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Suggested structure of a mathematics articleProbably the hardest part of writing a mathematical article (actually, any article) is the difficulty of addressing the level of mathematical knowledge on the part of the reader. For example, when writing about a field, do we assume that the reader already knows group theory? A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds. Article introductionThe article should start with an introductory paragraph (or two), which describes the subject in general terms. Name the field(s) of mathematics this concept belongs to and describe the mathematical context in which the term appears. Write the article title in bold. Include the historical motivation, provide some names and dates, etc. Here is an example. In topology and related branches of mathematics, a continuous function is, loosely speaking, a function from one topological space to another which preserves open sets. Originally, the idea of continuity was a generalization of the informal idea of smoothness, or lack of discontinuity. The first statement of the idea of continuity was by Euler in 1784, relating to plane curves. Other mathematicians, including Bolzano and Cauchy, then refined and extended the idea of continuity. Continuous functions are the raison d'être of topology itself.
In the case of real numbers, a continuous function corresponds to a graph that you can draw without lifting your pen from the paper; that is, without any gaps or jumps. The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach. If a physical or geometric analogy or diagram will help, use one: many of the readers may be non-mathematical scientists. It is quite helpful to have a section for motivation or applications, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics. Article bodyIf you want to introduce some notation, it should be in its own section. You should remember that not everyone understands that, for example, x^n = x**n = xn; so it is good to use standard notation if you can. If you need to use non-standard notations, or if you introduce new notations, define them in your article. There should be an exact definition, in mathematical terms; often in a Definition(s) section, for example: Let S and T be topological spaces, and let f be a function from S to T. Then f is called continuous if, for every open set O in T, the preimage f −1(O) is an open set in S. Using the term formal definition may seem rather empty to a mathematician (a formal definition is a mathematician's definition, as a formal proof is just a proof); but it may help to flag up where the actual definition is to be found, after some sections of motivation. (Cf. rigged Hilbert space.) Some representative examples would be nice to have, in a separate section, which could serve to both expand on the definition, and also provide some context as to why one might want to use the defined entity. You might also want to list non-examples — things which come close to satisfying the definition but do not — in order to refine the reader's intuition more precisely. A picture is a great way of bringing a point home, and often it could even precede the mathematical discussion of a concept. How to create graphs for Wikipedia articles has some hints on how to create graphs and other pictures, and how to include them in articles. A person editing a mathematics article should not fall into temptation that "this formula says it all". A non-mathematical reader will skip the formulas in most cases, and often a mathematician reading outside his research area will do the same. Careful thought should be given to each formula included, and words should be used instead if possible. In particular, the English words "for all", "exists", and "in" should be preferred to the ∀, ∃, and ∈ symbols. Similarly, highlight definitions with words such as "is defined by" in the text. If not included in the introductory paragraph, a section about the history of the concept is often useful and can provide additional insight and motivation. Concluding mattersMost mathematical ideas are amenable to some form of generalization. If appropriate, such material can be put under a Generalizations section. As an example, multiplication of the rational numbers can be generalized to other fields, etc. It is good to have a see also section, which connects to related subjects, or to pages which could provide more insight into the contents of the current article. Lastly, a well-written and complete article should have a references section. This topic will be discussed in detail below. Writing style in mathematicsThere are several issues of writing style that are particularly relevant in mathematical writing.
ProofsThis is an encyclopedia, not a collection of mathematical texts; but we often want to include proofs, as a way of really exposing the meaning of some theorem, definition, etc. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgement; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of result. Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section. Additional discussion and guidelines can be found at Wikipedia:WikiProject Mathematics/Proofs. Including literature and referencesIt is quite important for an article to have a well-chosen list of references and pointers to the literature. Some reasons for this are the following:
The Wikipedia:cite sources article has more information on this and also several examples for how the cited literature should look. Typesetting of mathematical formulasUsing LaTeX markupWikipedia allows editors to typeset mathematical formulas in (a subset of) LaTeX markup (see also TeX); the formulas are normally translated into PNG images, but may be rendered as HTML or MathML, depending on user preferences. For more details on this, see meta:Help:Formula. The LaTeX formulas can be displayed in-line (like this: <math>\mathbf{x}\in\mathbb{R}^2</math>), as well as on their own line:
Having LaTeX-based formulas in-line which render as PNG under the default user settings, as above, is generally discouraged, for the following reasons.
When displaying formulas on their own line, one should indent the line with one or more colons (:); the above was typeset as
If you find an article which indents lines with spaces in order to achieve some formula layout effect, you should convert the formula to LaTeX markup. If you plan on editing LaTeX formulas, it is helpful if you leave your preference settings (link in the upper right corner of this page, underneath your user name) in the "rendering math" section at the default "HTML if very simple or else PNG"; that way, you'll see the page like most users will see it. Very simple formulasIf you enter a very simple formula using the mathematical notations like <math>L^p</math> this will (in the default used by most users) not be displayed using a PNG image but using HTML, like this: Lp. This is different from typesetting it as
Either form is acceptable, but do not change one form to the other in other people's writing. They are likely to get annoyed since this seems to be a highly emotional issue. Changing to make an entire article consistent is acceptable. However, still try to avoid in-line PNG images. Even if you use If you want to force an image output for a simple formula, put, for example, a Using HTMLThe following sections cover the way of presenting simple in-line formulas in HTML, instead of using LaTeX. Font formattingVariablesTo start with, we generally use italic text for variables (but never for numbers or symbols). Most editors prefer to use
which results in:
Try to only italicize the variables. Things like parentheses, equal and plus signs should be kept outside of the double-quoted sections. Descriptive subscripts should not be in italics, because they are not variables. For example, mfoo is the mass of a foo. SI units are never italicized: x = 5 cm. FunctionsNames for standard functions, such as sin and cos, are not in italic font, but we use italic names such as f for functions in other cases; for example when we define the function as in f(x) = sin(x) cos(x). SetsSets are usually written in upper case italics; for example:
would be written:
Greek lettersGreek letters are not commonly italicized, so that one writes, for example, Common sets of numbersCommonly used sets of numbers are typeset in boldface, as in the set of real numbers R; see blackboard bold for the types in use. Again, typically we use wiki markup: three single quotes ( Superscripts and subscriptsSubscripts and superscripts should be wrapped in
Nested superscripts/subscripts are troublesome, due to a bug in the software (as of April 2006). Entering
will produce abc. Instead, you should wrap the inner subscript in
which will produce abc as desired. Special symbolsThere is a table of mathematical symbols and a list of the codes at Wikipedia:Mathematical symbols that may be useful when editing mathematics articles. One should keep in mind though, that not all of the symbols in these lists are displayed correctly on all browsers. It is generally better to be rather conservative with the use of symbols in order to reach a larger audience, for example by writing "x in Y" rather than "x ∈ Y". Less-than signAlthough the MediaWiki markup engine is fairly smart about differentiating between unescaped "<" characters that are used to denote the start of an embedded HTML or HTML-like tag and those that are just being used as literal less-than symbols, it is ideal to use
not
Multiplication signStandard algebraic notation is best for formulas, so two variables q and d being multiplied are best written as qd when presented in a formula. That is, when citing a formula, don't use However, when explaining, for a general audience (not just mathematicians), the formula or giving examples of its application, it is prudent to use the multiplication sign: "×", coded as
Another alternative to the However, do not use the letter x. PunctuationJust as in mathematics publications, a sentence which ends with a formula must have a period at the end of the formula. If the formula is written in LaTeX, that is, surrounded by the Choice of type styleIn mathematics notation, the names of trigonometric functions, logarithms, etc., should always be in an upright font, for example,
rather than
Some operator names do not have a pre-defined abbreviation; for these we may use \operatorname:
Similar consideration applies to text used as subscripted labels, such as
On the other hand, for the differential, imaginary unit, and Euler's number, Wikipedia articles usually use an italic font, so one writes
(Also, note the thin space before the dx.) Some authors prefer to use an upright (Roman) font for dx, and Roman boldface for i. What is important is to be consistent within an article, with deference to previous editors. (This is much the same as the colour/color spelling choice.) Also, it is considered inappropriate for an editor to go through articles doing mass changes from one style to another. See also
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