was said. Make the introduction and theseveral sections of the body roughly equal in length. When you title a section, strivefor conciseness, precision, and clarity; then readers will have an easier time jumping toa particular topic. Don’t simply insert a title, as is often done in newspaper articles, torecapture interest; rather, wind down the discussion in the first section in preparationfor a break, and then restart the discussion in the next section, after the title. When yourefer to Section 3, remember to capitalize the word “Section”; it is considered a propername. Don’t subsection a short paper; the breaks would make the flow too choppy.
Accent each main point via stylistic repetition, illustration, or language. Stylisticrepetition is the selective repetition of something important; for example, you shouldtalk about the important points once in the abstract, a second time in the introduction,and a third time in the body. When appropriate, repeat an important point in a figureor diagram. Finally, accent an important point with a linguistic device: italics, boldface,or quotation marks; a one-sentence paragraph; or a short sentence at the end of a longparagraph. In particular, set a technical term in italics or boldface—or enclose it inquotation marks if it is only moderately technical—once, at the time it is being defined.Do not underline when italics or boldface is available. Use headings such as Table 1-1, Figure 1-2, and Theorem 5-2, and refer to them as Table 1-1, Figure 1-2, and
Writing a Math Phase Two Paper 5
Theorem 5-2; note that the references are capitalized and set in roman. When youemploy linguistic devices, be consistent: always use the same device for the same job.
The list of references contains bibliographical information about each source cited.The style of the list is different in technical and nontechnical writing; so is the style ofcitation. In fact, there are several different styles used in technical writing, but theyare relatively minor variations of each other. The style used in this paper is commonlyused in contemporary mathematical writing.
The citation is treated somewhat like a parenthetical remark within a sentence, butthe reason for the citation must be immediately apparent. Footnotes are not used;neither are the abbreviations “loc. cit.,” “op. cit.,” and “ibid.” The reference key,trafitionally a numeral, is enclosed in square brackets. Within the brackets and after thereference key, place—as a service—specific page numbers, section numbers, or equationnumbers, preceded by a comma; see Gillman’s book [5, p. 9]. The reason for the citationmust be immediately apparent, and governs its placement, for example, after a mentionof an author’s name or work. If the citation comes at the end of a sentence, put theperiod after the citation, not before the brackets or inside them. In the list of reference,give the full page numbers of each article appearing in a journal, a proceedings volume,or other collection; do not give the numbers of the particular pages cited in the text.
3. Language. In the subject of writing, the word “language” means the choice ofwords and symbols, and their arrangement in phrases. It means the structuring of sen-tences and paragraphs, and the use of examples and analogies. When you write, watchyour language. When it falters, your readers stumble; if they stumble too often, they’lllose their patience and stop reading. Write, rewrite, then rewrite again, improving yourlanguage as you go; there is no short cut!
Alley [1, pp. 25–130] identifies seven goals of language: two primary goals—precisionand clarity—and five secondary goals—familiarity, forthrightness, conciseness, fluid-ity, and imagery. These goals often reinforce one another. For example, clarity andforthrightness promote conciseness; precision and familiarity promote clarity. We willnow consider these goals individually.
Being precise means using the right word. However, finding the right word can bedifficult. Consult a dictionary, not a thesaurus, because the dictionary explains thedifferences among words. For example, the American Heritage Dictionary is a goodchoice, because it has many notes on usage. Consult a book on usage, such as Web-ster’s Dictionary of English Usage. Always consider a word’s connotations (associatedmeanings) along with its denotations (explicit meanings); the wrong connotations cantrip up your readers by suggesting unintended ideas. For example, the word “adequate”means enough for what is required, but it gives you the feeling that there’s not quiteenough; its connotation is the exact opposite of its denotation. Strong writing doesnot require using synonyms, contrary to popular belief. Indeed, by repeating a word,you often strengthen the bond between two thoughts. Moreover, few words are exactsynonyms, and often, using an exact synonym adds nothing to the discussion.
Being precise means giving specific and concrete details. Without the details, readersstop and wonder needlessly. On the other hand, readers remember by means of thedetails. Being precise does not mean giving all the details, but giving the informativedetails. Giving the wrong details or giving the right ones at the wrong time makes thewriting boring and hard to follow. Being specific does not mean eradicating generalstatements. General statements are important, particularly in summaries. However,
6 MIT Undergraduate Journal of Mathematics
specific examples, illustrations, and analogies add meaning to the general statements.Being clear means using no wrong words. An ambiguous phrase or sentence will
disrupt the continuity and diminish the authority of an entire section. A commonmistake is to use overly complex prose. Don’t string adjectives together, especially ifthey are really nouns. Many high quality pure mathematics original research journalarticle sentences illustrate this problem.
Keep your sentences simple and to the point. Avoid long subjects. A sentence inwhich a lot goes on between the noun and the verb is hard to read. But a sentenceis easy to read when little goes on between the noun and the verb. Need to expressa complex idea? Then use several short sentences. Readers are thus led to stop andreflect. However, you do need some longer sentences to keep your writing from soundingchoppy and to provide variety and emphasis.
A pronoun normally refers to the first preceding noun. However, sometimes it refersbroadly to a preceding phrase, topic, or idea. This should be avoided. Make surethe reference is immediately clear, especially with “it,” “this,” and “which.” Considerrepeating the antecedent or summarizing it.
It is common to use a plural pronoun such as “their” to refer back to a singular, butindefinite, antecedent such as “reader.” This usage is still considered unacceptable informal writing; reformulate your sentence if necessary.
The pronouns “that” and “which” are not always interchangeable. Either may beused to introduce a restrictive clause, but use “that” ordinarily. Only “which” may beused to introduce a descriptive clause, and the clause must be set off with commas. Intheir classic guide to style [8, p. 47], Strunk and White recommend “which-hunting.”
Punctuation is used to eliminate ambiguities in language, and to ease the flow of thetext. Learn how to punctuate properly. Develop the habit of consulting a handbooklike The Chicago Manual of Style. When punctuation is optional, use it if it promotesclarity, but strive for consistency through out the paper. Here are a few rules.
Use periods only to end sentences. (A complete sentence within parentheses shouldbegin with a capital letter and end with a punctuation mark, unless the sentence is partof another and would end with a period.) Avoid abbreviations that require periods; forexample, write “MIT” instead of “M.I.T.” and use “that is” instead of “i.e.” Always usecommas to separate three or more items in a list and to set off contrasted elements (theyoften begin with “but” or “not”). Most of the time, use a comma after an introductoryword, phrase, or clause.
Use colons to introduce lists, explanations, and displays, but not lemmas, theorems,and corollaries. Do not use colons in continuing statements: if a statement is stopped atthe colon, then the introductory words should form a complete sentence. For example,don’t write, “Use colons to introduce: lists, explanations, and displays.” Use a semicolonto join two sentences to indicate that they are closely linked in content; however, if youinsert a conjunction, not an adverb, then use a comma.
Use a dash as a comma of extra strength—but use it sparingly—it carries a hint ofemotion. Place closing quotation marks (”) after commas and periods; it is a matter ofappearance, not logic. Enclose incidental material in parentheses; generally, footnotesand endnotes are discouraged in technical reports. Don’t use the apostrophe to formthe plurals of one or more digits and letters used as nouns, except to avoid confusion.For example, write this: the early 1970s, many YMCAs, several PhD’s, the x’s and y’s.
To inform, you must use language familiar to your readers. Define unfamiliar words,and familiar words used in unfamiliar ways. If the definition is short, then include it in
Writing a Math Phase Two Paper 7
the same sentence, preceding it by “or” or setting it off by commas or parentheses. Ifthe definition is complex or technical, then expand it in a sentence or two. Do not usewords like “capability,” “utilize,” and “implement”; they offer no precision, clarity, orcontinuity and smack of pseudo-intellectualism. Beware of words like “interface”; theyare precise in some contexts, yet imprecise and pretentious in others.
Jargon is vocabulary particular to a certain group, and it consists of abbreviationsand slang terms. Jargon is not inherently bad. Indeed, it is useful in internal memosand reports. However, jargon alienates external readers and may even mislead them. Sobeware. Clichés are figurative expressions that have been overused and have taken onundesirable connotations. Most are imprecise and unclear. Avoid them, or be laughedat. In addition, avoid numerals because they slow down the reading. Write numbers outif they can be expressed in one or two words and are used as adjectives, unless they areaccompanied by units, a percentage sign, or a monetary sign. For instance, write, “Theequation has two roots,” and “One root is 2.” Don’t begin a sentence with a numeralor a symbol; reformulate the sentence if necessary.
Be forthright: write in an unhesitating, straightforward, and friendly style, riddingyour language of needless and bewildering formality. Be wary of awkward and inefficientpassive constructions. Often the passive voice is used simply to avoid the first person.However, the pronoun “we” is now generally considered acceptable in contexts where itmeans the author and reader together, or less often, the author with the reader lookingon. Still, “we” should not be used as a formal equivalent of “I,” and “I” should be usedrarely, if at all.
For instance, don’t write, “By solving the equation, it is found that the roots arereal.” Instead write, “Solving the equation, we find the roots are real,” or “Solvingthe equation yields real roots.” It is acceptable, but less desirable, to write, “Solvingthe equation, one finds the roots are real”. The personal pronoun “one” is a sign offormality; save “one” for use as a number. Beware of dangling participles. It is wrongto write, “Solving the equation, the roots are real,” because “the roots” cannot solvethe equation.
Concise writing is vigorous; wordy writing is tedious. Conciseness comes from re-ducing sentences to their simplest forms. For instance, don’t write, “In order to find thesolution of the equation, we can use one of two alternative methods.” Instead, write,“To solve the equation, we can use one of two methods,” thus eliminating empty words(“in order), reducing fat phrases (“to find the solution of”), and eliminating needlessrepetition (“alternative”). If it goes without saying, don’t say it! Concise writing issimple and efficient, thus beautiful.
The flow of a paper is disturbed by weak transitions between sentences and para-graphs. To smooth out the flow, start a sentence where the preceding one left off. Useconnective words and phrases. Avoid gaps in the logic, and give ample details. Don’ttake needless jumps when deriving equations. Use parallel wording when discussingparallel concepts. Don’t raise questions implicitly, and leave them unanswered. Payattention to the tense, voice, and mode of verbs; prefer the active present indicative.
Some papers stagnate because they lack variety. The sentences begin the same way,run the same length, and are of the same type. The paragraphs have the same lengthand structure. Don’t worry about varying your sentences and paragraphs at first; waituntil you polish your writing. Remember though, if you have to choose between fluidityand clarity, then you must choose clarity.
The very structure of a sentence conveys meaning. Readers expect the stress to lie
8 MIT Undergraduate Journal of Mathematics
at the beginning and end. They take a breath at the beginning, but will run out ofbreath before the end if the structure is too complex, for instance, if the subject is toofar from the verb.
Most people think and remember images, not abstractions, and images are clarifiedby illustrations. Illustrations also provide pauses, so complex ideas can soak in. More-over, illustrations can make a paper more palatable and less forbiding. However, theuse of illustrations can be overdone; it must fit the audience and the subject.
Illustrations cannot stand alone; they must be introduced in the text. Assign themtitles, like Figure 5-1 or Table 5-1, for reference. Assign them captions that tell, in-dependently of the text, what they are and how they differ from one another, withoutbeing overly specific. In addition, clearly label the parts of your illustrations: label theaxes of graphs with words, not symbols; identify any unusual symbols of your diagramsin the text. Don’t put too much information into one illustration, because papers with-out white space tire readers. For the same reason, use adequate borders. Smooth thetransitions between your words and pictures. First, match the information in your textand illustrations. Second, place the illustrations closely after—never before—their firstmention in the text.
4. Mathematics. Mathematical writing tends to involve many abstract symbols andformal arguments, and they present special problems. To help you understand theseproblems and deal with them in your writing, here are some comments and guidlines.
Formulas are difficult to read because readers have to stop and work through themeaning of each term. Don’t merely list a sequence of formulas with no discernible goal,but give a running commentary. Define terms as they are introduced, state any assump-tions about their validity, and give examples to provide a feeling for them. Similarly,motivate and explain formal statements. Don’t simply call a statement “important,”“interesting,” or “remarkable,” but explain why it is so.
Display an important formula by centering it on a line by itself, and give a referencenumber in the margin if you need to refer to it. Also display any formula that’s morethan a quarter of a line long, that would be broken badly between lines, or that sticksout into the margin. Punctuate the display with commas, a period, and so forth as youwould if you had not displayed it; see Section 5 for some examples. Keep in mind thatthe display is not a figure, but an integral part of the sentence, and therefore needspunctuation.
Be clear about the status of every assertion; indicate whether it is a conjecture, theprevious theorem, or the next corollary. If it is not a standard result and you omit itsproof, then give a precise reference, in the text just before the statement. Tell whetherthe omitted proof is hard or easy to help readers decide whether to try to work it outfor themselves. If the theorem has a name, use it: say “by the First FundamentalTheorem,” not just “by Theorem 5-1.” State a theorem before proving it. Keep thestatement concise; put definitions and discussion elsewhere.
Prefer a conceptual proof to a computational one; ideas are easier to communi-cate, understand, and remember. Omit the details of purely routine computations andarguments—ones with no unexpected tricks and no new ideas. Beware of any proofby contradiction; often there’s a simpler direct argument. Finally, when the proof hasended, say so outright. For instance, say, “The proof is now complete,” or use the Hal-mos symbol ¤. In addition, surround the proof—and the statement as well—with someextra white space. (These matters are usually now handled by a LATEX style file.)
Writing a Math Phase Two Paper 9
Here are some more guidelines:1. Separate symbols in different formulas with words.
Bad: Consider Sq, q = 1, . . . , n.Good: Consider Sq for q = 1, . . . , n.
2. Don’t use such symbols as ∃, ∀, ∧, ⇒, ≈, =, > in text; replace them by words.They may, of course, be used in formulas placed in text.
Bad: Let S be the set of all numbers of absolute value < 1.Good: Let S be the set of all numbers of absolute value less than 1.Good: Let S be the set of all numbers x such that |x| < 1.
3. Don’t start a sentence with a symbol.Bad: ax2 + bx + c = 0 has real roots if b2 − 4ac ≥ 0.
Good: The quadratic equation ax2+bx+c = 0 has real roots if b2−4ac ≥ 0.
4. Beware of using symbols to convey too much information all at once.Very bad: If ∆ = b2 − 4ac ≥ 0, then the roots are real.
Bad: If ∆ = b2 − 4ac is nonnegative, then the roots are real.Good: Set ∆ = b2 − 4ac. If ∆ ≥ 0, then the roots are real.
5. If you introduce a condition with “if,” then introduce the conclusion with “then.”Very bad: If ∆ ≥ 0, ax2 + bx + c = 0 has real roots.
Bad: If ∆ ≥ 0, the roots are real.Good: If ∆ ≥ 0, then the roots are real.
6. Don’t set off by commas any symbol or formula used in text in apposition to anoun.
Bad: If the discriminant, ∆, is nonnegative, then the roots are real.Good: If the discriminant ∆ is nonnegative, then the roots are real.
7. Use consistent notation. Don’t say “Aj where 1 ≤ j ≤ n” one place and “Ak
where 1 ≤ k ≤ n” another place.
8. Keep the notation simple. For example, don’t write “xi is an element of X” if “xis an element of X” will do.
9. Precede a theorem, algorithm, and the like with a complete sentence.Bad: We now have the following
Theorem 4-1. H(x) is continuous.Good: We can now prove the following result.
Theorem 4-1. Let H(x) be the function defined by Formula (4-1).Then H(x) is continuous.
5. Example. As an example of mathematical writing, we discuss the two fundamentaltheorems of calculus. Our discussion is based on that in Apostol’s book [2, pp. 202–207].The First Fundamental Theorem says that the process of differentiation reverses thatof integration. This statement is remarkable because the two processes appear to be sodifferent: differentiation gives us the slope of a curve; integration, the area under thecurve. Here is a precise statement of the theorem.
Theorem 5-1 (First Fundamental Theorem of Calculus). Let f be a function de-fined and continuous on the closed interval [a, b] and let c be in [a, b]. Then for each x
10 MIT Undergraduate Journal of Mathematics
in the open interval (a, b), we have
d
dx
∫ x
c
f(t) dt = f(x).
Proof Take a positive number h such that x + h ≤ b. Then∫ x+h
c
f(t) dt−∫ x
c
f(t) dt =∫ x+h
x
f(t) dt.
By hypothesis, f is continuous. Hence there is some z in [x, x + h] for which this lastintegral is equal to h f(z) by the Mean Value Theorem for integrals [2, p. 154], whichis not hard to derive from the Intermediate Value Theorem. The setup is shown inFigure 5-1; the Mean Value Theorem says that the area under the graph of f is equalto the area of the rectangle. Therefore,
1h
(∫ x+h
c
f(t) dt−∫ x
c
f(t) dt
)= f(z).
Now, x ≤ z ≤ x + h. Hence, as h approaches 0, the difference quotient on the leftapproaches f(x). A similar argument holds for negative h. Thus the derivative of theintegral exists and is equal to f(x). ¤
z ba
f (z)f (x)
x+hx
f
Figure 5-1. Geometric setup of the proof of the First Fundamental Theorem.
The First Fundamental Theorem says that, given a continuous function f , thereexists a function F , namely, F (x) =
∫ x
cf(t) dt, whose derivative is equal to f :
F ′(x) = f(x).
Such a function F is called an integral, or a primitive, or an antiderivative, of f . Integralsare not unique: if F is an integral of f , then obviously so is F + C for any constant C.On the other hand, there is no further ambiguity: any two integrals F and G of f differby a constant. Indeed, their difference F −G has vanishing derivative: for every x,
(F −G)′(x) = F ′(x)−G′(x)= f(x)− f(x) = 0.
Writing a Math Phase Two Paper 11
Therefore, F −G is constant owing to the Mean Value Theorem for derivatives; see [2,Thm. 4.7(c), p. 187].
When we combine the First Fundamental Theorem with the fact that an integral isunique up to an additive constant, we obtain the following theorem.
Theorem 5-2 (Second Fundamental Theorem of Calculus). Let f be a function de-fined and continuous on the open interval I, and let F be an integral of f on I. Thenfor each c and x in I, ∫ x
c
f(t) dt = F (x)− F (c). (5-1)
Proof Set G(x) =∫ x
cf(t) dt. By the First Fundamental Theorem, G is an integral of
f . Now, any two integrals differ by a constant. Hence G(x)−F (x) = C for some constantC. Taking x = c yields −F (c) = C because G(c) = 0. Thus G(x)−F (x) = −F (c), andEquation (5-1) follows. ¤
The Second Fundamental Theorem is a powerful statement. It says that we cancompute the value of a definite integral merely by subtracting two values of any integralof the integrand. In practice, integrals are often found by reading a differentiationformula in reverse. For example, the integrals in Table 5-1 were found this way. The
Table 5-1A brief table of integrals
1.∫
xa dx = xa+1
a+1 + C, if a 6= −12.
∫x−1 dx = ln x + C
3.∫
sin x dx = − cos x + C
4.∫
cos x dx = sin x + C
5.∫
ex dx = ex + C
notation in the table is standard [9, p. 178]: the equation∫
f(x) dx = F (x) + C
is read, “The integral of f(x) dx is equal to F (x) plus C.” A longer table of integrals isfound on the endpapers of the calculus textbook [9].
Appendix. Appendix. Advanced mathematics
In many treatments of advanced mathematics, the key results are stated formallyas theorems, propositions, corollaries, and lemmas. However, these four terms are oftenused carelessly, robbing them of some useful information they have to convey: the natureof the result.
A theorem is a major result, one of the main goals of the work. Use the term“theorem” sparingly. Call a minor result a proposition if it is of independent interest.Call a minor result a corollary if it follows with relatively little proof from a theorem,a proposition, or another corollary. Sometimes a result could properly be called eithera proposition or a corollary. If so, then call it a proposition if it is relatively important,
12 MIT Undergraduate Journal of Mathematics
and call it a corollary if it is relatively unimportant. Call a subsidiary statement alemma if it is used in the proof of a theorem, a proposition, or another lemma. Thus alemma never has a corollary, although a lemma may be used, on occasion, in deriving acorollary. Normally, a lemma is stated and proved before it is used.
The terms “definition” and “remark” are also often abused. A formal definitionshould simply introduce some terminology or notation; there should be no accompanyingdiscussion of the new terms or symbols. It is traditional to use “if” instead of “if andonly if”; for example, a matrix is called symmetric if it is equal to its transpose. Aformal remark should be a brief comment made in passing; the main discussion should belogically independent of the content of the remark. Often it is better to weave definitionsand remarks into the general discussion rather than setting them apart formally.
Typographically, the statements of theorems, propositions, corollaries, and lemmasare traditionally set in italics, and the headings themselves are set in boldface or incaps and small caps (Theorem or Theorem, and so forth). The texts of definitionsand remarks are set as ordinary text; so are the texts of proofs, examples, and the like.These headings are traditionally set in italics, boldface, or small caps. (There is also atradition of treating definitions typographically like theorems, but this tradition is lesscommon today and less desirable.) All these formal statements and texts are usually setoff from the rest of the discussion by putting some extra white space before and afterthem.
Assign sequential reference numbers to these headings, irrespective of their differentnatures, and use a hierarchical scheme whose first component is the section number.Thus “Corollary 3-6” refers to the prominent statement in the sixth subsection of Sec-tion 3, and indicates that the statement is a corollary. If the statement is the secondcorollary of the third proposition in the paper, then it may seem more logical to namethe statement “Corollary 2,” but doing so may make the statement considerably moredifficult to locate.
References
[1] Alley, M., “The Craft of Scientific Writing,” Prentice-Hall, 1987.[2] Apostol, T. M., “Calculus,” Volume I, Second Edition, Blaisdell, 1967.[3] Committee on the Writing Requirement, “Guide to the MIT Writing Requirement,”
Undergraduate Academic Affairs, Room 20B–140, MIT, 1993.[4] Flanders, H., Manual for Monthly Authors, Amer. Math. Monthly 78 (1971), 1–10.[5] Gillman, L., “Writing Mathematics Well,” Math Association of America, 1987.[6] Knuth, D. E., Larrabee, T., and Roberts, P. M., “Mathematical writing,” MAA
Notes Series 14, Math Association of America, 1989.[7] Munkres, J. R., “Manual of style for mathematical writing,” Undergraduate Math-
ematics Office, Room 2–108, MIT, 1986.[8] Strunk Jr., W., and White, E. B., “The Elements of Style,” Macmillan Paperbacks
Edition, 1962.[9] Thomas, G. B., and Finney, R. L., “Calculus and Analytic Geometry,” Fifth edition,
Addison-Wesley, 1982.
